AMERICAN MATHEMATICS COMPETITION 8 - 2010

Problem 1

At Euclid High School, the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are 11 students in Mrs. Germain's class, 8 in Mr. Newton, and 9 in Mrs. Young's class are taking the AMC 8 this year. How many mathematics students at Euclid High School are taking the contest?
(A) 26
(B) 27
(C) 28
(D) 29
(E) 30

Answer:

(C) 28

Problem 2

If $a @ b=\frac{a \times b}{a+b}$, for $a, b$ positive integers, then what is $5 @ 10$ ?
(A) $\frac{3}{10}$
(B) 1
(C) 2
(D) $\frac{10}{3}$
(E) 50

Answer:

(D) $\frac{10}{3}$

Problem 3

3 The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price?


(A) 50
(B) 62
(C) 70
(D) 89
(E) 100

Answer:

(C) 70

Problem 4


What is the sum of the mean, median, and mode of the numbers, $2,3,0,3,1,4,0,3$ ?
(A) 6.5
(B) 7
(C) 7.5
(D) 8.5
(E) 9

Answer:

(C) 7.5

Problem 5


Alice needs to replace a light bulb located 10 centimeters below the ceiling of her kitchen. The ceiling is 2.4 meters above the floor. Alice is 1.5 meters tall and can reach 46 centimeters above her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters?
(A) 32
(B) 34
(C) 36
(D) 38
(E) 40

Answer:

(B) 34

Problem 6


Which of the following has the greatest number of line of symmetry?
(A) Equilateral Triangle (B) Non-square rhombus (C) Non-square rectangle (D) Isosceles Triangle (E) Square

Answer:

(E) Square

Problem 7


Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than one dollar?
(A) 6
(B) 10
(C) 15
(D) 25
(E) 99

Answer:

(B) 10


Problem 8


As Emily is riding her bike on a long straight road, she spots Ermenson skating in the same direction $1 / 2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1 / 2$ mile behind her. Emily rides at a constant rate of 12 miles per hour. Ermenson skates at a constant rate of 8 miles per hour. For how many minutes can Emily see Ermenson?
(A) 6
(B) 8
(C) 12
(D) 15
(E) 16

Answer:

(D) 15

Problem 9


Ryan got $80 \%$ of the problems on a 25 -problem test, $90 \%$ on a 40 -problem test, and $70 \%$ on a 10 -problem test. What percent of all problems did Ryan answer correctly?
(A) 64
(B) 75
(C) 80
(D) 84
(E) 86

Answer:

(D) 84

Problem 10


6 pepperoni circles will exactly fit across the diameter of a 12 -inch pizza when placed. If a total of 24 circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered with pepperoni?
(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) $\frac{3}{4}$
(D) $\frac{5}{6}$
(E) $\frac{7}{8}$

Answer:

(B) $\frac{2}{3}$


Problem 11


The top of one tree is 16 feet higher than the top of another tree. The height of the 2 trees are at a ratio of $3: 4$. In feet, how tall is the taller tree?
(A) 48
(B) 64
(C) 80
(D) 96
(E) 112

Answer:

(B) 64

Problem 12


12 & Of the 500 balls in a large bag, $80 \%$ are red and the rest are blue. How many of the red balls must be removed so that $75 \%$ of the remaining balls are red?
(A) 25
(B) 50
(C) 75
(D) 100
(E) 150

Answer:

(D) 100

Problem 13


The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shorter side is $30 \%$ of the perimeter. What is the length of the longest side?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

Answer:

(E) 11

Problem 14


What is the sum of the prime factors of 2010 ?
(A) 67
(B) 75
(C) 77
(D) 201
(E) 210

Answer:

(C) 77

Problem 15


A jar contains 5 different colors of gumdrops. $30 \%$ are blue, $20 \%$ are brown, $15 \%$ red, $10 \%$ yellow, and the other 30 gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown?
(A) 35
(B) 36
(C) 42
(D) 48
(E) 64

Answer:

(C) 42

Problem 16


A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle?
(B) $\sqrt{\pi}$
(C) $\pi$
(D) $2 \pi$
(E) $\pi^{2}$

Answer:

(B) $\sqrt{\pi}$

Problem 17


The diagram shows an octagon consisting of 10 unit squares. The portion below $\overline{P Q}$ is a unit square and a triangle with base 5 . If $\overline{P Q}$ bisects the area of the octagon, what is the ratio $\frac{X Q}{Q Y}$ ?


(A) $\frac{2}{5}$
(B) $\frac{1}{2}$
(C) $\frac{3}{5}$
(D) $\frac{2}{3}$
(E) $\frac{3}{4}$

Answer:

(D) $\frac{2}{3}$

Problem 18


A decorative window is made up of a rectangle with semicircles at either end. The ratio of $A D$ to $A B$ is $3: 2$. And $A B$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircle.


(A) $2: 3$
(B) $3: 2$
(C) $6: \pi$
(D) $9: \pi$
(E) $30: \pi$

Answer:

(C) $6: \pi$

Problem 19

The two circles pictured have the same center $C$. Chord $\overline{A D}$ is tangent to the inner circle at $B, A C$ is 10 , and chord $\overline{A D}$ has length 16 . What is the area between the two circles?


(A) $36 \pi$
(B) $49 \pi$
(C) $64 \pi$
(D) $81 \pi$
(E) $100 \pi$

Answer:

(C) $64 \pi$


Problem 20


In a room, $2 / 5$ of the people are wearing gloves, and $3 / 4$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove?
(A) 3
(B) 5
(C) 8
(D) 15
(E) 20

Answer:

(A) 3

Problem 21


Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, she read $1 / 5$ of the pages plus 12 more, and on the second day she read $1 / 4$ of the remaining pages plus 15 more. On the third day she read $1 / 3$ of the remaining pages plus 18 more. She then realizes she has 62 pages left, which she finishes the next day. How many pages are in this book?
(A) 120
(B) 180
(C) 240
(D) 300
(E) 360

Answer:

(C) 240

Problem 22

The hundreds digit of a three-digit number is 2 more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?
(A) 0
(B) 2
(C) 4
(D) 6
(E) 8

Answer:

(E) 8

Problem 23


Semicircles $P O Q$ and $R O S$ pass through the center of circle $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$ ?


(A) $\frac{\sqrt{2}}{4}$
(B) $\frac{1}{2}$
(C) $\frac{2}{\pi}$
(D) $\frac{2}{3}$
(E) $\frac{\sqrt{2}}{2}$

Answer:

(B) $\frac{1}{2}$


Problem 24

What is the correct ordering of the three numbers, $10^{8}, 5^{12}$, and $2^{24}$ ?
(A) $2^{24}<10^{8}<5^{12}$
(B) $2^{24}<5^{12}<10^{8}$
(C) $5^{12}<2^{24}<10^{8}$ (D) $10^{8}< 5^{12}<2^{24}$ (E) $10^{8}<2^{24}<5^{12}$

Answer:

(A) $2^{24}<10^{8}<5^{12}$


Problem 25


Everyday at school, Jo climbs a flight of 6 stairs. Joe can take the stairs 1,2, or 3 at a time. For example, Jo could climb 3, then 1 , then 2 . In how many ways can Jo climb the stairs?
(A) 13
(B) 18
(C) 20
(D) 22
(E) 24

Answer:

(E) 24

American Mathematics Competition - 2006

Problem 1

Mindy made three purchases for $\$ 1.98, \$ 5.04$ and $\$ 9.89$. What was her total, to the nearest dollar?
(A) $\$ 10$
(B) $\$ 15$
(C) $\$ 16$
(D) $\$ 17$
(E) $\$ 18$

Answer:

(D) $\$ 17$

Problem 2

On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5 . What is his score?
(A) 1
(B) 6
(C) 13
(D) 19
(E) 26

Answer:

(C) 13

Problem 3

Elisa swims laps in the pool. When she first started, she completed 10 laps in 25 minutes. Now she can finish 12 laps in 24 minutes. By how many minutes has she improved her lap time?
(A) $\frac{1}{2}$
(B) $\frac{3}{4}$
(C) 1
(D) 2
(E) 3

Answer:

(A) $\frac{1}{2}$

Problem 4

Initially, a spinner points west. Chenille moves it clockwise $2 \frac{1}{4}$ revolutions and then counterclockwise $3 \frac{3}{4}$ revolutions. In what direction does the spinner point after the two moves?


(A) north
(B) east
(C) south
(D) west
(E) northwest

Answer:

(B) east

Problem 5


Points $A, B, C$ and $D$ are midpoints of the sides of the larger square. If the larger square has area 60 , what is the area of the smaller square?

(A) 15
(B) 20
(C) 24
(D) 30
(E) 40

Answer:

(D) 30

Problem 6


The letter T is formed by placing two $2 \times 4$ inch rectangles next to each other, as shown. What is the perimeter of the T , in inches?


(A) 12
(B) 16
(C) 20
(D) 22
(E) 24

Answer:

(C) 20

Problem 7


Circle $X$ has a radius of $\pi$. Circle $Y$ has a circumference of $8 \pi$. Circle $Z$ has an area of $9 \pi$. List the circles in order from smallest to largest radius.
(A) $X, Y, Z$
(B) $Z, X, Y$
(C) $Y, X, Z$
(D) $Z, Y, X$
(E) $X, Z, Y$

Answer:

(B) $Z, X, Y$

Problem 8


The table shows some of the results of a survey by radiostation KAMC. What percentage of the males surveyed listen to the station?

