Question 1

Alicia had two containers. The first was $\frac{\mathbf{5}}{\mathbf{6}}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\frac{\mathbf{3}}{\mathbf{4}}$ full of water. What is the ratio of the volume of the first container to the volume of the second container?

Question 2

Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?

Question 3

In a high school with 500 students, $40 %$ of the seniors play a musical instrument, while $30 %$ of the non-seniors do not play a musical instrument. In all, $46.8 %$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?

Question 4

All lines with equation $a x+b y=c$ such that $a, b, c$ form an arithmetic progression pass through a common point. What are the coordinates of that point?

Question 5

Triangle $A B C$ lies in the first quadrant. Points $A, B$, and $C$ are reflected across the line $y=x$ to points $A^{\prime}, B^{\prime}$, and $C^{\prime}$, respectively. Assume that none of the vertices of the triangle lie on the line $y=x$. Which of the following statements is not always true?

Question 6

There is a real $n$ such that $$ (n+1)!+(n+2)!=n!\cdot 440 . $$ What is the sum of the digits of $n$ ?

Question 7

Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of $n$ ?

Question 8

The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?

Question 9

The function $f$ is defined by $$ f(x)=\lfloor|x|\rfloor-|\lfloor x\rfloor| $$ for all real numbers $x$, where $\lfloor r\rfloor$ denotes the greatest integer less than or equal to the real number $r$. What is the range of $f$ ?

Question 10

In a given plane, points $A$ and $B$ are 10 units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle A B C$ is 50 units and the area of $\triangle A B C$ is 100 square units?

Question 11

Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar 1 the ratio of blue to green marbles is $\mathbf{9 : 1}$, and the ratio of blue to green marbles in Jar 2 is 8:1. There are 95 green marbles in all. How many more blue marbles are in Jar 1 than in Jar 2?

Question 12

What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than 2019 ?

Question 13

What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17$, and $x$ is equal to the mean of those five numbers?

Question 14

The base-ten representation for 19 ! is $121,6 T 5,100,40 M, 832, H 00$, where $T, M$, and $H$ denote digits that are not given. What is $T+M+H$ ?

Question 15

Two right triangles, $T_{1}$ and $T_{2}$, have areas of 1 and 2 , respectively. One side length of one triangle is congruent to a different side length in the other, and another side length of the first triangle is congruent to yet another side length in the other. What is the product of the third side lengths of $T_{1}$ and $T_{2}$ ?

Question 16

In $\triangle A B C$ with a right angle at $C$, point $D$ lies in the interior of $\overline{A B}$ and point $E$ lies in the interior of $\overline{B C}$ so that $A C=C D, D E=E B$, and the ratio $A C: D E=4: 3$. What is the ratio $A D: D B$ ?

Question 17

A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k=1,2,3, \ldots$. What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?

Question 18

Henry decides one morning to do a workout, and he walks $\frac{3}{4}$ of the way from his home to his gym. The gym is 2 kilometers away from Henry's home. At that point, he changes his mind and walks $\frac{3}{4}$ of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks $\frac{3}{4}$ of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked $\frac{3}{4}$ of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point $A$ kilometers from home and a point $B$ kilometers from home. What is $|A-B|$ ?

Question 19

Let $S$ be the set of all positive integer divisors of 100,000 . How many numbers are the product of two distinct elements of $S$ ?

Question 20

As shown in the figure, line segment $\overline{A D}$ is trisected by points $B$ and $C$ so that $A B=B C=C D=2$. Three semicircles of radius $1, have their diameters on $\overline{A D}$, and are tangent to line $E G$ at $E, F$, and $G$, respectively. A circle of radius 2 has its center on $F$. The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form $\frac{a}{b} \cdot \pi-\sqrt{c}+d$, where $a, b, c$, and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d$ ?

Question 21

Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head?

Question 22

Raashan, Sylvia, and Ted play the following game. Each starts with \($\)1. A bell rings every 15 seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives \($\)1 to that player. What is the probability that after the bell has rung 2019 times, each player will have \($\)1 ? (For example, Raashan and Ted may each decide to give \($\)1 to Sylvia, and Sylvia may decide to give her her dollar to Ted, at which point Raashan will have \($\)0, Sylvia will have \($\)2, and Ted will have \($\)1, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their \($\)1 to, and the holdings will be the same at the end of the second round.)

Question 23

Points $A(6,13)$ and $B(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\omega$ ?

Question 24

Define a sequence recursively by $x_{0}=5$ and $$ x_{n+1}=\frac{x_{n}^{2}+5 x_{n}+4}{x_{n}+6} $$ for all nonnegative integers $n$. Let $m$ be the least positive integer such that $$ x_{m} \leq 4+\frac{1}{2^{20}} $$ In which of the following intervals does $m$ lie?

Question 25

How many sequences of 0 s and 1 s of length 19 are there that begin with a 0 , end with a 0 , contain no two consecutive 0 s , and contain no three consecutive 1s?

Question 1

Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy 60 square feet in each bedroom. How many square feet of walls must be painted?

Question 2

Define the operation \(\star\) by \(a\star b=(a+b)b\). What is \((3\star 5)-(5\star 3)\)?

Question 3

A college student drove his compact car 120 miles home for the weekend and averaged 30 miles per gallon. On the return trip the student drove his parents' SUV and averaged only 20 miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?

Question 4

The point \(O\) is the center of the circle circumscribed about \(\triangle ABC\), with \(\angle BOC=120^\circ\) and \(\angle AOB=140^\circ\), as shown. What is the degree measure of \(\angle ABC\)?

