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 $x \geq 0$ ?

Question 1

What is the value of \(2-(-2)^{-2}\)?

Question 2

Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?

Question 3

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?

Question 4

Four siblings ordered an extra large pizza. Alex ate \(\frac{1}{5}\), Beth \(\frac{1}{3}\), and Cyril \(\frac{1}{4}\) of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?

Question 5

David, Hikmet, Jack, Marta, Rand, and Todd were in a 12-person race with 6 other people. Rand finished 6 places ahead of Hikmet. Marta finished 1 place behind Jack. David finished 2 places behind Hikmet. Jack finished 2 places behind Todd. Todd finished 1 place behind Rand. Marta finished in 6th place. Who finished in 8th place?

Question 6

Marley practices exactly one sport each day of the week. She runs three days a week but never on two consecutive days. On Monday she plays basketball and two days later golf. She swims and plays tennis, but she never plays tennis the day after running or swimming. Which day of the week does Marley swim?

Question 7

Consider the operation "minus the reciprocal of,'' defined by \(a \diamond b=a-\frac{1}{b}\). What is \(((1 \diamond 2) \diamond 3)-(1 \diamond (2 \diamond 3))\)?

Question 8

The letter F shown below is rotated \(90^\circ\) clockwise around the origin, then reflected in the \(y\)-axis, and then rotated a half turn around the origin. What is the final image?

Question 9

The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius 3 and center \((0,0)\) that lies in the first quadrant, the portion of the circle with radius \(\frac{3}{2}\) and center \(\left(0,\frac{3}{2}\right)\) that lies in the first quadrant, and the line segment from \((0,0)\) to \((3,0)\). What is the area of the shark's fin falcata?

Question 10

What are the sign and units digit of the product of all the odd negative integers strictly greater than \(-2015\)?

Question 11

Among the positive integers less than 100, each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime?

Question 12

For how many integers \(x\) is the point \((x,-x)\) inside or on the circle of radius 10 centered at \((5,5)\)?

Question 13

The line \(12x+5y=60\) forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?

Question 14

Let \(a\), \(b\), and \(c\) be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation \[ (x-a)(x-b)+(x-b)(x-c)=0? \]

Question 15

The town of Hamlet has 3 people for each horse, 4 sheep for each cow, and 3 ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?

Question 16

Al, Bill, and Cal will each randomly be assigned a whole number from 1 to 10, inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's?

Question 17

The centers of the faces of the right rectangular prism shown below are joined to create an octahedron. What is the volume of the octahedron?

Question 18

Johann has 64 fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads?

Question 19

In \(\triangle ABC\), \(\angle C=90^\circ\) and \(AB=12\). Squares \(ABXY\) and \(ACWZ\) are constructed outside of the triangle. The points \(X\), \(Y\), \(Z\), and \(W\) lie on a circle. What is the perimeter of the triangle?

Question 20

Erin the ant starts at a given corner of a cube and crawls along exactly 7 edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?

Question 21

Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose that Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let \(s\) denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of \(s\)?

Question 22

In the figure shown below, \(ABCDE\) is a regular pentagon and \(AG=1\). What is \(FG+JH+CD\)?

Question 23

Let \(n\) be a positive integer greater than 4 such that the decimal representation of \(n!\) ends in \(k\) zeros and the decimal representation of \((2n)!\) ends in \(3k\) zeros. Let \(s\) denote the sum of the four least possible values of \(n\). What is the sum of the digits of \(s\)?

Question 24

Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin \(p_0=(0,0)\) facing to the east and walks one unit, arriving at \(p_1=(1,0)\). For \(n=1,2,3,\ldots\), right after arriving at the point \(p_n\), if Aaron can turn \(90^\circ\) left and walk one unit to an unvisited point \(p_{n+1}\), he does that. Otherwise, he walks one unit straight ahead to reach \(p_{n+1}\). Thus the sequence of points continues \(p_2=(1,1)\), \(p_3=(0,1)\), \(p_4=(-1,1)\), \(p_5=(-1,0)\), and so on in a counterclockwise spiral pattern. What is \(p_{2015}\)?

Question 25

A rectangular box measures \(a\times b\times c\), where \(a\), \(b\), and \(c\) are integers and \(1\le a\le b\le c\). The volume and surface area of the box are numerically equal. How many ordered triples \((a,b,c)\) are possible?

Question 1

Mary thought of a positive two-digit number. She multiplied it by 3 and added 11. Then she switched the digits of the result, obtaining a number between 71 and 75, inclusive. What was Mary's number?

Question 2

Sofia ran 5 laps around the 400 -meter track at her school. For each lap, she ran the first 100 meters at an average speed of 4 meters per second and the remaining 300 meters at an average speed of 5 meters per second. How much time did Sofia take running the 5 laps?

Question 3

Real numbers \(x, y\), and \(z\) satisfy the inequalities \(0<x<1,-1<y<0\), and \(1<z<2\). Which of the following numbers is necessarily positive?

Question 4

Supposed that \(x\) and \(y\) are nonzero real numbers such that \[ \frac{3 x+y}{x-3 y}=-2 . \] What is the value of \[ \frac{x+3 y}{3 x-y} ? \]

Question 5

Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have?

Question 6

What is the largest number of solid \(2-\mathrm{in}\) by \(2-\mathrm{in}\) by \(1-\mathrm{in}\) blocks that can fit in a \(3-\mathrm{in}\) by \(2-\) in by 3 -in box?

Question 7

Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all it took her 44 minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?