(A) 39
(B) 48
(C) 52
(D) 55
(E) 75

Answer:

(E) 75

Problem 9

What is the product of $\frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \times \cdots \times \frac{2006}{2005}$ ?
(A) 1
(B) 1002
(C) 1003
(D) 2005
(E) 2006

Answer:

(C) 1003

Problem 10


Jorge's teacher asks him to plot all the ordered pairs ( $w, l$ ) of positive integers for which $w$ is the width and $l$ is the length of a rectangle with area 12 . What should his graph look like?

Answer:

Problem 11

How many two-digit numbers have digits whose sum is a perfect square?
(A) 13
(B) 16
(C) 17
(D) 18
(E) 19

Answer:

(C) 17

Problem 12

Antonette gets $70 \%$ on a 10 -problem test, $80 \%$ on a 20 -problem test and $90 \%$ on a 30 -problem test. If the three tests are combined into one 60 -problem test, which percent is closest to her overall score?
(A) 40
(B) 77
(C) 80
(D) 83
(E) 87

Answer:

(D) 83

Problem 13


Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62 -mile route between Escanaba and Marquette. At what time in the morning do they meet?
(A) $10: 00$
(B) $10: 15$
(C) $10: 30$
(D) $11: 00$
(E) $11: 30$

Answer:

(D) $11: 00$

Problem 14

Problems 14, 15 and 16 involve Mrs. Reed's English assignment.

{A Novel Assignment}
The students in Mrs. Reed's English class are reading the same 760 -page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds.

If Bob and Chandra both read the whole book, Bob will spend how many more seconds reading than Chandra?

(A) 7,600
(B) 11,400
(C) 12,500
(D) 15,200
(E) 22,800

Answer:

(B) 11,400

Problem 15


The students in Mrs. Reed's English class are reading the same 760 -page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds.

Chandra and Bob, who each have a copy of the book, decide that they can save time by "team reading" the novel. In this scheme, Chandra will read from page 1 to a certain page and Bob will read from the next page through page 760, finishing the book. When they are through they will tell each other about the part they read. What is the last page that Chandra should read so that she and Bob spend the same amount of time reading the novel?
(A) 425
(B) 444
(C) 456
(D) 484
(E) 506

Answer:

(C) 456

Problem 16


The students in Mrs. Reed's English class are reading the same 760 -page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds.

Before Chandra and Bob start reading, Alice says she would like to team read
league Education Center
with them. If they divide the book into three sections so that each reads for the same length of time, how many seconds will each have to read?
(A) 6400
(B) 6600
(C) 6800
(D) 7000
(E) 7200

Answer:

(B) 6600

Problem 17


Jeff rotates spinners $P, Q$ and $R$ and adds the resulting numbers. What is the probability that his sum is an odd number?


(A) $\frac{1}{4}$
(B) $\frac{1}{3}$
(C) $\frac{1}{2}$
(D) $\frac{2}{3}$
(E) $\frac{3}{4}$

Answer:

(B) $\frac{1}{3}$

Problem 18


A cube with 3 -inch edges is made using 27 cubes with 1 -inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white?
(A) $\frac{1}{9}$
(B) $\frac{1}{4}$
(C) $\frac{4}{9}$
(D) $\frac{5}{9}$
(E) $\frac{19}{27}$

Answer:

(D) $\frac{5}{9}$

Problem 19


Triangle $A B C$ is an isosceles triangle with $\overline{A B}=\overline{B C}$. Point $D$ is the midpoint of both $\overline{B C}$ and $\overline{A E}$, and $\overline{C E}$ is 11 units long. Triangle $A B D$ is congruent to triangle $E C D$. What is the length of $\overline{B D}$ ?


(A) 4
(B) 4.5
(C) 5
(D) 5.5
(E) 6

Answer:

(D) 5.5

Problem 20


A singles tournament had six players. Each player played every other player only once, with no ties. If Helen won 4 games, Ines won 3 games, Janet won 2 games, Kendra won 2 games and Lara won 2 games, how many games did Monica win?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Answer:

(C) 2

Problem 21


An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm . The aquarium is tilled with water to a depth of 37 cm . A rock with volume $1000 \mathrm{~cm}^{3}$ is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise?
(A) 0.25
(B) 0.5
(C) 1
(D) 1.25
(E) 2.5

Answer:

(A) 0.25

Problem 22


Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell?

Answer:

(D) 26

Problem 23


A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 5

Answer:

(A) 0

Problem 24


In the multiplication problem below, $A, B, C$ and $D$ are different digits. What A B A\
is $A+B$ ?

Answer:

(A) 1

Problem 25


Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers?


(A) 13
(B) 14
(C) 15
(D) 16
(E) 17

Answer:

(B) 14

American Mathematics Competition - 2012

Problem 1

Rachelle uses 3 pounds of meat to make 8 hamburgers for her family. How many pounds of meat does she need to make 24 hamburgers for a neighborhood picnic?

Answer:

(E) 9.

Problem 2


In the country of East Westmore, statisticians estimate there is a baby born every 8 hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year?


(A) 600
(B) 700
(C) 800
(D) 900
(E) 1000

Answer:

(B) 700.

Problem 3


On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise as 6:57 am, and the sunset as 8:15 pm. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set?
(A) 5:10 PM
(B) 5:21 PM
(C) 5:41 PM
(D) 5: 57 PM
(E) 6:03 PM

Answer:

(B) 5:21 PM.

Problem 4


Peter's family ordered a 12 -slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat?


(A) $\frac{1}{24}$
(B) $\frac{1}{12}$
(C) $\frac{1}{8}$
(D) $\frac{1}{6}$
(E) $\frac{1}{4}$

Answer:

(C) $\frac{1}{8}$

Problem 5


In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is X, in centimeters?


(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer:

(E) 5.

Problem 6


A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inches high and 10 inches wide. What is the area of the border, in square inches?


(A) 36
(B) 40
(C) 64
(D) 72
(E) 88

Answer:

(E) 88.

Problem 7


Isabella must take four 100 -point tests in her math class. Her goal is to achieve an average grade of at least 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she realized that she could still reach her goal. What is the lowest possible score she could have made on the third test?


(A) 90
(B) 92
(C) 95
(D) 96
(E) 97

Answer:

(B) 92.

Problem 8

A shop advertises that everything is "half price in today's sale." In addition, a coupon gives a $20 \%$ discount on sale prices. Using the coupon, the price today represents what percentage discount off the original price?


(A) 10
(B) 33
(C) 40
(D) 60
(E) 70

Answer:

(D) 60.

Problem 9


The Fort Worth Zoo has a number of two-legged birds and a number of fourlegged mammals. On one visit to the zoo, Margie counted 200 heads and 522 legs. How many of the animals that Margie counted were two-legged birds?


(A) 61
(B) 122
(C) 139
(D) 150
(E) 161

Answer:

(C) 139.

Problem 10


How many 4 -digit numbers greater than 1000 are there that use the four digits of 2012?


(A) 6
(B) 7
(C) 8
(D) 9
(E) 12

Answer:

(D) 9.

Problem 11


The mean, median, and unique mode of the positive integers $3,4,5,6,6,7, x$ are all equal. What is the value of $x$ ?
(A) 5
(B) 6
(C) 7
(D) 11
(E) 12

Answer:

(D) 11.

Problem 12


What is the units digit of $13^{2012}$ ?
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

Answer:

(A) 1.

Problem 13


Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\$ 1.43$. Sharona bought some of the same pencils and paid $\$ 1.87$. How many more pencils did Sharona buy than Jamar?


(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Answer:

(C) 4.

Problem 14


In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

Answer:

(B) 7.

Problem 15


The smallest number greater than 2 that leaves a remainder of 2 when divided by $3,4,5$, or 6 lies between what numbers?


(A) 40 and 50
(B) 51 and 55
(C) 56 and 60
(D) 61 and 65
(E) 66 and 99

Answer:

(D) 61 and 65.

Problem 16


Each of the digits $0,1,2,3,4,5,6,7,8$, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers?


(A) 76531
(B) 86724
(C) 87431
(D) 96240
(E) 97403

Answer:

(C) 87431.

Problem 17


A square with an integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1 . What is the smallest possible value of the length of the side of the original square?


(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Answer:

(B) 4.

Problem 18


What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50 ?


(A) 3127
(B) 3133
(C) 3137
(D) 3139
(E) 3149

Answer:

(A) 3127.

Problem 19


In a jar of red, green, and blue marbles, all but 6 are red marbles, all but 8 are green, and all but 4 are blue. How many marbles are in the jar?


(A) 6
(B) 8
(C) 9
(D) 10
(E) 18

Answer:

(C) 9.

Problem 20


What is the correct ordering of the three numbers $\frac{5}{19}, \frac{7}{21}$, and $\frac{9}{23}$, in increasing order?