Question 5

In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?

Question 6

The 2007 AMC 10 will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the first 22 and leave only the last 3 unanswered. How many of the first 22 problems must she solve correctly in order to score at least 100 points?

Question 7

All sides of the convex pentagon \(ABCDE\) are of equal length, and \(\angle A=\angle B=90^\circ\). What is the degree measure of \(\angle E\)?

Question 8

On the trip home from the meeting where this AMC 10 was constructed, the Contest Chair noted that his airport parking receipt had digits of the form \(bbcac\), where \(0\le a<b<c\le 9\), and \(b\) was the average of \(a\) and \(c\). How many different five-digit numbers satisfy all these properties?

Question 9

A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is 1 place to its right in the alphabet, assuming that the letter \(A\) is one place to the right of the letter \(Z\). The second time this same letter appears in the given message, it is replaced by the letter that is \(1+2\) places to the right, the third time it is replaced by the letter that is \(1+2+3\) places to the right, and so on. For example, with this code the word ``banana'' becomes ``cbodqg''. What letter will replace the last letter \(s\) in the message ``Lee's sis is a Mississippi miss, Chriss!''?

Question 10

Two points \(B\) and \(C\) are in a plane. Let \(S\) be the set of all points \(A\) in the plane for which \(\triangle ABC\) has area 1. Which of the following describes \(S\)?

Question 11

A circle passes through the three vertices of an isosceles triangle that has two sides of length 3 and a base of length 2. What is the area of this circle?

Question 12

Tom's age is \(T\) years, which is also the sum of the ages of his three children. His age \(N\) years ago was twice the sum of their ages then. What is \(\frac{T}{N}\)?

Question 13

Two circles of radius 2 are centered at \((2,0)\) and at \((0,2)\). What is the area of the intersection of the interiors of the two circles?

Question 14

Some boys and girls are having a car wash to raise money for a class trip to China. Initially \(40%\) of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then \(30%\) of the group are girls. How many girls were initially in the group?

Question 15

The angles of quadrilateral \(ABCD\) satisfy \(\angle A=2\angle B=3\angle C=4\angle D\). What is the degree measure of \(\angle A\), rounded to the nearest whole number?

Question 16

A teacher gave a test to a class in which \(10%\) of the students are juniors and \(90%\) are seniors. The average score on the test was 84. The juniors all received the same score, and the average score of the seniors was 83. What score did each of the juniors receive on the test?

Question 17

Point \(P\) is inside equilateral \(\triangle ABC\). Points \(Q\), \(R\), and \(S\) are the feet of the perpendiculars from \(P\) to \(\overline{AB}\), \(\overline{BC}\), and \(\overline{CA}\), respectively. Given that \(PQ=1\), \(PR=2\), and \(PS=3\), what is \(AB\)?

Question 18

A circle of radius 1 is surrounded by 4 circles of radius \(r\) as shown. What is \(r\)?

Question 19

The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by 4, and the second number is divided by 5. The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square?

Question 20

A set of 25 square blocks is arranged into a \(5\times 5\) square. How many different combinations of 3 blocks can be selected from that set so that no two are in the same row or column?

Question 21

Right \(\triangle ABC\) has \(AB=3\), \(BC=4\), and \(AC=5\). Square \(XYZW\) is inscribed in \(\triangle ABC\) with \(X\) and \(Y\) on \(\overline{AC}\), \(W\) on \(\overline{AB}\), and \(Z\) on \(\overline{BC}\). What is the side length of the square?

Question 22

A player chooses one of the numbers 1 through 4. After the choice has been made, two regular four-sided tetrahedral dice are rolled, with the sides of the dice numbered 1 through 4. If the number chosen appears on the bottom of exactly one die after it is rolled, then the player wins \($1\). If the number chosen appears on the bottom of both of the dice, then the player wins \($2\). If the number chosen does not appear on the bottom of either of the dice, the player loses \($1\). What is the expected return to the player, in dollars, for one roll of the dice?

Question 23

A pyramid with a square base is cut by a plane that is parallel to its base and is 2 units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid?

Question 24

Let \(n\) denote the smallest positive integer that is divisible by both 4 and 9, and whose base-10 representation consists of only 4's and 9's, with at least one of each. What are the last four digits of \(n\)?

Question 25

How many pairs of positive integers \((a,b)\) are there such that \(\gcd(a,b)=1\) and \[ \frac{a}{b}+\frac{14b}{9a} \] is an integer?

Question 1

A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player?

Question 2

A \(4 \times 4\) block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?

Question 3

Assume that \(x\) is a positive real number. Which is equivalent to \(\sqrt[3]{x \sqrt{x}}\) ?

Question 4

A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least \($ 15,000\), and that the total of all players' salaries for each team cannot exceed \($ 700,000\). What is the maximum possiblle salary, in dollars, for a single player?

Question 5

For real numbers \(a\) and \(b\), define \(a $ b=(a-b)^{2}\). What is \((x-y)^{2} $(y-x)^{2}\) ?

Question 6

Points \(B\) and \(C\) lie on \(\overline{A D}\). The length of \(\overline{A B}\) is 4 times the length of \(\overline{B D}\), and the length of \(\overline{A C}\) is 9 times the length of \(\overline{C D}\). The length of \(\overline{B C}\) is what fraction of the length of \(\overline{A D}\) ?

Question 7

An equilateral triangle of side length 10 is completely filled in by non-overlapping equilateral triangles of side length 1 . How many small triangles are required?