Question 8

Points \(A(11,9)\) and \(B(2,-3)\) are vertices of \(\triangle A B C\) with \(A B=A C\). The altitude from \(A\) meets the opposite side at \(D(-1,3)\). What are the coordinates of point \(C\) ?

Question 9

A radio program has a quiz consisting of 3 multiple-choice questions, each with 3 choices. A contestant wins if he or she gets 2 or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?

Question 10

The lines with equations \(a x-2 y=c\) and \(2 x+b y=-c\) are perpendicular and intersect at \((1,-5)\). What is \(c\) ?

Question 11

At Typico High School, \(60 %\) of the students like dancing, and the rest dislike it. Of those who like dancing, \(80 %\) say that they like it, and the rest say that they dislike it. Of those who dislike dancing, \(90 %\) say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?

Question 12

Elmer's new car gives \(50 %\) percent better fuel efficiency. However, the new car uses diesel fuel, which is \(20 %\) more expensive per liter than the gasoline the old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip?

Question 13

There are 20 students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are 10 students taking yoga, 13 taking bridge, and 9 taking painting. There are 9 students taking at least two classes. How many students are taking all three classes?

Question 14

An integer \(N\) is selected at random in the range \(1 \leq N \leq 2020\). What is the probability that the remainder when \(N^{16}\) is divided by 5 is 1 ?

Question 15

Rectangle \(A B C D\) has \(A B=3\) and \(B C=4\). Point \(E\) is the foot of the perpendicular from \(B\) to diagonal \(\overline{A C}\). What is the area of \(\triangle A D E\) ?

Question 16

How many of the base-ten numerals for the positive integers less than or equal to 2017 contain the digit 0 ?

Question 17

Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3,23578 , and 987620 are monotonous, but 88,7434 , and 23557 are not. How many monotonous positive integers are there?

Question 18

In the figure below, 3 of the 6 disks are to be painted blue, 2 are to be painted red, and 1 is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?

Question 19

Let \(A B C\) be an equilateral triangle. Extend side \(\overline{A B}\) beyond \(B\) to a point \(B^{\prime}\) so that \(B B^{\prime}=3 A B\). Similarly, extend side \(\overline{B C}\) beyond \(C\) to a point \(C^{\prime}\) so that \(C C^{\prime}=3 B C\), and extend side \(\overline{C A}\) beyond \(A\) to a point \(A^{\prime}\) so that \(A A^{\prime}=3 C A\). What is the ratio of the area of \(\triangle A^{\prime} B^{\prime} C^{\prime}\) to the area of \(\triangle A B C\) ?

Question 20

The number \(21!=51,090,942,171,709,440,000\) has over 60,000 positive integer divisors. One of them is chosen at random. What is the probability that it is odd?

Question 21

In \(\triangle A B C, A B=6, A C=8, B C=10\), and \(D\) is the midpoint of \(\overline{B C}\). What is the sum of the radii of the circles inscribed in \(\triangle A D B\) and \(\triangle A D C\) ?

Question 22

The diameter \(A B\) of a circle of radius 2 is extended to a point \(D\) outside the circle so that \(B D=3\). Point \(E\) is chosen so that \(E D=5\) and line \(E D\) is perpendicular to line \(A D\). Segment \(A E\) intersects the circle at a point \(C\) between \(A\) and \(E\). What is the area of \(\triangle A B C\) ?

Question 23

Let \(N=123456789101112 \cdots 4344\) be the 79 -digit number that is formed by writing the integers from 1 to 44 in order, one after the other. What is the remainder when \(N\) is divided by 45 ?

Question 24

The vertices of an equilateral triangle lie on the hyperbola \(x y=1\), and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?

Question 25

Last year Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95 . What was her score on the sixth test?

Question 1

Kate bakes 20 -inch by 18 -inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?

Question 2

Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph . What was his average speed, in mph, during the last 30 minutes?

Question 3

In the expression ( ____ × ____ ) + ( ____ × ____ ) each blank is to be filled in with one of the digits $1,2,3$, or 4 , with each digit being used once. How many different values can be obtained?

Question 4

A three-dimensional rectangular box with dimensions $X, Y$, and $Z$ has faces whose surface areas are $24,24,48,48,72$, and 72 square units. What is $X+Y+Z$ ?

Question 5

How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?

Question 6

A box contains 5 chips, numbered $1,2,3,4$, and 5 . Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds 4 . What is the probability that 3 draws are required?

Question 7

In the figure below, $N$ congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the small semicircles. The ratio $A: B$ is $1: 18$. What is $N$ ?

Question 8

Sara makes a staircase out of toothpicks as shown:

This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?

Question 9

The faces of each of 7 standard dice are labeled with the integers from 1 to 6 . Let $p$ be the probability that when all 7 dice are rolled, the sum of the numbers on the top faces is 10 . What other sum occurs with the same probability $p$ ?

Question 10

In the rectangular parallelepiped shown, $A B=3, B C=1$, and $C G=2$. Point $M$ is the midpoint of $\overline{F G}$. What is the volume of the rectangular pyramid with base $B C H E$ and apex $M$ ?

Question 11

Which of the following expressions is never a prime number when $p$ is a prime number?

Question 12

Line segment $\overline{A B}$ is a diameter of a circle with $A B=24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle A B C$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?

Question 13

How many of the first 2018 numbers in the sequence $101,1001,10001,100001, \ldots$ are divisible by 101 ?

Question 14

A list of 2018 positive integers has a unique mode, which occurs exactly 10 times. What is the least number of distinct values that can occur in the list?

Question 15

A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$. What is the area of the sheet of wrapping paper?