(A) $\frac{9}{23}<\frac{7}{21}<\frac{5}{19}$
(B) $\frac{5}{19}<\frac{7}{21}<\frac{9}{23}$
(C) $\frac{9}{23}<\frac{5}{19}<\frac{7}{21}$
(D) $\frac{5}{19}<\frac{9}{23}<\frac{7}{21}$
(E) $\frac{7}{21}<\frac{5}{19}<\frac{9}{23}$

Answer:

(B) $\frac{5}{19}<\frac{7}{21}<\frac{9}{23}$

Problem 21


Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet?


(A) $5 \sqrt{2}$
(B) 10
(C) $10 \sqrt{2}$
(D) 50
(E) $50 \sqrt{2}$

Answer:

(D) 50.

Problem 22


Let $R$ be a set of nine distinct integers. Six of the elements of the set are 2, 3, 4, 6,9 , and 14 . What is the number of possible values of the median of $R$ ?


(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Answer:

(D) 7.

Problem 23


An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 4 , what is the area of the hexagon?
(A) 4
(B) 5
(C) 6
(D) $4 \sqrt{3}$
(E) $6 \sqrt{3}$

Answer:

(C) 6.

Problem 24
A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?


(A) $\frac{4-\pi}{\pi}$
(B) $\frac{1}{\pi}$
(C) $\frac{\sqrt{2}}{\pi}$
(D) $\frac{\pi-1}{\pi}$
(E) $\frac{3}{\pi}$

Answer:

(A) $\frac{4-\pi}{\pi}$

Problem 25


A square with area 4 is inscribed in a square with area 5 , with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$ and the other of length $b$. What is the value of $a b$ ?


(A) $\frac{1}{5}$
(B) $\frac{2}{5}$
(C) $\frac{1}{2}$
(D) 1
(E) 4

Answer:

(C) $\frac{1}{2}$

IOQM 2025 Questions, Answer Key, Solutions

Answer Key

Answer 1
40		Answer 2
17		Answer 3
18		Answer 4
5		Answer 5
36
Answer 6
18		Answer 7
576		Answer 8
44		Answer 9
28		Answer 10
15
Answer 11
80		Answer 12
38		Answer 13
13		Answer 14
11		Answer 15
75
Answer 16
8		Answer 17
8		Answer 18
1		Answer 19
72		Answer 20
42
Answer 21
80		Answer 22
7		Answer 23
19		Answer 24
66		Answer 25
9
Answer 26
6		Answer 27
37		Answer 28
12		Answer 29
33		Answer 30
97

Problem 1

If $60 \%$ of a number $x$ is 40 , then what is $x \%$ of 60 ?

Problem 2

Find the number of positive integers $n$ less than or equal to 100 , which are divisible by 3 but are not divisible by 2.

Problem 3

The area of an integer-sided rectangle is 20 . What is the minimum possible value of its perimeter?

Problem 4

How many isosceles integer-sided triangles are there with perimeter 23?

Problem 5

How many 3 -digit numbers $a b c$ in base 10 are there with $a \neq 0$ and $c=a+b$ ?

Problem 6

The height and the base radius of a closed right circular cylinder are positive integers and its total surface area is numerically equal to its volume. If its volume is $k \pi$ where $k$ is a positive integer, what is the smallest possible value of $k$ ?

Problem 7

A quadrilateral has four vertices $A, B, C, D$. We want to colour each vertex in one of the four colours red, blue, green or yellow, so that every side of the quadrilateral and the diagonal $A C$ have end points of different colours. In how many ways can we do this?

Problem 8

The sum of two real numbers is a positive integer $n$ and the sum of their squares is $n+1012$. Find the maximum possible value of $n$.

Problem 9

Four sides and a diagonal of a quadrilateral are of lengths $10, 20, 28, 50, 75$, not necessarily in that order. Which amongst them is the only possible length of the diagonal?

Problem 10

The age of a person (in years) in 2025 is a perfect square. His age (in years) was also a perfect square in 2012. His age (in years) will be a perfect cube $m$ years after 2025. Determine the smallest value of $m .=15$

Problem 11

There are six coupons numbered 1 to 6 and six envelopes, also numbered 1 to 6 . The first two coupons are placed together in any one envelope. Similarly, the third and the fourth are placed together in a different envelope, and the last two are placed together in yet another different envelope. How many ways can this be done if no coupon is placed in the envelope having the same number as the coupon?

Problem 12

Consider five-digit positive integers of the form $\overline{a b c a b}$ that are divisible by the two digit number $a b$ but not divisible by 13 . What is the largest possible sum of the digits of such a number?

Problem 13

A function $f$ is defined on the set of integers such that for any two integers $m$ and $n$,

$$
f(m n+1)=f(m) f(n)-f(n)-m+2
$$

holds and $f(0)=1$. Determine the largest positive integer $N$ such that $\sum_{k=1}^N f(k)<100$ .

Problem 14

Consider a fraction $\frac{a}{b} \neq \frac{3}{4}$, where $a, b$ are positive integers with $\operatorname{gcd}(a, b)=1$ and $b \leq 15$. If this fraction is chosen closest to $\frac{3}{4}$ amongst all such fractions, then what is the value of $a+b$ ?

Problem 15

Three sides of a quadrilateral are $a=4 \sqrt{3}, b=9$ and $c=\sqrt{3}$. The sides $a$ and $b$ enclose an angle of $30^{\circ}$, and the sides $b$ and $c$ enclose an angle of $90^{\circ}$. If the acute angle between the diagonals is $x^{\circ}$, what is the value of $x$ ?

Problem 16

$f(x)$ and $g(x)$ be two polynomials of degree 2 such that

$$
\frac{f(-2)}{g(-2)}=\frac{f(3)}{g(3)}=4
$$

If $g(5)=2, f(7)=12, g(7)=-6$, what is the value of $f(5)$ ?

Problem 17

The triangle $A B C, \angle B=90^{\circ}, A B=1$ and $B C=2$. On the side $B C$ there are two points $D$ and $E$ such that $E$ lies between $C$ and $D$ and $D E F G$ is a square, where $F$ lies on $A C$ and $G$ lies on the circle through $B$ with centre $A$. If the area of $D E F G$ is $\frac{m}{n}$ where $m$ and $n$ are positive integers with $\operatorname{gcd}(m, n)=1$, what is the value of $m+n$ ?

Problem 18

$M T A I$ is a parallelogram of area $\frac{40}{41}$ square units such that $M I=1 / M T$. If $d$ is the least possible length of the diagonal $M A$, and $d^2=\frac{a}{b}$, where $a, b$ are positive integers with $\operatorname{gcd}(a, b)=1$, find $|a-b|$.

Problem 19

Let $N$ be the number of nine-digit integers that can be obtained by permuting the digits of 223334444 and which have at least one 3 to the right of the right-most occurrence of 4 . What is the remainder when $N$ is divided by $100$?

Problem 20

Let $f$ be the function defined by

$$
f(n)=\text { remainder when } n^n \text { is divided by } 7,
$$

for all positive integers $n$. Find the smallest positive integer $T$ such that $f(n+T)=f(n)$ for all positive integers $n$.

Problem 21

Let $P(x)=x^{2025}, Q(x)=x^4+x^3+2 x^2+x+1$. Let $R(x)$ be the polynomial remainder when the polynomial $P(x)$ is divided by the polynomial $Q(x)$. Find $R(3)$.

Problem 22

Let $A B C D$ be a rectangle and let $M, N$ be points lying on sides $A B$ and $B C$, respectively. Assume that $M C= C D$ and $M D=M N$, and that points $C, D, M, N$ lic on a circle. If $(A B / B C)^2=m / n$ where $m$ and $n$ are positive integers with $\operatorname{gcd}(m, n)=1$, what is the value of $m+n$ ?

Problem 23

Let \(A B C D\) be a rectangle and let \(M, N\) be points lying on sides \(A B\) and \(B C\), respectively. Assume that \(M C= C D\) and \(M D=M N\), and that points \(C, D, M, N\) lie on a circle. If \((A B / B C)^2=m / n\) where \(m\) and \(n\) are positive integers with \(\operatorname{gcd}(m, n)=1\), what is the value of \(m+n\) ?

Problem 24

There are $m$ blue marbles and $n$ red marbles on a table. Armaan and Babita play a game by taking turns. In each turn the player has to pick a marble of the colour of his/her choice. Armaan starts first, and the player who picks the last red marble wins. For how many choices of $(m, n)$ with $1 \leq m, n \leq 11$ can Armaan force a win?

Problem 25

For some real numbers $m, n$ and a positive integer $a$, the list $(a+1) n^2, m^2, a(n+1)^2$ consists of three consecutive integers written in increasing order. What is the largest possible value of $m^2$ ?

Problem 26

Let $S$ be a circle of radius 10 with centre $O$. Suppose $S_1$ and $S_2$ are two circles which touch $S$ internally and intersect each other at two distinct points $A$ and $B$. If $\angle O A B=90^{\circ}$ what is the sum of the radii of $S_1$ and $S_2$ ?

Solution

Problem 27

A regular polygon with $n \geq 5$ vertices is said to be colourful if it is possible to colour the vertices using at most 6 colours such that each vertex is coloured with exactly one colour, and such that any 5 consecutive vertices have different colours. Find the largest number $n$ for which a regular polygon with $n$ vertices is not colourful.