Question 8

A class collects \($ 50\) to buy flowers for a classmate who is in the hospital. Roses cost \($ 3\) each, and carnations cost \($ 2\) each. No other flowers are to be used. How many different bouquets could be purchased for exactly \($ 50\) ?

Question 9

A quadratic equation \(a x^{2}-2 a x+b=0\) has two real solutions. What is the average of the solutions?

Question 10

Points \(A\) and \(B\) are on a circle of radius 5 and \(A B=6\). Point \(C\) is the midpoint of the minor \(\operatorname{arc} A B\). What is the length of the line segment \(A C\) ?

Question 11

Suppose that \(\left(u_{n}\right)\) is a sequence of real numbers satisfying \(u_{n+2}=2 u_{n+1}+u_{n}\), and that \(u_{3}=9\) and \(u_{6}=128\). What is \(u_{5}\) ?

Question 12

Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000 . During the year, the pedometer flips from 99999 to 00000 forty-four times. On December 31 the pedometer reads 50000 . Pete takes 1800 steps per mile. Which of the following is closest to the number of miles Pete walked during the year?

Question 13

For each positive integer \(n\), the mean of the first \(n\) terms of a sequence is \(n\). What is the 2008th term of the sequence?

Question 14

Triangle \(O A B\) has \(O=(0,0), B=(5,0)\), and \(A\) in the first quadrant. In addition, \(\angle A B O= 90^{\circ}\) and \(\angle A O B=30^{\circ}\). Suppose that \(\overline{O A}\) is rotated \(90^{\circ}\) counterclockwise about \(O\). What are the coordinates of the image of \(A\) ?

Question 15

How many right triangles have integer leg lengths \(a\) and \(b\) and a hypotenuse of length \(b+1\), where \(b<100\) ?

Question 16

Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, their sum is 0 .)

Question 17

A poll shows that \(70 %\) of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work?

Question 18

Bricklayer Brenda would take 9 hours to build a chimney alone, and bricklayer Brandon would take 10 hours to build it alone. When they work together they talk a lot, and their combined output is decreased by 10 bricks per hour. Working together, they build the chimney in 5 hours. How many bricks are in the chimney?

Question 19

A cylindrical tank with radius 4 feet and height 9 feet is lying on its side. The tank is filled with water to a depth of 2 feet. What is the volume of the water, in cubic feet?

Question 20

The faces of a cubical die are marked with the numbers \(1,2,2,3,3\), and 4 . The faces of a second cubical die are marked with the numbers \(1,3,4,5,6\), and 8 . Both dice are thrown. What is the probability that the sum of the two top numbers will be 5,7 , or 9 ?

Question 21

Ten chairs are evenly spaced around a round table and numbered clockwise from 1 through 10. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or directly across from his or her spouse. How many seating arrangements are possible?

Question 22

Three red beads, two white beads, and one blue bead are placed in a line in random order. What is the probability that no two neighboring beads are the same color?

Question 23

A rectangular floor measures \(a\) by \(b\) feet, where \(a\) and \(b\) are positive integers with \(b>a\). An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width 1 foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair \((a, b)\) ?

Question 24

Quadrilateral \(A B C D\) has \(A B=B C=C D, \angle A B C=70^{\circ}\), and \(\angle B C D=170^{\circ}\). What is the degree measure of \(\angle B A D\) ?

Question 25

Michael walks at the rate of 5 feet per second on a long straight path. Trash pails are located every 200 feet along the path. A garbage truck travels at 10 feet per second in the same direction as Michael and stops for 30 seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?

Question 1

Each morning of her five-day workweek, Jane bought either a 50 -cent muffin or a 75 -cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?

Question 2

Which of the following is equal to \(\frac{\frac{1}{3}-\frac{1}{4}}{\frac{1}{2}-\frac{1}{3}}\) ?

Question 3

Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell of her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms?

Question 4

A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. THe remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths 15 and 25 meters. What fraction of the yard is occupied by the flower beds?

Question 5

Twenty percent less than 60 is one-third more than what number?

Question 6

Kiana has two older twin brothers. The product of their ages is 128 . What is the sum of their three ages?

Question 7

By inserting parentheses, it is possible to give the expression \[2\times 3+4 \times 5 \] AMC 10 2009 several values. How many different values can be obtained?

Question 8

In a certain year the price of gasoline rose by \(20 %\) during January, fell by \(20 %\) during February, rose by \(25 %\) during March, and fell by \(x %\) during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is \(x\) ?

Question 9

Segment \(B D\) and \(A E\) intersect at \(C\), as shown, \(A B=B C=C D=C E\), and \(\angle A=\frac{5}{2} \angle B\). What is the degree measure of \(\angle D\) ?

Question 10

A flagpole is originally 5 meters tall. A hurricane snaps the flagpole at a point \(x\) meters above the ground so that the upper part, still attached to the stump, touches the ground 1 meter away from the base. What is \(x\) ?

Question 11

How many 7 digit palindromes (numbers that read the same backward as forward) can be formed using the digits \(2,2,3,3,5,5,5\) ?

Question 12

Distinct points \(A, B, C\), and \(D\) lie on a line, with \(A B=B C=C D=1\). Points \(E\) and \(F\) lie on a second line, parallel to the first, with \(E F=1\). A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?