Question 16

Let $a_{1}, a_{2}, \ldots, a_{2018}$ be a strictly increasing sequence of positive integers such that $a_{1}+a_{2}+\cdots+a_{2018}=2018^{2018}$. What is the remainder when $a_{1}^{3}+a_{2}^{3}+\cdots+a_{2018}^{3}$ is divided by 6 ?

Question 17

In rectangle $P Q R S, P Q=8$ and $Q R=6$. Points $A$ and $B$ lie on $\overline{P Q}$, points $C$ and $D$ lie on $\overline{Q R}$, points $E$ and $F$ lie on $\overline{R S}$, and points $G$ and $H$ lie on $\overline{S P}$ so that $A P=B Q<4$ and the convex octagon $A B C D E F G H$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m \sqrt{n}$, where $k, m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$ ?

Question 18

Three young brother-sister pairs from different families need tot take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?

Question 19

Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?

Question 20

A function $f$ is defined recursively by $f(1)=f(2)=1 \quad$ and $f(n)=f(n-1)-f(n-2)+n$ for all integers $n \geq 3$. What is $f(2018)$ ?

Question 21

Mary chose an even 4-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2, \ldots, \frac{n}{2}, n$. At some moment Mary wrote 323 as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of 323 .

Question 22

Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[0,1]$. Which of the following numbers is closest to the probability that $x, y$, and 1 are the side lengths of an obtuse triangle?

Question 23

How many ordered pairs $(a, b)$ of positive integers satisfy the equation $a \cdot b+63=20 \cdot \operatorname{lcm}(a, b)+12 \cdot \operatorname{gcd}(a, b)$,where $\operatorname{gcd}(a, b)$ denotes the greatest common divisor of $a$ and $b$, and $\operatorname{lcm}(a, b)$ denotes their least common multiple?

Question 24

Let $A B C D E F$ be a regular hexagon with side length 1 . Denote $X, Y$, and $Z$ the midpoints of sides $\overline{A B}, \overline{C D}$, and $\overline{E F}$, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle A C E$ and $\triangle X Y Z$ ?

Question 25

How many $x$ satisfy the equation $x^{2}+10,000\lfloor x\rfloor=10,000 x_{\text {? }}$

Question 1

What is the value of $$ 1-(-2)-3-(-4)-5-(-6) ? $$

Question 2

Carl has 5 cubes each having side length 1 , and Kate has 5 cubes each having side length 2 . What is the total volume of these 10 cubes?

Question 3

The ratio of $w$ to $x$ is $4: 3$, the ratio of $y$ to $z$ is $3: 2$, and the ratio of $z$ to $x$ is $1: 6$. What is the ratio of $w$ to $y$ ?

Question 4

The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$ ?

Question 5

How many distinguishable arrangements are there of 1 brown tile, 1 purple tile, 2 green tiles, and 3 yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.)

Question 6

Driving along a highway, Megan noticed that her odometer showed 15951 (miles). This number is a palindrome-it reads the same forward and backward. Then 2 hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this 2 -hour period?

Question 7

How many positive even multiples of 3 less than 2020 are perfect squares?

Question 8

Points $P$ and $Q$ lie in a plane with $P Q=8$. How many locations for point $R$ in this plane are there such that the triangle with vertices $P$, $Q$, and $R$ is a right triangle with area 12 square units?

Question 9

How many ordered pairs of integers $(x, y)$ satisfy the equation $$ x^{2020}+y^{2}=2 y ? $$

Question 10

A three-quarter sector of a circle of radius 4 inches together with its interior can be rolled up to form the lateral surface of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?

Question 11

Ms. Carr asks her students to read any 5 of the 10 books on a reading list. Harold randomly selects 5 books from this list, and Betty does the same. What is the probability that there are exactly 2 books that they both select?

Question 12

The decimal representation of $$ \frac{1}{20^{20}} $$ consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How many zeros are in that initial string of zeros after the decimal point?

Question 13

Andy the Ant lives on a coordinate plane and is currently at ( $-20,20$ ) facing east (that is, in the positive $x$-direction). Andy moves 1 unit and then turns $90^{\circ}$ left. From there, Andy moves 2 units (north) and then turns $90^{\circ}$ left. He then moves 3 units (west) and again turns $90^{\circ}$ left. Andy continues this process, increasing his distance each time by 1 unit and always turning left. What is the location of the point at which Andy makes the 2020th left turn?

Question 14

As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region-inside the hexagon but outside all of the semicircles?

Question 15

Steve wrote the digits $1,2,3,4$, and 5 in order repeatedly from left to right, forming a list of 10,000 digits, beginning $123451234512 \ldots$. He then erased every third digit from his list (that is, the 3rd, 6th, 9th, ... digits from the left), then erased every fourth digit from the resulting list (that is, the 4th, 8th, 12th, ... digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in positions 2019, 2020, and 2021 ?

Question 16

Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than 4 . They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?

Question 17

There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other?

Question 18

An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?

Question 19

In a certain card game, a player is dealt a hand of 10 cards from a deck of 52 distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158 A 00 A 4 A A 0$. What is the digit $A$ ?

Question 20

Let $B$ be a right rectangular prism (box) with edge lengths 1,3 , and 4 , together with its interior. For real $r \geq 0$, let $S(r)$ be the set of points in 3-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $a r^{3}+b r^{2}+c r+d$, where $a$, $b, c$, and $d$ are positive real numbers. What is $\frac{b c}{a d}$ ?

