Find the number of positive integer palindromes written in base 10, with no zero digits, and whose digits add up to 13. For example, 42124 has these properties. Recall that a palindrome is a number whose representation reads the same from left to right as from right to left.
A hemisphere with radius 200 sits on top of a horizontal circular disk with radius 200, and the hemisphere and disk have the same center. Let T be the region of points P in the disk such that a sphere of radius 42 can be placed on top of the disk at P and lie completely inside the hemisphere. The area of T divided by the area of the disk is qp, where p and q are relatively prime positive integers. Find p+q.
A plane contains points A and B with AB=1. Point A is rotated in the plane counterclockwise through an acute angle θ around point B to point A′. Then B is rotated in the plane clockwise through angle θ around point A′ to point B′. Suppose AB′=34. The value of cosθ can be written as nm, where m and n are relatively prime positive integers. Find m+n.
Joanne has a blank fair six-sided die and six stickers each displaying a different integer from 1 to 6. Joanne rolls the die and then places the sticker labeled 1 on the top face of the die. She then rolls the die again, places the sticker labeled 2 on the top face, and continues this process to place the rest of the stickers in order. If the die ever lands with a sticker already on its top face, the new sticker is placed to cover the old sticker.
Let p be the conditional probability that at the end of the process exactly one face has been left blank, given that all the even-numbered stickers are visible on faces of the die. Then p can be written as nm, where m and n are relatively prime positive integers. Find m+n.
Let △ABC have side lengths AB=13, BC=14, and CA=15. Triangle A′B′C′ is obtained by rotating △ABC about its circumcenter so that A′C′ is perpendicular to BC, with A′ and B not on the same side of line B′C′. Find the integer closest to the area of hexagon AA′CC′BB′.
The integers from 1 to 64 are placed in some order into an 8×8 grid of cells with one number in each cell. Let ai,j be the number placed in the cell in row i and column j, and let M be the sum of the absolute differences between adjacent cells. That is, M=i=1∑8j=1∑7(∣ai,j+1−ai,j∣+∣aj+1,i−aj,i∣). Find the remainder when the maximum possible value of M is divided by 1000.
Triangle △ABC lies in plane D with AB=6, AC=4, and ∠BAC=90∘. Let D be the reflection across BC of the centroid of △ABC. For four spheres, all on the same side of P, have radii 1,2,3, and r and are tangent to D at points A,B,C, and D, respectively. The four spheres are also each tangent to a second plane T and are all on the same side of T. The value of r can be written as nm, where m and n are relatively prime positive integers. Find m+n.
For each nonnegative integer r less than 502, define Sr=m≥0∑(502m+r10000), where (n10000) is defined to be 0 when n>10,000. That is, Sr is the sum of all binomial coefficients of the form (k10000) for which 0≤k≤10,000 and k−r is a multiple of 502. Find the number of integers in the list S0,S1,S2,…,S501 that are multiples of the prime number 503.
In an equiangular pentagon, the sum of the squares of the side lengths equals 308, and the sum of the squares of the diagonal lengths equals 800. The square of the perimeter of the pentagon can be expressed as mn, where m and n are positive integers and n is not divisible by the square of any prime. Find m+n.
Let a, b, and n be positive integers with both a and b greater than or equal to 2 and less than or equal to 2n. Define an a×b cell loop in a 2n×2n grid of cells to be the 2a+2b−4 cells that surround an (a−2)×(b−2) (possibly empty) rectangle of cells in the grid. For example, the following diagram shows a way to partition a 6×6 grid of cells into 4 cell loops.
Find the number of ways to partition a 10×10 grid of cells into 5 cell loops so that every cell of the grid belongs to exactly one cell loop.
The figure below shows a grid of 10 squares in a row. Each square has a diagonal connecting its lower left vertex to its upper right vertex. A bug moves along the line segments from vertex to vertex, never traversing the same segment twice and never moving from right to left along a horizontal or diagonal segment. Let N be the number of paths the bug can take from the lower left corner (A) to the upper right corner (B). One such path from A to B is shown by the thick line segments in the figure. Find N.
Let ABCDE be a nonconvex pentagon with internal angles ∠A=∠E=90∘ and ∠B=∠D=45∘. Suppose that DE<AB, AE=20, BC=142, and points B, C, and D lie on the same side of line AE. Suppose further that AB is an integer with AB<2026 and the area of pentagon ABCDE is an integer multiple of 16. Find the number of possible values of AB.
For each positive integer n let f(n) be the value of the base-ten numeral n viewed in base b, where b is the least integer greater than the greatest digit in n. For example, if n=72, then b=8, and 72 as a numeral in base 8 equals 7⋅8+2=58; therefore f(72)=58. Find the number of positive integers n less than 1000 such that f(n)=n.
An urn contains n marbles. Each marble is either red or blue, and there are at least 7 marbles of each color. When 7 marbles are drawn randomly from the urn without replacement, the probability that exactly 4 of them are red equals the probability that exactly 5 of them are red. Find the sum of the five least values of n for which this is possible.
Find the sum of all real numbers r such that there is at least one point where the circle with radius r centered at (4,39) is tangent to the parabola with equation 2y=x2−8x+12.
Isosceles triangle △ABC has AB=BC. Let I be the incenter of △ABC. The perimeters of △ABC and △AIC are in the ratio 125:6, and all the sides of both triangles have integer lengths. Find the minimum possible value of AB.
Let △ABC be a triangle with D on BC such that AD bisects ∠BAC. Let ω be the circle that passes through A and is tangent to segment BC at D. Let E=A and F=A be the intersections of ω with segments AB and AC, respectively. Suppose that AB=200, AC=225, and all of AE,AF,BD, and CD are positive integers. Find the greatest possible value of BC.