Question 13

As shown below, convex pentagon \(A B C D E\) has sides \(A B=3, B C=4, C D=6, D E=3\), and \(E A=7\). The pentagon is originally positioned in the plane with vertex \(A\) at the origin and vertex \(B\) on the positive \(x\)-axis. The pentagon is then rolled clockwise to the right along the \(x\)-axis. Which side will touch the point \(x=2009\) on the \(x\)-axis? AMC 10 2009

Question 14

On Monday, Millie puts a quart of seeds, \(25 %\) of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only \(25 %\) of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?

Question 15

When a bucket is two-thirds full of water, the bucket and water weigh a kilograms. When the bucket is one-half full of water the total weight is \(b\) kilograms. In terms of \(a\) and \(b\), what is the total weight in kilograms when the bucket is full of water?

Question 16

Points \(A\) and \(C\) lie on a circle centered at \(O\), each of \(\overline{B A}\) and \(\overline{B C}\) are tangent to the circle, and \(\triangle A B C\) is equilateral. The circle intersects \(\overline{B O}\) at \(D\). What is \(\frac{B D}{B O}\) ?

Question 17

Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from \((a, 0)\) to \((3,3)\), divides the entire region into two regions of equal area. What is \(a\) ?

Question 18

Rectangle \(A B C D\) has \(A B=8\) and \(B C=6\). Point \(M\) is the midpoint of diagonal \(\overline{A C}\), and E is on \(\overline{A B}\) with \(\overline{M E} \perp \overline{A C}\). What is the area of \(\triangle A M E\) ?

Question 19

A particular 12-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a 1 , it mistakenly displays a 9 . For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?

Question 20

Triangle \(A B C\) has a right angle at \(B, A B=1\), and \(B C=2\). The bisector of \(\angle B A C\) meets \(\overline{B C}\) at \(D\). What is \(B D\) ?

Question 21

What is the remainder when \(3^{0}+3^{1}+3^{2}+\ldots+3^{2009}\) is divided by 8 ?

Question 22

A cubical cake with edge length 2 inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where \(M\) is the midpoint of a top edge. The piece whose top is triangle \(B\) contains \(c\) cubic inches of cake and \(s\) square inches of icing. What is \(c+s\) ?

Question 23

Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the start line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?

Question 24

The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with 9 trapezoids, let \(x\) be the angle measure in degrees of the larger interior angle of the trapezoid. What is \(x\) ?

Question 25

Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?

Question 1

What is \(100(100-3)-(100 \cdot 100-3)\) ?

Question 2

Makayla attended two meetings during her 9 -hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?

Question 3

A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?

Question 4

For a real number \(x\), define \(\varnothing(x)\) to be the average of \(x\) and \(x^{2}\). What is \(\varnothing(1)+\varnothing(2)+\varnothing(3)\) ?

Question 5

A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?

Question 6

A circle is centered at \(O, \overline{A B}\) is a diameter and \(C\) is a point on the circle with \(\angle C O B=50^{\circ}\). What is the degree measure of \(\angle C A B\) ?

Question 7

A triangle has side lengths 10,10 , and 12 . A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle?

Question 8

A ticket to a school play costs \(x\) dollars, where \(x\) is a whole number. A group of 9th graders buys tickets costing a total of \($ 48\), and a group of 10th graders buys tickets costing a total of \($ 64\). How many values of \(x\) are possible?

Question 9

Lucky Larry's teacher asked him to substitute numbers for \(a, b, c, d\), and \(e\) in the expression \(a-(b-(c-(d+e)))\) and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincedence. The numbers Larry AMC 10 2010 substituted for \(a, b, c\), and \(d\) were \(1,2,3\), and 4 , respectively. What number did Larry substitute for \(e\) ?

Question 10

Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?

Question 11

A shopper plans to purchase an item that has a listed price greater than \($ 100\) and can use any one of the three coupons. Coupon A gives \(15 %\) off the listed price, Coupon B gives \($ 30\) the listed price, and Coupon C gives \(25 %\) off the amount by which the listed price exceeds \($ 100\). Let \(x\) and \(y\) be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is \(y-x\) ?

Question 12

At the beginning of the school year, \(50 %\) of all students in Mr. Well's math class answered "Yes" to the question "Do you love math", and \(50 %\) answered "No." At the end of the school year, \(70 %\) answered "Yes" and \(30 %\) answered "No." Altogether, \(x %\) of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of \(x\) ?

Question 13

What is the sum of all the solutions of \(x=|2 x-|60-2 x||\) ?

Question 14

The average of the numbers \(1,2,3, \ldots, 98,99\), and \(x\) is \(100 x\). What is \(x\) ?

Question 15

On a 50 -question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and -1 point for an incorrect answer. Jesse's total score on the contest was 99. What is the maximum number of questions that Jesse could have answered correctly?

Question 16

A square of side length 1 and a circle of radius \(\sqrt{3} / 3\) share the same center. What is the area inside the circle, but outside the square?

Question 17

Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed 37th and 64th, respectively. How many schools are in the city?

Question 18

Positive integers \(a, b\), and \(c\) are randomly and independently selected with replacement from the set \(\{1,2,3, \ldots, 2010\}\). What is the probability that \(a b c+a b+a\) is divisible by 3 ?

Question 19

A circle with center \(O\) has area \(156 \pi\). Triangle \(A B C\) is equilateral, \(\overline{B C}\) is a chord on the circle, \(O A=4 \sqrt{3}\), and point \(O\) is outside \(\triangle A B C\). What is the side length of \(\triangle A B C\) ?