Question 21

In square $A B C D$, points $E$ and $H$ lie on $\overline{A B}$ and $\overline{D A}$, respectively, so that $A E=A H$. Points $F$ and $G$ lie on $\overline{B C}$ and $\overline{C D}$, respectively, and points $I$ and $J$ lie on $\overline{E H}$ so that $\overline{F I} \perp \overline{E H}$ and $\overline{G J} \perp \overline{E H}$. See the figure below. Triangle $A E H$, quadrilateral $B F I E$, quadrilateral $D H J G$, and pentagon $F C G J I$ each has area 1 . What is $F I^{2}$ ?

Question 22

What is the remainder when $2^{202}+202$ is divided by $2^{101}+2^{51}+1$ ?

Question 23

Square $A B C D$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1)$, and $D(1,-1)$. Consider the following four transformations:

• $L$, a rotation of $90^{\circ}$ counterclockwise around the origin;
• R, a rotation of $90^{\circ}$ clockwise around the origin;
• $H$, a reflection across the $x$-axis; and
• $V$, a reflection across the $y$-axis.
  Each of these transformations maps the square onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of 20 transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of 4 transformations that will send the vertices back to their original positions.)
  (a) $2^{37}$
  (b) $3 \cdot 2^{36}$
  (c) $2^{38}$
  (d) $3 \cdot 2^{37}$
  (e) $2^{39}$

Question 24

How many positive integers $n$ satisfy $$ \frac{n+1000}{70}=\lfloor\sqrt{n}\rfloor ? $$ (Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)

Question 25

Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product $$ n=f_{1} \cdot f_{2} \cdots f_{k} \text {, } $$ where $k \geq 1$, the $f_{i}$ are integers strictly greater than 1 , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number 6 can be written as $6,2 \cdot 3$, and $3 \cdot 2$, so $D(6)=3$. What is $D(96)$ ?

Question 1

How many integer values of $x$ satisfy $|x|<3 \pi$ ?

Question 2

What is the value of $\sqrt{(3-2 \sqrt{3})^{2}}+\sqrt{(3+2 \sqrt{3})^{2}}$ ?

Question 3

In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. Among the 28 students in the program, $25 %$ of the juniors and $10 %$ of the seniors are on the debate team. How many juniors are in the program?

Question 4

At a math contest, 57 students are wearing blue shirts, and another 75 students are wearing yellow shirts. The 132 students are assigned into 66 pairs. In exactly 23 of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?

Question 5

The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give 24 , while the other two multiply to 30 . What is the sum of the ages of Jonie's four cousins?

Question 6

Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is 84 , and the afternoon class's mean score is 70 . The ratio of the number of students in the morning class to the number of students in the afternoon class is $\frac{3}{4}$. What is the mean of the scores of all the students?

Question 7

In a plane, four circles with radii $1,3,5$, and 7 are tangent to line $\ell$ at the same point $A$, but they may be on either side of $\ell$. Region $S$ consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region $S$ ?

Question 8

Mr. Zhou places all the integers from 1 to 225 into a 15 by 15 grid. He places 1 in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row from the top?

Question 9

The point $P(a, b)$ in the $x y$-plane is first rotated counterclockwise by $90^{\circ}$ around the point ( 1,5 ) and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b-a$ ?

Question 10

An inverted cone with base radius 12 cm and height 18 cm is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of 24 cm . What is the height in centimeters of the water in the cylinder?

Question 11

Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce?

Question 12

Let $N=34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$ ?

Question 13

Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $\underline{32 d}$ in base $n$ equals 263 , and the value of the numeral $\underline{324}$ in base $n$ equals the value of the numeral $\underline{11 d 1}$ in base six. What is $n+d$ ?

Question 14

Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38,38 , and 34 . What is the distance between two adjacent parallel lines?

Question 15

The real number $x$ satisfies the equation $x+\frac{1}{x}=\sqrt{5}$. What is the value of $x^{11}-7 x^{7}+x^{3}$ ?

Question 16

Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, 1357, 89 , and 5 are all uphill integers, but 32,1240 , and 466 are not. How many uphill integers are divisible by 15 ?

Question 17

Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given 2 cards out of a set of 10 cards numbered $1,2,3, \ldots, 10$. The score of a player is the sum of the numbers of their cards. The scores of the players are as follows: Ravon- 11 , Oscar- 4 , Aditi- 7 , Tyrone- 16 , Kim- 17 . Which of the following statements is true?

Question 18

A fair 6 -sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?

Question 19

Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is 32 . If the least integer in $S$ is also removed, then the average value of the integers remaining is 35 . If the greatest integer is then returned to the set, the average value of the integers rises to 40 . The greatest integer in the original set $S$ is 72 greater than the least integer in $S$. What is the average value of all the integers in the set $S$ ?

Question 20

The figure is constructed from 11 line segments, each of which has length 2 . The area of pentagon $A B C D E$ can be written as $\sqrt{m}+\sqrt{n}$, where $m$ and $n$ are positive integers. What is $m+n$ ?

Question 21

A square piece of paper has side length 1 and vertices $A, B, C$, and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\overline{A D}$ at point $C^{\prime}$, and edge $\overline{A B}$ at point $E$. Suppose that $C^{\prime} D=\frac{1}{3}$. What is the perimeter of triangle $\triangle A E C^{\prime}$ ?

Question 22

Ang, Ben, and Jasmin each have 5 blocks, colored red, blue, yellow, white, and green; and there are 5 empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives 3 blocks all of the same color is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?