Find the greatest integer n such that the cubic polynomial x3−6nx2+(n−11)x−400 has roots α2,β2, and γ2, where α,β, and γ are complex numbers, and there are exactly seven different possible values for α+β+γ.
Consider a tetrahedron with two isosceles triangle faces with side lengths 510,510,10 and two isosceles triangle faces with side lengths 510,510,18. The four vertices of the tetrahedron lie on a sphere with center S, and the four faces of the tetrahedron are tangent to a sphere with center R. The distance RS can be written as nm, where m and n are relatively prime positive integers. Find m+n.
- S and T have the same number of elements, - S and T are disjoint, and - the elements of S can be paired with the elements of T so that the elements in each pair differ by exactly 1.
For example, {1,2,5} and {0,3,4} are cousins. Suppose that the set S has exactly 4040 cousins. Find the least number of elements the set S can have.
For integers a and b, let a∘b=a−b if a is odd and b is even, and a+b otherwise. Find the number of sequences a1,a2,a3,…,an of positive integers such that a1+a2+a3+⋯+an=12 and a1∘a2∘a3∘⋯∘an=0 where the operations are performed from left to right; that is, a1∘a2∘a3 means (a1∘a2)∘a3.
A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form T=aL+bG, where a and b are constants, T is the time in minutes, L is the length of the trail in miles, and G is the altitude gain in feet. The model estimates that it will take 69 minutes to hike to the top if a trail is 1.5 miles long and ascends 800 feet, as well as if a trail is 1.2 miles long and ascends 1100 feet. How many minutes does the model estimate it will take to hike to the top if the trail is 4.2 miles long and ascends 4000 feet?
The number 2024 is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
What is the minimum number of successive swaps of adjacent letters in the string ABCDEF that are needed to change the string to FEDCBA? (For example, 3 swaps are required to change ABC to CBA; one such sequence of swaps is ABC→BAC→BCA→CBA.)
Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at 1:00 PM and were able to pack 4, 3, and 3 packages, respectively, every 3 minutes. At some later time, Daria joined the group, and Daria was able to pack 5 packages every 4 minutes. Together, they finished packing 450 packages at exactly 2:45 PM. At what time did Daria join the group?
Consider the following operation. Given a positive integer n, if n is a multiple of 3, then you replace n by 3n. If n is not a multiple of 3, then you replace n by n+10. For example, beginning with n=4, this procedure gives 4→14→24→8→18→6→2→12→⋯. Suppose you start with n=100. What value results if you perform this operation exactly 100 times?
Zelda played the Adventures of Math game on August 1 and scored 1700 points. She continued to play daily over the next 5 days. The bar chart below shows the daily change in her score compared to the day before. (For example, Zelda's score on August 2 was 1700+80=1780 points.) What was Zelda's average score in points over the 6 days?
Two transformations are said to commute if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane:
• a translation 2 units to the right,
• a 90∘-rotation counterclockwise about the origin,
• a reflection across the x-axis, and
• a dilation centered at the origin with scale factor 2.
Of the 6 pairs of distinct transformations from this list, how many commute?
One side of an equilateral triangle of height 24 lies on line ℓ. A circle of radius 12 is tangent to line ℓ and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line ℓ can be written as ab−cπ, where a, b, and c are positive integers and b is not divisible by the square of any prime. What is a+b+c?
All of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of the rectangle. What is length AB?
Two teams are in a best-two-out-of-three playoff: the teams will play at most 3 games, and the winner of the playoff is the first team to win 2 games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a 32 chance of winning at home, and its probability of winning when playing away from home is p. Outcomes of the games are independent. The probability that Team A wins the playoff is 21. Then p can be written in the form 21(m−n), where m and n are positive integers. What is m+n?
There are exactly K positive integers 5≤b≤2024 such that the base-b integer 2024b is divisible by 16 (where 16 is in base ten). What is the sum of the digits of K?
The first three terms of a geometric sequence are the integers a, 720, and b, where a<720<b. What is the sum of the digits of the least possible value of b?
The numbers, in order, of each row and the numbers, in order, of each column of a 5×5 array of integers form an arithmetic progression of length 5. The numbers in positions (5,5), (2,4), (4,3) and (3,1) are 0, 48, 16, and 12, respectively. What number is in position (1,2)?
Let K be the kite formed by joining two right triangles with legs 1 and 3 along a common hypotenuse. Eight copies of K are used to form the polygon shown below. What is the area of triangle △ABC?
A bee is moving in three-dimensional space. A fair six-sided die with faces labeled A+, A−, B+, B−, C+, and C− is rolled. Suppose the bee occupies the point (a,b,c). If the die shows A+, then the bee moves to the point (a+1,b,c) and if the die shows A−, then the bee moves to the point (a−1,b,c). Analogous moves are made with the other four outcomes. Suppose the bee starts at the point (0,0,0) and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube?
The figure below shows a dotted grid 8 cells wide and 3 cells tall consisting of 1′′×1′′ squares. Carl places 1-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks?
A data set containing 20 numbers, some of which are 6, has mean 45. When all the 6s are removed, the data set has mean 66. How many 6s were in the original data set?
In △ABC, ∠ABC=90∘ and BA=BC=2. Points P1,P2,…,P2024 lie on hypotenuse AC so that AP1=P1P2=P2P3=⋯=P2023P2024=P2024C. What is the length of the vector sum BP1+BP2+BP3+⋯+BP2024?
Let α be the radian measure of the smallest angle in a 3−4−5 right triangle. Let β be the radian measure of the smallest angle in a 7−24−25 right triangle. In terms of α, what is β?