Question 20

Two circles lie outside regular hexagon \(A B C D E F\). The first is tangent to \(\overline{A B}\), and the second is tangent to \(\overline{D E}\). Both are tangent to lines \(B C\) and \(F A\). What is the ratio of the area of the second circle to that of the first circle?

Question 21

A palindrome between 1000 and 10,000 is chosen at random. What is the probability that it is divisible by 7 ?

Question 22

Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?

Question 23

The entries in a \(3 \times 3\) array include all the digits from 1 through 9 , arranged so that the entries in every row and column are in increasing order. How many such arrays are there?

Question 24

A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than 100 points. What was the total number of points scored by the two teams in the first half?

Question 25

Let \(a>0\), and let \(P(x)\) be a polynomial with integer coefficients such that \[ \begin{gathered} P(1)=P(3)=P(5)=P(7)=a, \text { and } P(2)=P(4)=P(6)=P(8)=-a \end{gathered} \] What is the smallest possible value of \(a\) ?

Question 1

What is \[ \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? \]

Question 2

Josanna's test scores to date are \(90,80,70,60,\) and 85. Her goal is to raise her test average at least 3 points with her next test. What is the minimum test score she would need to accomplish this goal?

Question 3

At a store, when a length is reported as \(x\) inches, that means the length is at least \(x-0.5\) inches and at most \(x+0.5\) inches. Suppose the dimensions of a rectangular tile are reported as 2 inches by 3 inches. In square inches, what is the minimum area for the rectangle?

Question 4

LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid \(A\) dollars and Bernardo had paid \(B\) dollars, where \(A<B\). How many dollars must LeRoy give to Bernardo so that they share the costs equally?

Question 5

In multiplying two positive integers \(a\) and \(b\), Ron reversed the digits of the two-digit number \(a\). His erroneous product was 161. What is the correct value of the product \(ab\)?

Question 6

On Halloween Casper ate \(\frac{1}{3}\) of his candies and then gave 2 candies to his brother. The next day he ate \(\frac{1}{3}\) of his remaining candies and then gave 4 candies to his sister. On the third day he ate his final 8 candies. How many candies did Casper have at the beginning?

Question 7

The sum of two angles of a triangle is \(\frac{6}{5}\) of a right angle, and one of these two angles is \(30^\circ\) larger than the other. What is the degree measure of the largest angle in the triangle?

Question 8

At a certain beach, if it is at least \(80^\circ\text{F}\) and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be said about the weather conditions on June 10?

Question 9

The area of \(\triangle EBD\) is one third of the area of the 3-4-5 triangle \(\triangle ABC\). Segment \(\overline{DE}\) is perpendicular to segment \(\overline{AB}\). What is \(BD\)?

Question 10

Consider the set of numbers \(\{1,10,10^2,10^3,\ldots,10^{10}\}\). The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?

Question 11

There are 52 people in a room. What is the largest value of \(n\) such that the statement ``At least \(n\) people in this room have birthdays falling in the same month'' is always true?

Question 12

Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width 6 meters, and it takes her 36 seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?

Question 13

Two real numbers are selected independently at random from the interval \([-20,10]\). What is the probability that the product of those numbers is greater than zero?

Question 14

A rectangular parking lot has a diagonal of 25 meters and an area of 168 square meters. In meters, what is the perimeter of the parking lot?

Question 15

Let \(\text{@}\) denote the ``averaged with'' operation: \[ a \text{@} b=\frac{a+b}{2}. \] Which of the following distributive laws hold for all numbers \(x,y,\) and \(z\)?

Question 16

A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?

Question 17

In the given circle, the diameter \(\overline{EB}\) is parallel to \(\overline{DC}\), and \(\overline{AB}\) is parallel to \(\overline{ED}\). The angles \(\angle AEB\) and \(\angle ABE\) are in the ratio \(4:5\). What is the degree measure of \(\angle BCD\)?

Question 18

Rectangle \(ABCD\) has \(AB=6\) and \(BC=3\). Point \(M\) is chosen on side \(AB\) so that \(\angle AMD=\angle CMD\). What is the degree measure of \(\angle AMD\)?

Question 19

What is the product of all the roots of the equation \[ \sqrt{5|x|+8}=\sqrt{x^2-16}? \]

Question 20

Rhombus \(ABCD\) has side length 2 and \(\angle B=120^\circ\). Region \(R\) consists of all points inside the rhombus that are closer to vertex \(B\) than any of the other three vertices. What is the area of \(R\)?

Question 21

Brian writes down four integers \(w>x>y>z\) whose sum is 44. The pairwise positive differences of these numbers are \(1,3,4,5,6,\) and 9. What is the sum of the possible values for \(w\)?

Question 22

A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?

Question 23

What is the hundreds digit of \(2011^{2011}\)?

Question 24

A lattice point in an \(xy\)-coordinate system is any point \((x,y)\) where both \(x\) and \(y\) are integers. The graph of \(y=mx+2\) passes through no lattice point with \(0<x\le 100\) for all \(m\) such that \(\frac{1}{2}<m<a\). What is the maximum possible value of \(a\)?

Question 25

Let \(T_1\) be a triangle with sides 2011, 2012, and 2013. For \(n\ge 1\), if \(T_n=\triangle ABC\) and \(D,E,\) and \(F\) are the points of tangency of the incircle of \(\triangle ABC\) to the sides \(AB\), \(BC\), and \(AC\), respectively, then \(T_{n+1}\) is a triangle with side lengths \(AD\), \(BE\), and \(CF\), if it exists. What is the perimeter of the last triangle in the sequence \((T_n)\)?