Question 23

A square with side length 8 is colored white except for 4 black isosceles right triangular regions with legs of length 2 in each corner of the square and a black diamond with side length $2 \sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter 1 is dropped onto the square and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}(a+b \sqrt{2}+\pi)$, where $a$ and $b$ are positive integers. What is $a+b$ ?

Question 24

Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes 4 and 2 can be changed into any of the following by one move: $(3,2),(2,1,2),(4),(4,1),(2,2)$, or $(1,1,2)$.

Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?

Question 25

Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between 1 and 30 , inclusive. Exactly 300 points in $S$ lie on or below a line with equation $y=m x$. The possible values of $m$ lie in an interval of length $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is $a+b$ ?

Question 1

What is the value of $1234+2341+3412+4123$ ?

Question 2

What is the area of the shaded figure shown below?

Question 3

The expression $\frac{2021}{2020}-\frac{2020}{2021}$ is equal to the fraction $\frac{p}{q}$ in which $p$ and $q$ are positive integers whose greatest common divisor is 1 . What is $p$ ?

Question 4

At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At 4:00 the temperature in Minneapolis has fallen by 5 degrees while the temperature in St. Louis has risen by 3 degrees, at which time the temperatures in the two cities differ by 2 degrees. What is the product of all possible values of $N$ ?

Question 5

Let $n=8^{2022}$. Which of the following is equal to $\frac{n}{4}$ ?

Question 6

The least positive integer with exactly 2021 distinct positive divisors can be written in the form $m \cdot 6^{k}$, where $m$ and $k$ are integers and 6 is not a divisor of $m$. What is $m+k$ ?

Question 7

Call a fraction $\frac{a}{b}$, not necessarily in simplest form, special if $a$ and $b$ are positive integers whose sum is 15 . How many distinct integers can be written as the sum of two, not necessarily different, special fractions?

Question 8

The greatest prime number that is a divisor of 16,384 is 2 because $16,384= 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of 16,383 ?

Question 9

The knights in a certain kingdom come in two colors: $\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is 2 times the fraction of blue knights who are magical. What fraction of red knights are magical?

Question 10

Forty slips of paper numbered 1 to 40 are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by 100 and add my number, the result is a perfect square." What is the sum of the two numbers drawn from the hat?

Question 11

A regular hexagon of side length 1 is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these 6 reflected arcs?

Question 12

Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation $$ x(x-y)+y(y-z)+z(z-x)=1 ? $$

Question 13

A square with side length 3 is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length 2 has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle?

Question 14

Una rolls 6 standard 6 -sided dice simultaneously and calculates the product of the 6 numbers obtained. What is the probability that the product is divisible by 4 ?

Question 15

In square $A B C D$, points $P$ and $Q$ lie on $\overline{A D}$ and $\overline{A B}$, respectively. Segments $\overline{B P}$ and $\overline{C Q}$ intersect at right angles at $R$, with $B R=6$ and $P R=7$. What is the area of the square?

Question 16

Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions?

Question 17

Distinct lines $\ell$ and $m$ lie in the $x y$-plane. They intersect at the origin. Point $P(-1,4)$ is reflected about line $\ell$ to point $P^{\prime}$, and then $P^{\prime}$ is reflected about line $m$ to point $P^{\prime \prime}$. The equation of line $\ell$ is $5 x-y=0$, and the coordinates of $P^{\prime \prime}$ are $(4,1)$. What is the equation of line $m$ ?

Question 18

Three identical square sheets of paper each with side length 6 are stacked on top of each other. The middle sheet is rotated clockwise $30^{\circ}$ about its center and the top sheet is rotated clockwise $60^{\circ}$ about its center, resulting in the 24 -sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b \sqrt{c}$, where $a, b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c$ ?

Question 19

Let $N$ be the positive integer $7777 \ldots 777$, a 313-digit number where each digit is a 7 . Let $f(r)$ be the leading digit of the $r$ th root of $N$. What is $f(2)+f(3)+f(4)+f(5)+f(6)$ ?

Question 20

In a particular game, each of 4 players rolls a standard 6 -sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a 5 , given that he won the game?

Question 21

Regular polygons with 5, 6, 7, and 8 sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?

Question 22

For each integer $n \geq 2$, let $S_{n}$ be the sum of all products $j k$, where $j$ and $k$ are integers and $1 \leq j<k \leq n$. What is the sum of the 10 least values of $n$ such that $S_{n}$ is divisible by 3 ?

Question 23

Each of the 5 sides and the 5 diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color?

Question 24

A cube is constructed from 4 white unit cubes and 4 blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)

Question 25

A rectangle with side lengths 1 and 3 , a square with side length 1 , and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?

MAA Partner Organizations

We acknowledge the generosity of the following organizations in supporting the MAA AMC and Invitational Competitions:

AMC 10 B

DO NOT OPEN until Tuesday, November 16, 2021

${ }^{* *}$ Administration on an earlier date will disqualify your school's results.**

Question 26

All the information needed to administer this competition is contained in the AMC 10/12 Teacher's Manual. PLEASE READ THE MANUAL BEFORE TUESDAY, NOVEMBER 16, 2021.

Question 27

Answer sheets must be returned to the MAA AMC office within 24 hours of the competition administration. Use an overnight or 2-day shipping service, with a tracking number, to guarantee timely arrival of these answer sheets. FedEx, UPS, or USPS overnight are strongly recommended.

Question 28

The 40th annual American Invitational Mathematics Exam will be held on Tuesday, February 8, 2022, with an alternate date on Wednesday, February 16, 2022. It is a 15 -question, 3 -hour, integer-answer competition. Students who achieve a high score on the AMC 10 will be invited to participate. Top-scoring students on the AMC 10/12 and AIME will be selected to take the USA (Junior) Mathematical Olympiad.