Solution

Problem 28

Find the number of ordered triples $(a, b, c)$ of positive integers such that $1 \leq a, b, c \leq 50$ which satisfy the relation

$$
\frac{\operatorname{lcm}(a, c)+\operatorname{lcm}(b, c)}{a+b}=\frac{26 c}{27}
$$

Here, by $\operatorname{lcm}(x, y)$ we mean the LCM, that is, least common multiple of $x$ and $y$.

Problem 29

Consider a sequence of real numbers of finite length. Consecutive four term averages of this sequence are strictly increasing, but consecutive seven term averages are strictly decreasing. What is the maximum possible length of such a sequence?

Problem 30

Assume $a$ is a positive integer which is not a perfect square. Let $x, y$ be non-negative integers such that $\sqrt{x-\sqrt{x+a}}=\sqrt{a}-y$. What is the largest possible value of $a$ such that $a<100 ?$

BHASKARA Contest - NMTC - Screening Test – 2025

Problem 1

The greatest 4 -digit number such that when divided by 16,24 and 36 leaves 4 as remainder in each case is
А) 9994
B) 9940
C) 9094
D) 9904

Problem 2

\(A B C D\) is a rectangle whose length \(A B\) is 20 units and breadth is 10 units. Also, given \(A P=8\) units. The area of the shaded region is \(\frac{p}{q}\) sq unit, where \(p, q\) are natural numbers with no common factors other than 1 . The value of \(p+q\) is
A) 167
В) 147
C) 157
D) 137

Problem 3

The solution of \(\frac{\sqrt[7]{12+x}}{x}+\frac{\sqrt[7]{12+x}}{12}=\frac{64}{3}(\sqrt[7]{x})\) is of the form \(\frac{a}{b}\) where \(a, b\) are natural numbers with \(\operatorname{GCD}(a, b)=1\); then \((b-a)\) is equal to
A) 115
B) 114
C) 113
D) 125

Problem 4

The value of \((52+6 \sqrt{43})^{3 / 2}-(52-6 \sqrt{43})^{3 / 2}\) is
A) 858
В) 918
C) 758
D) 828

Problem 5

In the adjoining figure \(\angle D C E=10^{\circ}\), \(\angle C E D=98^{\circ}, \angle B D F=28^{\circ}\)
Then the measure of angle \(x\) is
A) \(72^{\circ}\)
B) \(76^{\circ}\)
C) \(44^{\circ}\)
D) \(82^{\circ}\)

Problem 6

\(A B C\) is a right triangle in which \(\angle \mathrm{B}=90^{\circ}\). The inradius of the triangle is \(r\) and the circumradius of the triangle is R . If \(\mathrm{R}: r=5: 2\), then the value of \(\cot ^2 \frac{A}{2}+\cot ^2 \frac{C}{2}\) is
A) \(\frac{25}{4}\)
B) 17
C) 13
D) 14

Problem 7

If \((\alpha, \beta)\) and \((\gamma, \beta)\) are the roots of the simultaneous equations:

\[
|x-1|+|y-5|=1 ; \quad y=5+|x-1|
\]

then the value of \(\alpha+\beta+\gamma\) is
A) \(\frac{15}{2}\)
B) \(\frac{17}{2}\)
C) \(\frac{14}{3}\)
D) \(\frac{19}{2}\)

Problem 8

Three persons Ram, Ali and Peter were to be hired to paint a house. Ram and Ali can paint the whole house in 30 days, Ali and Peter in 40 days while Peter and Ram can do it in 60 days. If all of them were hired together, in how many days can they all three complete $50 \%$ of the work?
A) $24 \frac{1}{3}$
B) $25 \frac{1}{2}$
C) $26 \frac{1}{3}$
D) $26 \frac{2}{3}$

Problem 9

$\frac{\sqrt{a+3 b}+\sqrt{a-3 b}}{\sqrt{a+3 b}-\sqrt{a-3 b}}=x$, then the value of $\frac{3 b x^2+3 b}{a x}$ is
A) 1
B) 2
C) 3
D) 4

Problem 10

The number of integral solutions of the inequation $\left|\frac{2}{x-13}\right|>\frac{8}{9}$ is
A) 1
B) 2
C) 3
D) 4

Problem 11

In the adjoining figure, $P$ is the centre of the first circle, which touches the other circle in C . PCD is along the diameter of the second circle. $\angle \mathrm{PBA}=20^{\circ}$ and $\angle \mathrm{PCA}=30^{\circ}$.

The tangents at B and D meet at E . The measure of the angle $x$ is
A) $75^{\circ}$
B) $80^{\circ}$
C) $70^{\circ}$
D) $85^{\circ}$

Problem 12

If $\alpha, \beta$ are the values of $x$ satisfying the equation $3 \sqrt{\log _2 x}-\log _2 8 x+1=0$, where $\alpha<\beta$, then the value of $\left(\frac{\beta}{\alpha}\right)$ is
A) 2
B) 4
C) 6
D) 8

Problem 13

When a natural number is divided by 11 , the remainder is 4 . When the square of this number is divided by 11 , the remainder is
A) 4
B) 5
C) 7
D) 9

Problem 14

The unit's digit of a 2-digit number is twice the ten's digit. When the number is multiplied by the sum of the digits the result is 144 . For another 2-digit number, the ten's digit is twice the unit's digit and the product of the number with the sum of its digits is 567 . Then the sum of the two 2 -digit numbers is
A) 68
В) 86
C) 98
D) 87

Problem 15

$A B C D E$ is a pentagon. $\angle A E D=126^{\circ}, \angle B A E=\angle C D E$ and $\angle A B C$ is $4^{\circ}$ less than $\angle B A E$ and $\angle B C D$ is $6^{\circ}$ less than $\angle C D E . P R, Q R$ the bisectors of $\angle B P C, \angle E Q D$ respectively, meet at $R$. Points $\mathrm{P}, \mathrm{C}, \mathrm{D}, \mathrm{Q}$ are collinear. Then measure of $\angle P R Q$ is
A) $151^{\circ}$
B) $137^{\circ}$
C) $141^{\circ}$
D) $143^{\circ}$

Problem 16

$a, b, c$ are real numbers such that $b-c=8$ and $b c+a^2+16=0$.
The numerical value of $a^{2025}+b^{2025}+c^{2025}$ is $\rule{2cm}{0.2mm}$.

Problem 17

Given $f(x)=\frac{2025 x}{x+1}$ where $x \neq-1$. Then the value of $x$ for which $f(f(x))=(2025)^2$ is $\rule{2cm}{0.2mm}$.

Problem 18

The sum of all the roots of the equation $\sqrt[3]{16-x^3}=4-x$ is $\rule{2cm}{0.2mm}$.

Problem 19

In the adjoining figure, two
Quadrants are touching at $B$.
$C E$ is joined by a straight line, whose mid-point is $F$.

The measure of $\angle C E D$ is $\rule{2cm}{0.2mm}$.

Problem 20

The value of $k$ for which the equation $x^3-6 x^2+11 x+(6-k)=0$ has exactly three positive integer solutions is $\rule{2cm}{0.2mm}$.

Problem 21

The number of 3-digit numbers of the form $a b 5$ (where $a, b$ are digits) which are divisible by 9 is $\rule{2cm}{0.2mm}$.

Problem 22

If $a=\sqrt{(2025)^3-(2023)^3}$, the value of $\sqrt{\frac{a^2-2}{6}}$ is $\rule{2cm}{0.2mm}$.

Problem 23

In a math Olympiad examination, $12 \%$ of the students who appeared from a class did not solve any problem; $32 \%$ solved with some mistakes. The remaining 14 students solved the paper fully and correctly. The number of students in the class is $\rule{2cm}{0.2mm}$.

Problem 24

When $a=2025$, the numerical value of
$\left|2 a^3-3 a^2-2 a+1\right|-\left|2 a^3-3 a^2-3 a-2025\right|$ is $\rule{2cm}{0.2mm}$.

Problem 25

A circular hoop and a rectangular frame are standing on the level ground as shown. The diagonal $A B$ is extended to meet the circular hoop at the highest point $C$. If $A B=18 \mathrm{~cm}, B C=32 \mathrm{~cm}$, the radius of the hoop (in cm ) is $\rule{2cm}{0.2mm}$.

Problem 26

' $n$ ' is a natural number. The number of ' $n$ ' for which $\frac{16\left(n^2-n-1\right)^2}{2 n-1}$ is a natural number is $\rule{2cm}{0.2mm}$.

Problem 27

The number of solutions $(x, y)$ of the simultaneous equations $\log _4 x-\log _2 y=0, \quad x^2=8+2 y^2$ is $\rule{2cm}{0.2mm}$.

Problem 28

In the adjoining figure,
$P A, P B$ are tangents.
$A R$ is parallel to $P B$

$P Q=6 ; Q R=18 .$

Length $S B= \rule{2cm}{0.2mm}$.

Problem 29

A large watermelon weighs 20 kg with $98 \%$ of its weight being water. It is left outside in the sunshine for some time. Some water evaporated and the water content in the watermelon is now $95 \%$ of its weight in water. The reduced weight in kg is $\rule{2cm}{0.2mm}$.