Question 1

Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?

Question 2

A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is \(2:1\). What is the area of the rectangle?

Question 3

The point in the \(xy\)-plane with coordinates \((1000,2012)\) is reflected across the line \(y=2000\). What are the coordinates of the reflected point?

Question 4

When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?

Question 5

Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is \(10%\). She leaves a \(15%\) tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of \($27.50\) for dinner. What is the cost of her dinner without tax or tip?

Question 6

In order to estimate the value of \(x-y\), where \(x\) and \(y\) are real numbers with \(x>y>0\), Xiaoli rounded \(x\) up by a small amount, rounded \(y\) down by the same amount, and then subtracted her values. Which of the following statements is necessarily correct?

Question 7

For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?

Question 8

What is the sum of all integer solutions to \(1<(x-2)^2<25\)?

Question 9

Two integers have a sum of 26. When two more integers are added to the first two integers, the sum is 41. Finally, when two more integers are added to the sum of the previous four integers, the sum is 57. What is the minimum number of even integers among the 6 integers?

Question 10

How many ordered pairs of positive integers \((M,N)\) satisfy the equation \(\frac{M}{6}=\frac{6}{N}\)?

Question 11

A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?

Question 12

Point \(B\) is due east of point \(A\). Point \(C\) is due north of point \(B\). The distance between points \(A\) and \(C\) is \(10\sqrt{2}\) meters, and \(\angle BAC=45^\circ\). Point \(D\) is 20 meters due north of point \(C\). The distance \(AD\) is between which two integers?

Question 13

It takes Clea 60 seconds to walk down an escalator when it is not operating and only 24 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it?

Question 14

Two equilateral triangles are contained in a square whose side length is \(2\sqrt{3}\). The bases of these triangles are opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus?

Question 15

In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament?

Question 16

Three circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?

Question 17

Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the smaller having a central angle of \(120^\circ\). He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?

Question 18

Suppose that one of every 500 people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a \(2%\) false positive rate. In other words, for such people, \(98%\) of the time the test will turn out negative, but \(2%\) of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let \(p\) be the probability that a person who is chosen at random from the population and gets a positive test result actually has the disease. Which of the following is closest to \(p\)?

Question 19

In rectangle \(ABCD\), \(AB=6\), \(AD=30\), and \(G\) is the midpoint of \(\overline{AD}\). Segment \(\overline{AB}\) is extended 2 units beyond \(B\) to point \(E\), and \(F\) is the intersection of \(\overline{ED}\) and \(\overline{BC}\). What is the area of \(BFDG\)?

Question 20

Bernardo and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernardo. The winner is the last person who produces a number less than 1000. Let \(N\) be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of \(N\)?

Question 21

Four distinct points are arranged in a plane so that the segments connecting them have lengths \(a,a,a,a,2a,\) and \(b\). What is the ratio of \(b\) to \(a\)?

Question 22

Let \((a_1,a_2,\ldots,a_{10})\) be a list of the first 10 positive integers such that for each \(2\le i\le 10\), either \(a_i+1\) or \(a_i-1\) or both appear somewhere before \(a_i\) in the list. How many such lists are there?

Question 23

A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?

Question 24

Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?

Question 25

A bug travels from \(A\) to \(B\) along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?

Question 1

What is \[ \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? \]

Question 2

Mr.\ Green measures his rectangular garden by walking two of the sides and finds that it is 15 steps by 20 steps. Each of Mr.\ Green's steps is 2 feet long. Mr.\ Green expects half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr.\ Green expect from his garden?

Question 3

On a particular January day, the high temperature in Lincoln, Nebraska, was 16 degrees higher than the low temperature, and the average of the high and low temperatures was \(3^\circ\). In degrees, what was the low temperature in Lincoln that day?

Question 4

When counting from 3 to 201, 53 is the \(51^\text{st}\) number counted. When counting backwards from 201 to 3, 53 is the \(n^\text{th}\) number counted. What is \(n\)?

Question 5

Positive integers \(a\) and \(b\) are each less than 6. What is the smallest possible value for \(2a-ab\)?

Question 6

The average age of 33 fifth-graders is 11. The average age of 55 of their parents is 33. What is the average age of all of these parents and fifth-graders?

Question 7

Six points are equally spaced around a circle of radius 1. Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle?

Question 8

Ray's car averages 40 miles per gallon of gasoline, and Tom's car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?

Question 9

Three positive integers are each greater than 1, have a product of 27000, and are pairwise relatively prime. What is their sum?

Question 10

A basketball team's players were successful on \(50%\) of their two-point shots and \(40%\) of their three-point shots, which resulted in 54 points. They attempted \(50%\) more two-point shots than three-point shots. How many three-point shots did they attempt?

Question 11

Real numbers \(x\) and \(y\) satisfy the equation \[ x^{2}+y^{2}=10x-6y-34. \] What is \(x+y\)?

Question 12

Let \(S\) be the set of sides and diagonals of a regular pentagon. A pair of elements of \(S\) are selected at random without replacement. What is the probability that the two chosen segments have the same length?

Question 13

Jo and Blair take turns counting from 1 to one more than the last number said by the other person. Jo starts by saying "1,'' so Blair follows by saying "1, 2.'' Jo then says "1, 2, 3,'' and so on. What is the 53rd number said?

Question 14

Define \[ a \star b = a^{2}b-ab^{2}. \] Which of the following describes the set of points \((x,y)\) for which \(x \star y = y \star x\)?