Question 29

The publication, reproduction, or communication of the problems or solutions of this competition during the period when students are eligible to participate seriously jeopardizes the integrity of the results. Dissemination via phone, email, or digital media of any type during this period is a violation of the competition rules.

Question 1

Define $x \diamond y$ to be $|x-y|$ for all real numbers $x$ and $y$. What is the value of $$ (1 \diamond(2 \diamond 3))-((1 \diamond 2) \diamond 3) ? $$

Question 2

In rhombus $A B C D$, point $P$ lies on segment $\overline{A D}$ so that $\overline{B P} \perp \overline{A D}, A P=3$, and $P D=2$. What is the area of $A B C D$ ? (Note: The figure is not drawn to scale.)

Question 3

How many three-digit positive integers have an odd number of even digits?

Question 4

A donkey suffers an attack of hiccups and the first hiccup happens at 4:00 one afternoon. Suppose that the donkey hiccups regularly every 5 seconds. At what time does the donkey's 700th hiccup occur?

Question 5

What is the value of $$ \frac{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}{\sqrt{\left(1-\frac{1}{3^{2}}\right)\left(1-\frac{1}{5^{2}}\right)\left(1-\frac{1}{7^{2}}\right)}} ? $$

Question 6

How many of the first ten numbers of the sequence $121,11211,1112111, \ldots$ are prime numbers?

Question 7

For how many values of the constant $k$ will the polynomial $x^{2}+k x+36$ have two distinct integer roots?

Question 8

Consider the following 100 sets of 10 elements each: $$ \begin{aligned} & \{1,2,3, \ldots, 10\}, & \{11,12,13, \ldots, 20\}, & \{21,22,23, \ldots, 30\}, & \vdots & \{991,992,993, \ldots, 1000\} . \end{aligned} $$ How many of these sets contain exactly two multiples of 7 ?

Question 9

The sum $$ \frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{2021}{2022!} $$ can be expressed as $a-\frac{1}{b!}$, where $a$ and $b$ are positive integers. What is $a+b$ ?

Question 10

Camila writes down five positive integers. The unique mode of these integers is 2 greater than their median, and the median is 2 greater than their arithmetic mean. What is the least possible value for the mode?

Question 11

All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement "No school bigger than Euclid HS sold more T-shirts than Euclid HS"?

Question 12

A pair of fair 6 -sided dice is rolled $n$ times. What is the least value of $n$ such that the probability that the sum of the numbers face up on a roll equals 7 at least once is greater than $\frac{1}{2}$ ?

Question 13

The positive difference between a pair of primes is equal to 2 , and the positive difference between the cubes of the two primes is equal to 31106 . What is the sum of the digits of the least prime that is greater than those two primes?

Question 14

Suppose that $S$ is a subset of $\{1,2,3, \ldots, 25\}$ such that the sum of any two (not necessarily distinct) elements of $S$ is never an element of $S$. What is the maximum number of elements $S$ may contain?

Question 15

Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence that has a common difference of 2 . The quotient $\frac{S_{3 n}}{S_{n}}$ does not depend on $n$. What is $S_{20}$ ?

Question 16

The diagram below shows a rectangle with side lengths 4 and 8 and a square with side length 5 . Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?

Question 17

One of the following numbers is not divisible by any prime number less than 10 . Which is it?

Question 18

Consider systems of three linear equations with unknowns $x, y$, and $z$, $$ \begin{aligned} & a_{1} x+b_{1} y+c_{1} z=0 & a_{2} x+b_{2} y+c_{2} z=0 & a_{3} x+b_{3} y+c_{3} z=0 \end{aligned} $$ where each of the coefficients is either 0 or 1 and the system has a solution other than $x=y=z=0$. For example, one such system is $\langle 1 x+1 y+0 z=0,0 x+1 y+1 z=0,0 x+0 y+0 z=0\rangle$ with a nonzero solution of $(x, y, z)=(1,-1,1)$. How many such systems of equations are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.)

Question 19

Each square in a $5 \times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:

• Any filled square with two or three filled neighbors remains filled.
• Any empty square with exactly three filled neighbors becomes a filled square.
• All other squares remain empty or become empty.

  A sample transformation is shown in the figure below.

Suppose the $5 \times 5$ grid has a border of empty squares surrounding a $3 \times 3$ subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)

Question 20

Let $A B C D$ be a rhombus with $\angle A D C=46^{\circ}$. Let $E$ be the midpoint of $\overline{C D}$, and let $F$ be the point on $\overline{B E}$ such that $\overline{A F}$ is perpendicular to $\overline{B E}$. What is the degree measure of $\angle B F C$ ?

Question 21

Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^{2}+x+1$, the remainder is $x+2$, and when $P(x)$ is divided by the polynomial $x^{2}+1$, the remainder is $2 x+1$. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?

Question 22

Let $S$ be the set of circles that are tangent to each of the three circles in the coordinate plane whose equations are $x^{2}+y^{2}=4, x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all the circles in $S$ ?

Question 23

Ant Amelia starts on the number line at 0 and crawls in the following manner. For $n=1,2,3$, Amelia chooses a time duration $t_{n}$ and an increment $x_{n}$ independently and uniformly at random from the interval $(0,1)$. During the $n$th step of the process, Amelia moves $x_{n}$ units in the positive direction, using up $t_{n}$ minutes. If the total elapsed time has exceeded 1 minute during the $n$th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most 3 steps in all. What is the probability that Amelia's position when she stops will be greater than 1 ?