Problem 30

In a geometric progression, the fourth term exceeds the third term by 24 and the sum of the second and third term is 6 . Then, the sum of the second, third and fourth terms is $\rule{2cm}{0.2mm}$.

Australian Mathematics Competition - 2019 - Upper Primary Division - Grades 5 & 6- Questions and Solutions

Problem 1:

$201-9=$

(A) 111 (B) 182 (C) 188 (D) 192 (E) 198

Problem 2:

A Runnyball team has 5 players. This graph shows the number of goals each player scored in a tournament. Who scored the second-highest number of goals?


(A) Ali (B) Beth (C) Caz (D) Dan (E) Evan

Problem 3:

Six million two hundred and three thousand and six would be written as

(A) 62036 (B) 6230006 (C) 6203006 (D) 6203600 (E) 6200306

Problem 4:

These cards were dropped on the table, one at a time. In which order were they dropped?

Problem 5:

Sophia is at the corner of 1st Street and 1st Avenue. Her school is at the corner of 4th Street and 3rd Avenue. To get there, she walks



(A) 4 blocks east, 3 blocks north (B) 3 blocks west, 4 blocks north (C) 4 blocks west, 2 blocks north (D) 3 blocks east, 2 blocks north (E) 2 blocks north, 2 blocks south

Problem 6:

Jake is playing a card game, and these are his cards. Elena chooses one card from Jake at random. Which of the following is Elena most likely to choose?

Problem 7:

Which 3D shape below has 5 faces and 9 edges?

Problem 8:

We're driving from Elizabeth to Renmark, and as we leave we see this sign. We want to stop at a town for lunch and a break, approximately halfway to Renmark. Which town is the best place to stop?

(A) Gawler (B) Nuriootpa (C) Truro (D) Blanchetown (E) Waikerie

Problem 9:

What is the difference between the heights of the two flagpoles, in metres?


(A) 16.25 (B) 16.75 (C) 17.25 (D) 17.75 (E) 33.25

Problem 10:

Most of the numbers on this scale are missing.

Which number should be at position $P$ ?
(A) 18 (B) 33 (C) 34 (D) 36 (E) 42

Problem 11:

In a game, two ten-sided dice each marked 0 to 9 are rolled and the two uppermost numbers are added. For example, with the dice as shown, $0+9=9$. How many different results can be obtained?

(A) 17 (B) 18 (C) 19 (D) 20 (E) 21

Problem 12:

Every row and every column of this $3 \times 3$ square must contain each of the numbers 1,2 and 3 . What is the value of $N+M$ ?

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

Problem 13:

Ada Lovelace and Charles Babbage were pioneering researchers into early mechanical computers. They were born 24 years apart.

To the nearest year, how much longer did Charles Babbage live than Ada Lovelace?

(A) 29 (B) 32 (C) 35 (D) 37 (E) 43

Problem 14:

You have 12 metres of ribbon. Each decoration needs $\frac{2}{5}$ of a metre of ribbon. How many decorations can you make?

(A) 6 (B) 7 (C) 10 (D) 24 (E) 30

Problem 15:

Andrew and Bernadette are clearing leaves from their backyard. Bernadette can rake the backyard in 60 minutes, while Andrew can do it in 30 minutes with the vacuum setting on the leaf blower. If they work together, how many minutes will it take?

(A) 10 (B) 20 (C) 24 (D) 30 (E) 45

Problem 16:

A carpet tile measures 50 cm by 50 cm . How many of these tiles would be needed to cover the floor of a room 6 m long and 4 m wide?

(A) 24 (B) 20 (C) 40 (D) 48 (E) 96

Problem 17:

In how many different ways can you place the numbers 1 to 4 in these four circles so that no two consecutive numbers are side by side?

(A) 2 (B) 4 (C) 6 (D) 8 (E) 12

Problem 18:

John, Chris, Anne, Holly and Mike are seated around a round table, each with a card with a number on it in front of them. Each person can see the numbers in front of their two neighbours. Each person calls out the sum of the two numbers in front of their neighbours. John says 30, Chris says 33, Anne says 31, Holly says 38 and Mike says 36. Holly has the number 21 in front of her. What number does Anne have in front of her?

(A) 9 (B) 13 (C) 15 (D) 18 (E) 19

Problem 19:

Annabel has 2 identical equilateral triangles. Each has an area of $9 \mathrm{~cm}^2$. She places one triangle on top of the other as shown to form a star, as shown. What is the area of the star in square centimetres?


(A) 10 (B) 12 (C) 14 (D) 16 (E) 18

Problem 20:

Lola went on a train trip. During her journey she slept for $\frac{3}{4}$ of an hour and stayed awake for $\frac{3}{4}$ of the journey. How long did the trip take?

(A) 1 hour (B) 2 hours (C) $2 \frac{1}{2}$ hours (D) 3 hours (E) 4 hours

Problem 21:

My sister and I are playing a game where she picks two counting numbers and I have to guess them. When I tell her a number, she multiplies my number by her first number and then adds her second number. When I say 15 , she says 50 . When I say 2 , she says 11 . If I say 6 , what should she say?

(A) 23 (B) 27 (C) 35 (D) 41 (E) 61

Problem 22:

Once the muddy water from the 2018 Ingham floods had drained from Harry's house, he found this folded map that had been standing in the floodwater at an angle. He unfolded it and laid it out to dry, but it was still mud-stained. What could it look like now?

Problem 23:

A tower is built from exactly 2019 equal rods. Starting with 3 rods as a triangular base, more rods are added to form a regular octahedron with this base as one of its faces. The top face is then the base of the next octahedron. The diagram shows the construction of the first three octahedra. How many octahedra are in the tower when it is finished?

(A) 2016 (B) 1008 (C) 336 (D) 224 (E) 168

Problem 24:

These three cubes are labelled in exactly the same way, with the 6 letters A, M, C, D, E and F on their 6 faces:


The cubes are now placed in a row so that the front looks like this:

When we look at the cubes from the opposite side, we will see

Problem 25:

In Jeremy's hometown of Windar, people live in either North, East, South, West or Central Windar. Jeremy is putting together a chart showing where the students in his class live, but unfortunately his dog chewed his survey results before he managed to label the five columns.


He only remembers two things about the survey: South Windar is more common than both East and Central Windar, and the number of students in North and Central Windar combined is the same as the total of the other three regions.
Using only this information, how many columns can Jeremy correctly label with \(100 \%\) certainty?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 5

Problem 26:

Pip starts with a large square sheet of paper and makes two straight cuts to form four smaller squares. She then takes one of these smaller squares and makes two more straight cuts to make four even smaller ones, as shown.

Continuing in this way, how many cuts does Pip need to make to get a total of 1000 squares of various sizes?

Problem 27:

Seven of the numbers from 1 to 9 are placed in the circles in the diagram in such a way that the products of the numbers in each vertical or horizontal line are the same. What is this product?

Problem 28:

A hare and a tortoise compete in a 10 km race. The hare runs at \(30 \mathrm{~km} / \mathrm{h}\) and the tortoise walks at \(3 \mathrm{~km} / \mathrm{h}\). Unfortunately, at the start, the hare started running in the opposite direction. After some time, it realised its mistake and turned round, catching the tortoise at the halfway mark. For how many minutes did the hare run in the wrong direction?

Problem 29:

I want to place the numbers 1 to 10 in this diagram, with one number in each circle. On each of the three sides, the four numbers add to a side total, and the three side totals are all the same. What is the smallest number that this side total could be?

Problem 30:

The sum of two numbers is 11.63 . When adding the numbers together, Oliver accidentally shifted the decimal point in one of the numbers one position to the left. Oliver got an answer of 5.87 instead. What is one hundred times the difference between the two original numbers?

Australian Mathematics Competition - 2020 - Upper Primary Division - Grades 5 & 6- Questions and Solutions

Problem 1:

How many pieces have been placed in the jigsaw puzzle so far?

(A) 25 (B) 27 (C) 30 (D) 33 (E) 35

Problem 2:

What is half of 2020 ?

(A) 20 (B) 101 (C) 110 (D) 1001 (E) 1010

Problem 3:

What is the perimeter of this triangle?

(A) 33 m (B) 34 m (C) 35 m (D) 36 m (E) 37 m

Problem 4:

Which fraction is the largest?

(A) one-half (B) one-quarter (C) one-third (D) three-quarters (E) six-tenths

Problem 5:

A protractor is used to measure angle (P X Q). The angle is

(A) $45^{\circ}$ (B) $55^{\circ}$ (C) $135^{\circ}$ (D) $145^{\circ}$ (E) $180^{\circ}$

Problem 6:

Some friends are walking to a lake in the mountains. First they climb a hill before they walk down to the lake. Which graph most accurately represents their journey?

Problem 7:

How many tenths are in 6.2 ?

(A) 62 (B) 8 (C) 4 (D) 12 (E) 36

Problem 8:

The graph shows the number of eggs laid by backyard chickens Nony and Cera for the first six months of the year.

In how many months did Nony lay more eggs than Cera?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Problem 9:

A class of 24 students, all of different heights, is standing in a line from tallest to shortest. Mary is the 8th tallest and John is the 6 th shortest. How many students are standing between them in the line?