Question 15

A wire is cut into two pieces, one of length \(a\) and the other of length \(b\). The piece of length \(a\) is bent to form an equilateral triangle, and the piece of length \(b\) is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is \(\frac{a}{b}\)?

Question 16

In \(\triangle ABC\), medians \(\overline{AD}\) and \(\overline{CE}\) intersect at \(P\), \(PE=1.5\), \(PD=2\), and \(DE=2.5\). What is the area of quadrilateral \(AEDC\)?

Question 17

Alex has 75 red tokens and 75 blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?

Question 18

The number 2013 has the property that its units digit is the sum of its other digits, that is, \(2+0+1=3\). How many integers less than 2013 but greater than 1000 share this property?

Question 19

The real numbers \(c,b,a\) form an arithmetic sequence with \(a\ge b\ge c\ge 0\). The quadratic \(ax^{2}+bx+c\) has exactly one root. What is this root?

Question 20

The number 2013 is expressed in the form \[ 2013=\frac{a_{1}!a_{2}!\cdots a_{m}!}{b_{1}!b_{2}!\cdots b_{n}!}, \] where \(a_{1}\ge a_{2}\ge \cdots \ge a_{m}\) and \(b_{1}\ge b_{2}\ge \cdots \ge b_{n}\) are positive integers and \(a_{1}+b_{1}\) is as small as possible. What is \(|a_{1}-b_{1}|\)?

Question 21

Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is \(N\). What is the smallest possible value of \(N\)?

Question 22

The regular octagon \(ABCDEFGH\) has its center at \(J\). Each of the vertices and the center are to be associated with one of the digits 1 through 9, with each digit used once, in such a way that the sums of the numbers on the lines \(AJE\), \(BJF\), \(CJG\), and \(DJH\) are equal. In how many ways can this be done?

Question 23

In triangle \(ABC\), \(AB=13\), \(BC=14\), and \(CA=15\). Distinct points \(D\), \(E\), and \(F\) lie on segments \(\overline{BC}\), \(\overline{CA}\), and \(\overline{DE}\), respectively, such that \(\overline{AD}\perp \overline{BC}\), \(\overline{DE}\perp \overline{AC}\), and \(\overline{AF}\perp \overline{BF}\). The length of segment \(\overline{DF}\) can be written as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. What is \(m+n\)?

Question 24

A positive integer \(n\) is nice if there is a positive integer \(m\) with exactly four positive divisors, including 1 and \(m\), such that the sum of the four divisors is equal to \(n\). How many numbers in the set \(\{2010,2011,2012,\ldots,2019\}\) are nice?

Question 25

Bernardo chooses a three-digit positive integer \(N\) and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer \(S\). For example, if \(N=749\), Bernardo writes the numbers 10444 and 3245, and LeRoy obtains the sum \(S=13689\). For how many choices of \(N\) are the two rightmost digits of \(S\), in order, the same as those of \(2N\)?

Question 1

Leah has 13 coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth?

Question 2

What is \(\frac{2^{3}+2^{3}}{2^{-3}+2^{-3}}\)?

Question 3

Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip?

Question 4

Susie pays for 4 muffins and 3 bananas. Calvin spends twice as much as Susie paying for 2 muffins and 16 bananas. A muffin is how many times as expensive as a banana?

Question 5

Doug constructs a square window using 8 equal-size panes of glass, as shown. The ratio of the height to width for each pane is \(5:2\), and the borders around and between the panes are 2 inches wide. In inches, what is the side length of the square window?

Question 6

Orvin went to the store with just enough money to buy 30 balloons. When he arrived, he discovered that the store had a special sale on balloons: buy 1 balloon at the regular price and get a second at \(\frac{1}{3}\) off the regular price. What is the greatest number of balloons Orvin could buy?

Question 7

Suppose \(A>B>0\) and \(A\) is \(x%\) greater than \(B\). What is \(x\)?

Question 8

A truck travels \(\frac{b}{6}\) feet every \(t\) seconds. There are 3 feet in a yard. How many yards does the truck travel in 3 minutes?

Question 9

For real numbers \(w\) and \(z\), \[ \frac{\frac{1}{w}+\frac{1}{z}}{\frac{1}{w}-\frac{1}{z}}=2014. \] What is \(\frac{w+z}{w-z}\)?

Question 10

In the addition shown below, \(A,B,C,\) and \(D\) are distinct digits. How many different values are possible for \(D\)? \[ \begin{array}{r} ABBCB +B C A D A DBDDD \end{array} \]

Question 11

For the consumer, a single discount of \(n%\) is more advantageous than any of the following discounts:

(1)  two successive \(15%\) discounts  (2)  three successive \(10%\) discounts  (3)  a \(25%\) discount followed by a \(5%\) discount. What is the smallest possible positive integer value of \(n\)?

Question 12

The largest divisor of \(2,014,000,000\) is itself. What is its fifth largest divisor?

Question 13

Six regular hexagons surround a regular hexagon of side length 1 as shown. What is the area of \(\triangle ABC\)?

Question 14

Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, \(abc\) miles were displayed on the odometer, where \(abc\) is a 3-digit number with \(a\ge 1\) and \(a+b+c\le 7\). At the end of the trip, the odometer showed \(cba\) miles. What is \(a^{2}+b^{2}+c^{2}\)?

Question 15

In rectangle \(ABCD\), \(DC=2CB\) and points \(E\) and \(F\) lie on \(\overline{AB}\) so that \(\overline{ED}\) and \(\overline{FD}\) trisect \(\angle ADC\) as shown. What is the ratio of the area of \(\triangle DEF\) to the area of rectangle \(ABCD\)?