Question 24

Consider functions $f$ that satisfy $|f(x)-f(y)| \leq \frac{1}{2}|x-y|$ for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300)=f(900)$, what is the greatest possible value of $$ f(f(800))-f(f(400)) ? $$

Question 25

Let $x_{0}, x_{1}, x_{2}, \ldots$ be a sequence of numbers, where each $x_{k}$ is either 0 or 1 . For each positive integer $n$, define $$ S_{n}=\sum_{k=0}^{n-1} x_{k} 2^{k} $$ Suppose $7 S_{n} \equiv 1\left(\bmod 2^{n}\right)$ for all $n \geq 1$. What is the value of the sum $$ x_{2019}+2 x_{2020}+4 x_{2021}+8 x_{2022} ? $$

Question 1

Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely full but runs out of juice when the fourth glass is only $\frac{1}{3}$ full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?

Question 2

Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20 %$ on every pair of shoes. Carlos also knew that he had to pay a $7.5 %$ sales tax on the discounted price. He had $43. What is the original (before discount) price of the most expensive shoes he could afford to buy?

Question 3

A 3-4-5 right triangle is inscribed in circle $A$, and a 5-12-13 right triangle is inscribed in circle $B$. What is the ratio of the area of circle $A$ to the area of circle $B$ ?

Question 4

Jackson's paintbrush makes a narrow strip with a width of 6.5 millimeters. Jackson has enough paint to make a strip 25 meters long. How many square centimeters of paper could Jackson cover with paint?

Question 5

Maddy and Lara see a list of numbers written on a blackboard. Maddy adds 3 to each number in the list and finds that the sum of her new numbers is 45 . Lara multiplies each number in the list by 3 and finds that the sum of her new numbers is also 45 . How many numbers are written on the blackboard?

Question 6

Let $L_{1}=1, L_{2}=3$, and $L_{n+2}=L_{n+1}+L_{n}$ for $n \geq 1$. How many terms in the sequence $L_{1}, L_{2}, L_{3}, \cdots, L_{2023}$ are even?

Question 7

Square $A B C D$ is rotated $20^{\circ}$ clockwise about its center to obtain square $E F G H$, as shown below. What is the degree measure of $\angle E A B$ ?

Question 8

What is the units digit of $$ 2022^{2023}+2023^{2022} ? $$

Question 9

The numbers 16 and 25 are a pair of consecutive positive perfect squares whose difference is 9 . How many pairs of consecutive positive perfect squares have a difference of less than or equal to 2023 ?

Question 10

You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensue that at least one of your guessed squares is covered by the rectangle?

Question 11

Suzanne went to the bank and withdrew 800. The teller gave her this amount using 20 bills, 50 bills, and 100 bills, with at least one of each denomination. How many different collections of bills could Suzanne have received?

Question 12

When the roots of the polynomial $$ P(x)=\prod_{i=1}^{10}(x-i)^{i} $$ are removed from the real number line, what remains is the union of 11 disjoint open intervals. On how many of those intervals is $P(x)$ positive?

Question 13

What is the area of the region in the coordinate plane defined by the inequality $$ ||x|-1|+||y|-1| \leq 1 ? $$

Question 14

How many ordered pairs of integers ( $m, n$ ) satisfy the equation $$ m^{2}+m n+n^{2}=m^{2} n^{2} ? $$

Question 15

What is the least positive integer $m$ such that $m \cdot 2!\cdot 3!\cdot 4!\cdot 5!\cdots 16!$ is a perfect square?

Question 16

Define an upno to be a positive integer of 2 or more digits where the digits are strictly increasing moving left to right. Similarly, define a downno to be a positive integer of 2 or more digits where the digits are strictly decreasing moving left to right. For instance, the number 258 is an upno and 8620 is a downno. Let $U$ equal the total number of upnos and let $d$ equal the total number of downnos. What is $|U-D|$ ?

Question 17

A rectangular box $\mathcal{P}$ has distinct edge lengths $a, b$, and $c$. The sum of the lengths of all 12 edges of $\mathcal{P}$ is 13 , the sum of the areas of all 6 faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$ ?

Question 18

Suppose $a, b$, and $c$ are positive integers such that $$ \frac{a}{14}+\frac{b}{15}=\frac{c}{210} . $$ Which of the following statements are necessarily true?

Question 19

Sonya the frog chooses a point uniformly at random lying within the square $[0,6] \times[0,6]$ in the coordinate plane and hops to that point. She then randomly chooses a distance uniformly at random from $[0,1]$ and a direction uniformly at random from \{north, south east, west\}. All he choices are independent. She now hops the distance in the chosen direction. What is the probability that she lands outside the square?

Question 20

Four congruent semicircles are drawn on the surface of a sphere with radius 2 , as shown, creating a close curve that divides the surface into two congruent regions. The length of the curve is $\pi \sqrt{n}\}$. What is $n$ ?

Question 21

Each of 2023 balls is placed in on of 3 bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?

Question 22

How many distinct values of $x$ satisfy $$ [x]^{2}-3 x+2=0 $$ where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$ ?

Question 23

An arithmetic sequence has $n \geq 3$ terms, initial term $a$ and common difference $d>$ 1. Carl wrote down all the terms in this sequence correctly except for one term which was off by 1 . The sum of the terms was 222 . What was $a+d+n$ ?

Question 24

What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2 u-3 w, v+4 w)$ with $0 \leq u \leq 1,0 \leq v \leq 1$, and $0 \leq w \leq$ 1?

Question 25

A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?