(A) 6 (B) 7 (C) 8 (D) 9 (E) 10

Problem 10:

Maria divided a rectangle into a number of identical squares and coloured some of them in, as shown. She wants three-quarters of the rectangle's area to be coloured in altogether. How many more squares does she need to colour in?


(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

Problem 11:

At the end of a game of marbles, Lei has 15 marbles, Dora has 8 and Omar has 4 . How many marbles must Lei give back to his friends if they want to start the next game with an equal number each?

(A) 5 (B) 6 (C) 7 (D) 8 (E) 9

Problem 12:

In the grid, the total of each row is given at the end of the row, and the total of each column is given at the bottom of the column.
The value of $N$ is

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Problem 13:

At his birthday party, Ricky and his friends wear stripy paper hats in the shape of a cone, as shown on the left. After the party, Ricky makes a straight cut in one of the hats all the way up to the point at the top, as shown on the right.

Which of the following best matches what the hat will look like when Ricky flattens it out on the table?

Problem 14:

Emma is going to write all the numbers from 1 to 50 in order. She writes 25 digits on the first line of her page. What was the last number she wrote on this line?

(A) 13 (B) 15 (C) 17 (D) 19 (E) 21

Problem 15:

The playing card shown is flipped over along edge $b$ and then flipped over again along edge $c$. What does it look like now?

Problem 16:

Which labelled counter should you remove so that no two rows have the same number of counters and no two columns have the same number of counters?

Problem 17:

Aidan puts a range of 3D shapes on his desk at school. This is the view from his side of the desk:

Nadia is sitting on the opposite side of the desk facing Aidan. Which of the following diagrams best represents the view from Nadia's side of the desk?

Problem 18:

The area of each of the five equilateral triangles in the diagram is 1 square metre. What is the shaded area?

(A) $1.5 \mathrm{~m}^2$ (B) $2 \mathrm{~m}^2$ (C) $2.5 \mathrm{~m}^2$ (D) $3 \mathrm{~m}^2$ (E) $3.5 \mathrm{~m}^2$

Problem 19:

Kayla is 5 years old and Ryan is 13 years younger than Cody. One year ago, Cody's age was twice the sum of Kayla's and Ryan's age. Find the sum of the three children's current ages.

(A) 10 (B) 22 (C) 26 (D) 30 (E) 36

Problem 20:

Mary has a piece of paper. She folds it exactly in half. Then she folds it in half again. She finishes up with this shape.

Which of the shapes $P, Q$ and $R$ shown below could have been her starting shape?

(A) only $P$ (B) only $Q$ (C) only $R$ (D) only $P$ and $R$ (E) all three

Problem 21:

Four positive whole numbers are placed at the vertices of a square. On each edge, the difference between the two numbers at the vertices is written. The four edge numbers are $1,2,3$ and 4 in some order. What is the smallest possible sum of the numbers at the vertices?

(A) 10 (B) 11 (C) 12 (D) 13 (E) 14

Problem 22:

The large rectangle shown has been divided into 4 smaller rectangles. The perimeters of three of these are $10 \mathrm{~cm}, 16 \mathrm{~cm}$ and 20 cm . The fourth rectangle does not have the largest or the smallest perimeter of the four smaller rectangles.

What, in centimetres, is the perimeter of the large rectangle?

(A) 26 (B) 30 (C) 32 (D) 36 (E) 46

Problem 23:

A bale of hay can be eaten by a horse in 2 days, by a cow in 3 days and by a sheep in 12 days. A farmer has 22 bales of hay and one horse, one cow and one sheep to feed. How many days will his bales last?

(A) 20 (B) 22 (C) 24 (D) 26 (E) 28

Problem 24:

This rectangle is 36 cm long. It is cut into two pieces and rearranged to form a square, as shown.


What is the height of the original rectangle?

(A) 14 cm (B) 16 cm (C) 18 cm (D) 20 cm (E) 24 cm

Problem 25:

A bottle with a sealed lid contains some water. The diagram shows this bottle up the right way and upside down. How full is the bottle?


(A) $\frac{1}{2}$ (B) $\frac{4}{7}$ (C) $\frac{5}{7}$ (D) $\frac{2}{3}$ (E) $\frac{9}{14}$

Problem 26:

A number is oddtastic if all of its digits are odd. For example, 9,57 and 313 are oddtastic. However, 50 and 787 are not oddtastic, since 0 and 8 are even digits. How many of the numbers from 1 to 999 are oddtastic?

Problem 27:

On my chicken farm where I have 24 pens, the pens were a bit crowded. So I built 6 more pens, and the number of chickens in each pen reduced by 6 . How many chickens do I have?

Problem 28:

How many even three-digit numbers are there where the digits add up to $8 ?$

Problem 29:

Madeleine types her three-digit Personal Identification Number (PIN) into this keypad. All three digits are different, but the buttons for the first and second digits share an edge, and the buttons for the second and third digits share an edge. For instance, 563 is a possible PIN, but 536 is not, since 5 and 3 do not share an edge. How many possibilities are there for Madeleine's PIN?

Problem 30:

Writing one digit every second, you have half an hour to list as many of the counting numbers as you can, starting $1,2,3, \ldots$. At the end of half an hour, what number have you just finished writing?

Australian Mathematics Competition - 2021 - Upper Primary Division - Grades 5 & 6- Questions and Solutions

Problem 1:

This Nigerian flag is white and green. What fraction of it is green?

(A) one-third (B) one-quarter (C) one-half (D) two-fifths (E) two-thirds

Problem 2:

Which number makes this number sentence true? $\square$ -5=9

(A) 0 (B) 4 (C) 12 (D) 9 (E) 14

Problem 3:

What is the perimeter of the quadrilateral shown?

(A) 13 cm (B) 15 cm (C) 17 cm (D) 19 cm (E) 21 cm

Problem 4:

Which of the following decimal numbers has the smallest value?

(A) 0.0002 (B) 0.002 (C) 0.02 (D) 0.2 (E) 2.0

Problem 5:

$\frac{1}{2}+\frac{2}{4}-\frac{4}{8}=$

(A) $\frac{1}{2}$ (B) 1 (C) $1 \frac{1}{2}$ (D) 2 (E) 4

Problem 6:

Suri has a number of 20-cent and 50-cent coins. Which of the following amounts of money is it NOT possible for her to make?

(A) 50 cents (B) 60 cents (C) 80 cents (D) 30 cents (E) 70 cents

Problem 7:

A square of paper is rolled up, pressed flat, and then cut as shown.

What could the sheet of paper look like when unrolled and leid flat?



Problem 8:

Leo is waiting in line at school. There are four students ahead of him and twice as many behind him. How many students are in this line?

(A) 4 (B) 8 (C) 9 (D) 12 (E) 13

Problem 9:

Cassandra makes a healing potion from a mixture of herbs. She uses this balance to weigh out the herbs. If she uses 5 grams of fennel, how many grams of mint will she need?


(A) 5 (B) 10 (C) 15 (D) 20 (E) 40

Problem 10:

There are 14 pieces of fruit in a bowl. There are twice as many nectarines as pears, and half as many nectarines as apples. There are no other types of fruit. How many apples are there?


(A) 2 (B) 4 (C) 6 (D) 8 (E) 10

Problem 11:

I am shuffling a deck of cards but I accidentally drop a card on the ground every now and then. After a while, I notice that I have dropped five cards. From above, the five cards look like one of the following pictures. Which picture could it be?

Problem 12:

This rectangle has been made by joining two squares together. Each square has an area of $25 \mathrm{~cm}^2$. What is the perimeter of the rectangle?

(A) 18 cm (B) 20 cm (C) 26 cm (D) 30 cm (E) 50 cm

Problem 13:

A kangaroo is chasing a wallaby that is 42 metres ahead. For every 4 -metre hop the kangaroo makes, the wallaby makes a 1-metre hop. How many hops will the kangaroo have to make to catch up with the wallaby?

(A) 8 (B) 10 (C) 11 (D) 14 (E) 21

Problem 14:

A piece of straight wire is 50 cm long. Six right-angled bends are made in the wire, so that it ends up looking like the diagram shown:

The lengths of two sections are shown. What is the length marked $x$ ?

(A) 28 cm (B) 31 cm (C) 34 cm (D) 36 cm (E) 39 cm

Problem 15:

Margie and Rosie both live near Lawson train station. Each plans to catch the 10 am train. Margie thinks her watch is 10 minutes fast, but in fact it is 10 minutes slow. Rosie thinks her watch is 10 minutes slow, but in fact it is 5 minutes fast. Each of them leaves home to catch the train without having to wait on the platform. Who misses the train, and by how much?

(A) Margie by 10 minutes (B) Margie by 20 minutes (C) Rosie by 5 minutes (D) Rosie by 15 minutes (E) Neither of them

Problem 16:

Sally was playing with block patterns and came up with this one she called Hollow Squares. They all follow the same pattern.

How many blocks would she need to make Hollow Square 7 ?

(A) 28 (B) 30 (C) 32 (D) 34 (E) 53

Problem 17:

I have a jug containing 100 mL of liquid, which is half vinegar and half olive oil. How much vinegar must I add to make a mixture which is one-third olive oil?