Question 16

Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?

Question 17

What is the greatest power of 2 that is a factor of \(10^{1002}-4^{501}\)?

Question 18

A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in the list?

Question 19

Two concentric circles have radii 1 and 2. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?

Question 20

For how many integers is the number \(x^{4}-51x^{2}+50\) negative?

Question 21

Trapezoid \(ABCD\) has parallel sides \(\overline{AB}\) of length 33 and \(\overline{CD}\) of length 21. The other two sides are of lengths 10 and 14. The angles at \(A\) and \(B\) are acute. What is the length of the shorter diagonal of \(ABCD\)?

Question 22

Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?

Question 23

A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?

Question 24

The numbers \(1,2,3,4,5\) are to be arranged in a circle. An arrangement is bad if it is not true that for every \(n\) from 1 to 15 one can find a subset of the numbers that appear consecutively on the circle that sum to \(n\). Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?

Question 25

In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad \(N\), \(0<N<10\), it will jump to pad \(N-1\) with probability \(\frac{N}{10}\) and to pad \(N+1\) with probability \(1-\frac{N}{10}\). Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake?

Question 1

What is the value of $$ \frac{2 a^{-1}+\frac{a^{-1}}{2}}{a} $$ when $a=\frac{1}{2}$ ?

Question 2

If $n \circlearrowleft m=n^{3} m^{2}$, what is $\frac{294}{49^{2}}$ ?

Question 3

Let $x=-2016$. What is the value of $|||x|-x|-|x||-x$ ?

Question 4

Zoey read 15 books, one at a time. The first book took her 1 day to read, the second book took her 2 days to read, the third book took her 3 days to read, and so on, with each book taking her 1 more day to read than the previous book. Zoey finished the first book on a Monday and the second on a Wednesday. On what day of the week did she finish her 15th book?

Question 5

The mean age of Amanda's 4 cousins is 8 , and their median age is 5 . What is the sum of the ages of Amanda's youngest and oldest cousins?

Question 6

Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$ ?

Question 7

The ratio of the measures of two acute angles is $5: 4$, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?

Question 8

What is the tens digit of $2015^{2016}-2017$ ?

Question 9

All three vertices of $\triangle A B C$ lie on the parabola defined by $y=x^{2}$, with $A$ at the origin and $\overline{B C}$ parallel to the $x$-axis. The area of the triangle is 64 . What is the length $B C$ ?

Question 10

A thin piece of wood of uniform density in the shape of an equilateral triangle with side length 3 inches weighs 12 ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length 5 inches. Which of the following is closest to the weight, in ounces, of the second piece?

Question 11

Carl decided to fence in his rectangular garden. He bought 20 fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly 4 yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl's garden?

Question 12

Two different numbers are selected at random from $\{1,2,3,4,5\}$ and multiplied together. What is the probability that the product is even?

Question 13

At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for 1000 of the babies born. There were four times as many sets of triplets as sets of quadruplets, and three times as many sets of twins as sets of triplets. How many of these 1000 babies were in sets of quadruplets?

Question 14

How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$, and the line $x=5.1$ ?

Question 15

All the numbers $1,2,3,4,5,6,7,8,9$ are written in a $3 \times 3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to 18 . What number is in the center?

Question 16

The sum of an infinite geometric series is a positive number $S$, and the second term in the series is 1 . What is the smallest possible value of $S$ ?

Question 17

All the numbers $2,3,4,5,6,7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?

Question 18

In how many ways can 345 be written as the sum of an increasing sequence of two or more consecutive positive integers?

Question 19

Rectangle $A B C D$ has $A B=5$ and $B C=4$. Point $E$ lies on $\overline{A B}$ so that $E B=1$, point $G$ lies on $\overline{B C}$ so that $C G=1$, and point $F$ lies on $\overline{C D}$ so that $D F=2$. Segments $\overline{A G}$ and $\overline{A C}$ intersect $\overline{E F}$ at $Q$ and $P$, respectively. What is the value of $\frac{P Q}{E F}$ ?

Question 20

A dilation of the plane-that is, a size transformation with a positive scale factor-sends the circle of radius 2 centered at $A(2,2)$ to the circle of radius 3 centered at $A^{\prime}(5,6)$. What distance does the origin $O(0,0)$ move under this transformation?

Question 21

What is the area of the region enclosed by the graph of the equation $x^{2}+y^{2}= |x|+|y| ?$

Question 22

A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won 10 games and lost 10 games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B, B$ beat $C$, and $C$ beat $A$ ?

Question 23

In regular hexagon $A B C D E F$, points $W, X, Y$, and $Z$ are chosen on sides $\overline{B C}$, $\overline{C D}, \overline{E F}$, and $\overline{F A}$, respectively, so that lines $A B, Z W, Y X$, and $E D$ are parallel and equally spaced. What is the ratio of the area of hexagon $W C X Y F Z$ to the area of hexagon $A B C D E F$ ?

Question 24

How many four-digit positive integers $a b c d$, with $a \neq 0$, have the property that the three two-digit integers $a b<b c<c d$ form an increasing arithmetic sequence? One such number is 4692 , where $a=4, b=6, c=9$, and $d=2$.

Question 25

Let $f(x)=\sum_{k=2}^{10}(\lfloor k x\rfloor-k\lfloor x\rfloor)$, where $\lfloor r\rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \geq 0$ ?