Question 1

In a long line of people arranged left to right, the $1013^{\text {th }}$ person from the left is also the $1010^{\text {th }}$ person from the right. How many people are in the line?

Question 2

What is $10!-7!\cdot 6!$ ?

Question 3

For how many integer values of $x$ is $|2 x| \leq 7 \pi$ ?

Question 4

Balls numbered $1,2,3, \ldots$ are deposited in 5 bins, labeled $A, B, C, D$, and $E$, using the following procedure. Ball 1 is deposited in bin $A$, and balls 2 and 3 are deposited in bin $B$. The next 3 balls are deposited in bin $C$, the next 4 in bin $D$, and so on, cycling back to bin $A$ after balls are deposited in bin $E$. (For example, balls numbered $22,23, \ldots, 28$ are deposited in bin $B$ at step 7 of this process.) In which bin is ball 2024 deposited?

Question 5

In the following expression, Melanie changed some of the plus signs to minus signs: $$ 1+3+5+7+\cdots+97+99 $$ When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?

Question 6

A rectangle has integer length sides and an area of 2024 . What is the least possible perimeter of the rectangle?

Question 7

What is the remainder when $7^{2024}+7^{2025}+7^{2026}$ is divided by 19 ?

Question 8

Let $N$ be the product of all the positive integer divisors of 42 . What is the units digit of $N$ ?

Question 9

Real numbers $a, b$, and $c$ have arithmetic mean 0 . The arithmetic mean of $a^{2}, b^{2}$, and $c^{2}$ is 10 . What is the arithmetic mean of $a b, a c$, and $b c$ ?

Question 10

Quadrilateral $A B C D$ is a parallelogram, and $E$ is the midpoint of the side $\overline{A D}$. Let $F$ be the intersection of lines $E B$ and $A C$. What is the ratio of the area of quadrilateral $C D E F$ to the area of triangle $C F B$ ?

Question 11

In the figure below, $W X Y Z$ is a rectangle with $W X=4$ and $W Z=8$. Point $M$ lies on $\overline{X Y}$, point $A$ lies on $\overline{Y Z}$, and $\angle W M A$ is a right angle. The areas of triangles $\triangle W X M$ and $\triangle W A Z$ are equal. What is the area of $\triangle W M A$ ?

Question 12

A group of 100 students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students $A$ and $B$, student $A$ speaks some language that student $B$ does not speak, and student $B$ speaks some language that student $A$ does not speak. What is the least possible total number of languages spoken by all the students?

Question 13

Positive integers $x$ and $y$ satisfy the equation $\sqrt{x}+\sqrt{y}=\sqrt{1183}$. What is the minimum possible value of $x+y$ ?

Question 14

A dartboard is the region $B$ in the coordinate plane consisting of points ( $x, y$ ) such that $|x|+|y| \leq 8$. A target $T$ is the region where $\left(x^{2}+y^{2}-25\right)^{2} \leq 49$. A dart is thrown and lands at a random point in $B$. The probability that the dart lands in $T$ can be expressed as $\frac{m}{n} \cdot \pi$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?

Question 15

A list of 9 real numbers consists of $1,2.2,3.2,5.2,6.2$, and 7 , as well as $x, y$, and $z$ with $x \leq y \leq z$. The range of the list is 7 , and the mean and the median are both positive integers. How many ordered triples $(x, y, z)$ are possible?

Question 16

Jerry likes to play with numbers. One day, he wrote all the integers from 1 to 2024 on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them by either their sum or their product. (For example, Jerry's first step might have been to erase $1,2,3$, and 5 , and then write either 11 , their sum, or 30 , their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the whiteboard were odd. What is the maximum possible number of integers on the whiteboard at that time?

Question 17

In a race among 5 snails, there is at most one tie, but that tie can involve any number of snails. For example, the result of the race might be that Dazzler is first; Abby, Cyrus, and Elroy are tied for second; and Bruna is fifth. How many different results of the race are possible?

Question 18

How many different remainders can result when the $100^{\text {th }}$ power of an integer is divided by 125 ?

Question 19

In the following table, each question mark is to be replaced by "Possible" or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points (points both of whose coordinates are integers). How many of the 12 entries will be "Possible"?

Question 20

Three different pairs of shoes are placed in a row so that no left shoe is next to a right shoe from a different pair. In how many ways can these six shoes be lined up?

Question 21

Two straight pipes (circular cylinders), with radii 1 and $\frac{1}{4}$, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?

Question 22

A group of 16 people will be partitioned into 4 indistinguishable 4 -person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r} M$, where $r$ and $M$ are positive integers and $M$ is not divisible by 3 . What is $r$ ?

Question 23

The Fibonacci numbers are defined by $F_{1}=1, F_{2}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 3$. What is $$ \frac{F_{2}}{F_{1}}+\frac{F_{4}}{F_{2}}+\frac{F_{6}}{F_{3}}+\cdots+\frac{F_{20}}{F_{10}} ? $$

Question 24

How many of the values $P(2022), P(2023), P(2024)$, and $P(2025)$ are integers?

Question 25

Each of 27 bricks (right rectangular prisms) has dimensions $a \times b \times c$, where $a, b$, and $c$ are pairwise relatively prime positive integers. These bricks are arranged to form a $3 \times 3 \times 3$ block, as shown on the left below. A $28^{\text {th }}$ brick with the same dimensions is introduced, and these bricks are reconfigured into a $2 \times 2 \times 7$ block, shown on the right. The new block is 1 unit taller, 1 unit wider, and 1 unit deeper than the old one. What is $a+b+c$ ?