(A) 30 mL (B) 40 mL (C) 50 mL (D) 60 mL (E) 100 mL

Problem 18:

It is 10 am now. What time will it be in 2021 hours time?

(A) 11 am (B) 1 pm (C) 3 pm (D) 4 pm (E) 5 pm

Problem 19:

Alexander's pen leaked on his addition homework, covering up three of the digits in the calculation shown. How many different possibilities are there for the correct working?


(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

Problem 20:

Our school is organising a quiz night. They are expecting from 25 to 35 people to come. The people will be arranged in teams of 6 to 8 people. What is the range of possible numbers of teams to expect?

(A) 4 to 5 (B) 4 to 6 (C) 5 to 6 (D) 3 to 6 (E) 3 to 5

Problem 21:

In an election for school captain, there were 4 candidates and 453 students each voted for one candidate. The winner's margins over the other candidates were 31,25 and 19. How many votes did the winner receive?

(A) 113 (B) 127 (C) 129 (D) 131 (E) 132

Problem 22:

Three blocks with rectangular faces are placed together to form a larger rectangular prism. All blocks have side lengths which are whole numbers of centimetres. The areas of some of the faces are shown, as is the length of one edge.

In cubic centimetres, what is the volume of the combined prism?

(A) 360 (B) 540 (C) 600 (D) 720 (E) 900

Problem 23:

Three gears are connected as shown. The two larger gears have 20 teeth each and the smaller gear has 10 teeth.
The middle gear is rotated half a turn in the direction of the arrows, turning the M upside down.

What do the three gears look like after this rotation?

Problem 24:

Anna has a large number of tiles of three types:


She wants to build a green rectangle with a white frame similar to those below.

She builds such a rectangle using as many tiles as possible while using exactly 20 completely green tiles. How many tiles will she use altogether?

(A) 80 (B) 66 (C) 48 (D) 42 (E) 39

Problem 25:

In Jeremy's hometown of Windar, people live in either North, East, South, West or Central Windar. Jeremy is putting together a chart showing where the students in his class live, but unfortunately his dog chewed his survey results before he managed to label the five columns.


He only remembers two things about the survey: South Windar is more common than both East and Central Windar, and the number of students in North and Central Windar combined is the same as the total of the other three regions. Using only this information, how many columns can Jeremy correctly label with \(100 \%\) certainty?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 5

Problem 26:

Pip starts with a large square sheet of paper and makes two straight cuts to form four smaller squares. She then takes one of these smaller squares and makes two more straight cuts to make four even smaller ones, as shown.

Continuing in this way, how many cuts does Pip need to make to get a total of 1000 squares of various sizes?

Problem 27:

Seven of the numbers from 1 to 9 are placed in the circles in the diagram in such a way that the products of the numbers in each vertical or horizontal line are the same. What is this product?

Problem 28:

A hare and a tortoise compete in a 10 km race. The hare runs at \(30 \mathrm{~km} / \mathrm{h}\) and the tortoise walks at \(3 \mathrm{~km} / \mathrm{h}\). Unfortunately, at the start, the hare started running in the opposite direction. After some time, it realised its mistake and turned round, catching the tortoise at the halfway mark. For how many minutes did the hare run in the wrong direction?

Problem 29:

I want to place the numbers 1 to 10 in this diagram, with one number in each circle. On each of the three sides, the four numbers add to a side total, and the three side totals are all the same. What is the smallest number that this side total could be?

Problem 30:

The sum of two numbers is 11.63 . When adding the numbers together, Oliver accidentally shifted the decimal point in one of the numbers one position to the left. Oliver got an answer of 5.87 instead. What is one hundred times the difference between the two original numbers?

Australian Mathematics Competition - 2022 - Upper Primary Division - Grades 5 & 6- Questions and Solutions

Problem 1:

What number is two hundred and five thousand, one hundred and fifty?

(A) 150 (B) 205 (C) 20150 (D) 25150 (E) 205150

Problem 2:

What fraction of this picture is shaded?


(A) $\frac{1}{2}$ (B) $\frac{2}{3}$ (C) $\frac{3}{4}$ (D) $\frac{4}{9}$ (E) $\frac{5}{9}$

Problem 3:

$2220-2022=$

(A) 18 (B) 188 (C) 198 (D) 200 (E) 202

Problem 4:

Audrey wrote these three numbers in order from smallest to largest:

$$
\begin{array}{llll}
1.03 & 0.08 & 0.4
\end{array}
$$

In which order did she write them?

(A) $0.08,1.03,0.4$ (B) $0.08,0.4,1.03$ (C) $0.4,0.08,1.03$
(D) $0.4,1.03, .008$ (E) $1.03,0.4,0.08$

Problem 5:

I was 7 years old when my brother turned 3. How old will I be when
he turns 7?

(A) 9 (B) 10 (C) 11 (D) 12 (E) 13

Problem 6:

This shape is built from 29 squares, each $1 \mathrm{~cm} \times 1 \mathrm{~cm}$. What is its perimeter in centimetres?

(A) 52 (B) 58 (C) 60 (D) 68 (E) 72

Problem 7:

A tachometer indicates how fast the crankshaft in a car's engine is spinning, in thousands of revolutions per minute (rpm). What is the reading on the tachometer shown?



(A) 2.2 rpm (B) 2.4 rpm (C) 240 rpm (D) 2200 rpm (E) 2400 rpm

Problem 8:

Joseph had a full, one-litre bottle of water. He drank 320 millilitres of it. How much was left?

(A) 660 mL (B) 670 mL (C) 680 mL (D) 730 mL (E) 780 mL

Problem 9:

Which of these rectangles has an area of 24 square centimetres?



(A) Q only (B) Q and R only (C) R only (D) S only (E) P, Q, R and S

Problem 10:

This table shows Jai's morning routine. If he needs to be at school by $8: 55 \mathrm{am}$ what is the latest time he can start his shower?


(A) 7:35 am (B) 7: 50 am (C) 8:05 am (D) 8:20 am (E) 8:35 am

Problem 11:

Which spinner is twice as likely to land on red as white?

Problem 12:

Starting at 0 on the number line, Clement walks back and forth in the following pattern: 3 to the right, 2 to the left, 3 to the right, 2 to the left, and so on.

How many times does he walk past the position represented by $4 \frac{1}{2}$ ?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Problem 13:

Three digits are missing from this sum. Toby worked out the missing numbers and added them together. What was his answer?

(A) 11 (B) 13 (C) 15 (D) 17 (E) 19

Problem 14:

I have three cardboard shapes: a square, a circle and a triangle. I glue them on top of each other as shown in this diagram.

I then flip the glued-together shapes over. What could they look like?

Problem 15:

What is the missing number needed to make this number sentence true? $270 \div 45=\square \div 15$

(A) 3 (B) 6 (C) 60 (D) 90 (E) 150

Problem 16:

Three different squares are arranged as shown. The perimeter of the largest square is 32 centimetres. The area of the smallest square is 9 square centimetres. What is the perimeter of the mediumsized square?

(A) 12 cm (B) 14 cm (C) 20 cm (D) 24 cm (E) 30 cm

Problem 17:

Huang has a bag of marbles. Mei takes out one-third of them. Huang then takes out one-half of those left, leaving 8 marbles in the bag. How many marbles were originally in the bag?

(A) 12 (B) 16 (C) 18 (D) 24 (E) 36

Problem 18:

A different positive whole number is placed at each vertex of a cube. No two numbers joined by an edge of the cube can have a difference of 1.

What is the smallest possible sum of the eight numbers?

(A) 36 (B) 37 (C) 38 (D) 39 (E) 40

Problem 19:

George is 78 this year. He has three grandchildren: Michaela, Tom and Lucy. Michaela is 27 , Tom is 23 and Lucy is 16 . After how many years will George's age be equal to the sum of his grandchildren's ages?

(A) 3 (B) 6 (C) 9 (D) 10 (E) 12

Problem 20:

Ms Graham asked each student in her Year 5 class how many television sets they each have This graph shows the results.

How many television sets do the students have altogether?

(A) 9 (B) 29 (C) 91 (D) 99 (E) 101

Problem 21:

In a mathematics competition, 70 boys and 80 girls competed. Prizes were won by 6 boys and $15 \%$ of the girls. What percentage of the students were prize winners?

(A) $10 \%$ (B) $12 \%$ (C) $15 \%$ (D) $18 \%$ (E) $20 \%$

Problem 22:

Ariel writes the letters of the alphabet on a piece of paper as shown She turns the page upside down and looks at it in her bathroom mirror. How many of the letters appear unchanged when viewed this way?

(A) 0 (B) 3 (C) 4 (D) 6 (E) 9

Problem 23:

The Australian Mathematical College (AMC) has 1000 students. Each student takes 6 classes a day. Each teacher teaches 5 classes per day with 25 students in each class. How many teachers are there at the AMC?

(A) 40 (B) 48 (C) 50 (D) 200 (E) 240

Problem 24:

This list pqrs, pqsr, prqs, prsq, … can be continued to include all 24 possible arrangements of the four letters $p, q, r$ and $s$. The arrangements are listed in alphabetical order. Which one of the following is 19th in this list?

(A) $s p q r$ (B) $s r p q$ (C) $q p s r$ (D) $q r p s$ (E) $r p s q$

Problem 25:

In this puzzle, each circle should contain an integer. Each of the five lines of four circles should add to 40. When the puzzle is completed, what is the largest number used?

(A) 15 (B) 16 (C) 17 (D) 18 (E) 19

Problem 26:

Nguyen writes down some numbers according to the following rules. Starting with the number 1, he doubles the number and adds 4 , so the second number he writes is 6 . He now repeats this process, starting with the last number written, doubling and then adding 4, but he doesn't write the hundreds digit if the number is bigger than 100 . What is the 2022nd number that Nguyen writes down?

Problem 27:

Karen's mother made a cake for her birthday. After it was iced on the top and the 4 vertical faces, it was a cube with 20 cm sides. Darren was asked to decorate the cake with chocolate drops. He arranged them all over the icing in a square grid pattern, spaced with centres 2 cm apart. Those near the edges of the cube had centres 2 cm from the edge. The diagram shows one corner of the cake.

How many chocolate drops did Darren use to decorate Karen's cake?

Problem 28:

I choose three different numbers out of this list and add them together:

$$
1,3,5,7,9, \ldots, 105
$$

How many different totals can I get?

Problem 29:

The Athletics clubs of Albury and Wodonga agree to send a combined team to the regional championships. They have 11 sprinters on the combined team, 5 from Albury and 6 from Wodonga. For the $4 \times 100$ metre relay, they agree to have a relay team with two sprinters from the Albury club and two sprinters from the Wodonga club. How many relay teams are possible?

Problem 30:

The following is a net of a rectangular prism with some dimensions, in centimetres, given.

What is the volume of the rectangular prism in cubic centimetres?

Australian Mathematics Competition - 2023 - Upper Primary Division - Grades 5 & 6- Questions and Solutions

Problem 1:

This shape is made from 7 squares, each 1 cm by 1 cm . What is its perimeter?

(A) 7 cm (B) 12 cm (C) 14 cm (D) 16 cm (E) 28 cm

Problem 2:

There are five shapes here. How many are quadrilaterals?


(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Problem 3:

In a board game, Nik rolls three standard dice, one at a time. He needs his three rolls to add to 12 . His first two dice rolls are 5 and 3 . What does he need his third roll to be?

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

Problem 4:

In this diagram, how many of the small squares need to be shaded for the large rectangle to be one-quarter shaded?

(A) 2 (B) 3 (C) 4 (D) 6 (E) 12

Problem 5:

Petra left for school at 8:51 am. She got to school at 9:09 am. How long did it take Petra to get to school?

(A) 9 minutes (B) 10 minutes (C) 18 minutes (D) 42 minutes (E) 1 hour

Problem 6:

Which letter marks where 25 is on this number line?

Problem 7:

This bottle holds 4 glasses of water.

Which one of the following holds the most water?

Problem 8:

Two pizzas are shared equally between 3 students. What fraction of a whole pizza does each student get?

(A) $\frac{1}{2}$ (B) $\frac{1}{3}$ (C) $\frac{1}{4}$ (D) $\frac{2}{3}$ (E) $\frac{3}{4}$

Problem 9:

A piece of card is cut out and labelled as shown in the diagram.

It is folded along the dotted lines to make a box without a top. Which letter is on the bottom of the box?

(A) A (B) B (C) C (D) D (E) E

Problem 10:

Doughnuts come in bags of 3 and boxes of 8 . I bought exactly 25 doughnuts for my party.What do I get when I add the number of boxes I bought and the number of bags I bought?


(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

Problem 11:

This line graph shows the temperature each hour during a day.

Roughly for how long was the temperature above $20^{\circ} \mathrm{C}$ ?

(A) 7 hours (B) 8 hours (C) 9 hours (D) 10 hours (E) 11 hours

Problem 12:

VIV takes her three children, HANNAH, OTTO and IZZI, out shopping. Each is wearing a t-shirt with their name on the front in capital letters. When they stand in front of the shop mirror, which names appear the same in the reflection as on the shirts?


(A) VIV and OTTO (B) VIV, OTTO and IZZI (C) VIV, HANNAH and IZZI (D) HANNAH and OTTO (E) All four of them

Problem 13:

This regular hexagon has angles of $120^{\circ}$ and the square has angles of $90^{\circ}$.

What is the angle $x^{\circ}$ in the diagram?

(A) $90^{\circ}$ (B) $120^{\circ}$ (C) $135^{\circ}$ (D) $150^{\circ}$ (E) $180^{\circ}$

Problem 14:

Syed's mother had some money to share with her family. She gave one-quarter of her money to Syed. Then she gave one-third of what was left to Ahmed. Then she gave one-half of what was left to Raiyan. She was left with $\$ 15$, which she kept for herself. How much money did Syed's mother have to start with?

(A) $\$ 30$ (B) $\$ 45$ (C) $\$ 60$ (D) $\$ 90$ (E) $\$ 120$

Problem 15:

The rectangle shown has a side length of 9 cm . It is divided into 3 identical rectangles as shown. What is the area, in square centimetres, of the original rectangle?

(A) 45 (B) 50 (C) 52 (D) 54 (E) 63

Problem 16:

This diagram shows a rectangle with a perimeter of 30 cm . It has been divided by 2 lines into 4 small rectangles. Three of the small rectangles have the perimeters shown. What is the perimeter of fourth small rectangle?

(A) 10 cm (B) 12 cm (C) 14 cm (D) 16 cm (E) 18 cm

Problem 17:

There are 10 questions in a test. Each correct answer scores 5 points, each wrong answer loses 3 points, and if a question is left blank it scores 0 points. Tycho did this test and scored 27 points. How many questions did Tycho leave blank?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Problem 18:

Estelle is making decorations shaped like the 8-pointed star shown. She folds a square of paper to make a triangle with 8 layers as shown.

How could she cut the triangle so that the unfolded shape is the star?

Problem 19:

Earlier this year Ben said, 'Next year I will turn 13 , but 2 days ago I was $10 . '$ Ben's birthday is

(A) 1st January (B) 2nd January (C) 29th December (D) 30th December (E) 31st December

Problem 20:

Peyton, Luka and Dan have 180 stickers in total. Peyton has half as many stickers as Luka. Dan has three times as many as Luka.
How many stickers does Peyton have?

(A) 20 (B) 24 (C) 30 (D) 40 (E) 54

Problem 21:

Mrs Graaf invents a game for her students to practise arithmetic. They roll two 10 -sided dice to pick two random numbers. Starting at one of the numbers, they keep adding the other number until they reach a 3-digit number. Ian rolls a 5 and an 8 . If he chooses to start with 5 and then add 8 again and again, his list is $5,13,21, \ldots$, 93,101 . If he chooses to start with 8 and add 5 , his list is $8,13,18, \ldots, 98,103$ On Nara's turn, she makes a list that ends with 107. What pair of numbers could she have rolled?

(A) 4 and 8 (B) 5 and 7 (C) 3 and 4 (D) 6 and 9 (E) 3 and 8

Problem 22:

At a school concert, the tickets cost $\$ 20$ per adult and $\$ 2$ per child. The total paid by the 100 people who attended was $\$ 920$. How many were children?

(A) between 25 and 35 (B) between 35 and 45 (C) between 45 and 55 (D) between 55 and 65 (E) between 65 and 75

Problem 23:

Meena has a standard dice, with each pair of opposite faces adding to 7 . At first, the three faces she can see add to 6 , as shown. She holds the dice between a pair of opposite faces and rotates it $180^{\circ}$, keeping these opposite faces facing the same direction. She puts the dice back down and adds up the three faces she can now see.

What is the smallest possible total she could get?

(A) 6 (B) 8 (C) 10 (D) 12 (E) 14

Problem 24:

I have 4 whole numbers that add up to 98. If I were to add 6 to the first number, subtract 6 from the second number, multiply the third number by 6 and divide the fourth number by 6, the four answers would all be the same. What is the sum of the largest two of my original four numbers?

(A) 72 (B) 86 (C) 88 (D) 90 (E) 94

Problem 25:

When I ride my bike at 20 kilometres per hour, each wheel turns at 2 revolutions per second. When I ride 1 kilometre, how many revolutions does each wheel make?

(A) 40 (B) 240 (C) 320 (D) 360 (E) 420

Problem 26:

Problem 27:

Li attempted to multiply a single-digit number by 36 , but he accidentally multiplied by 63 instead. His answer was 189 larger than the correct answer. What was the correct answer to the multiplication?

Problem 28:

Using 9 out of the 10 possible digits Safia writes 3 numbers, each between 100 and 999. She adds her 3 numbers together. What is the smallest possible sum?

Problem 29:

Yifan has a construction set consisting of red, blue and yellow rods. All rods of the same colour are the same length, but differently coloured rods are different lengths. She wants to make quadrilaterals using these rods.

What number do you get when you multiply the lengths of one red rod, one blue rod and one yellow rod?

Problem 30:

Janus is making patterns using square tiles. Each pattern is made by copying the previous pattern, then adding a tile to every grid square that shares an edge with the copied pattern.

His last pattern is the largest one that can be made with fewer than 1000 tiles. How many tiles are in this last pattern?