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Patrick started walking at a constant speed along a straight road from his school to the park. One hour after Patrick left, Tanya started running at a constant speed of $2$ miles per hour faster than Patrick walked, following the same straight road from the school to the park. One hour after Tanya left, José started bi... | 277 |
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. | 62 |
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 $\mathcal{T}$ 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 ... | 79 |
Find the number of integers less than or equal to $100$ that are equal to $a+b+ab$ for some choice of distinct positive integers $a$ and $b$. | 70 |
A plane contains points $A$ and $B$ with $AB=1$. Point $A$ is rotated in the plane counterclockwise through an acute angle $\theta$ around point $B$ to point $A'$. Then $B$ is rotated in the plane clockwise through angle $\theta$ around point $A'$ to point $B'$. Suppose $AB'=\frac{4}{3}$. The value of $\cos\theta$ can ... | 65 |
The product of all positive real numbers $x$ satisfying the equation $\sqrt[20]{x^{\log_{2026}x}}=26x$ is an integer $P$. Find the number of positive integer divisors of $P$. | 441 |
Find the number of functions $\pi$ mapping the set $A=\{1,2,3,4,5,6\}$ onto $A$ such that for every $a\in A$, $\pi(\pi(\pi(\pi(\pi(\pi(a))))))=a$. | 396 |
Let $N$ be the number of positive integer divisors of $17017^{17}$ that leave a remainder of $5$ upon division by $12$. Find the remainder when $N$ is divided by $1000$. | 244 |
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 ... | 29 |
Let $\triangle ABC$ have side lengths $AB=13$, $BC=14$, and $CA=15$. Triangle $\triangle A'B'C'$ is obtained by rotating $\triangle ABC$ about its circumcenter so that $\overline{A'C'}$ is perpendicular to $\overline{BC}$, with $A'$ and $B$ not on the same side of line $B'C'$. Find the integer closest to the area of he... | 156 |
The integers from $1$ to $64$ are placed in some order into an $8\times 8$ grid of cells with one number in each cell. Let $a_{i,j}$ 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=\sum^{8}_{i=1}\sum^{7}_{j=1}\left(|a_{i,... | 896 |
Triangle $\triangle ABC$ lies in plane $\mathcal{P}$ with $AB=6$, $AC=4$, and $\angle BAC=90^{\circ}$. Let $D$ be the reflection across $\overline{BC}$ of the centroid of $\triangle ABC$. Four spheres, all on the same side of $\mathcal{P}$, have radii $1$, $2$, $3$, and $r$ and are tangent to $\mathcal{P}$ at points $A... | 161 |
For each nonnegative integer $r$ less that $502$, define $S_r=\sum_{m\geq 0}\binom{10,000}{502m+r}$, where $\binom{10,000}{n}$ is defined to be $0$ when $n>10,000$. That is, $S_r$ is the sum of all the binomial coefficients of the form $\binom{10,000}{k}$ for which $0\leq k\leq 10,000$ and $k-r$ is a multiple of $502$.... | 39 |
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 $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. ... | 681 |
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 $a \times b$ cell loop in a $2n \times 2n$ grid of cells to be the $2a + 2b - 4$ cells that surround an $(a - 2) \times (b - 2)$ (possibly empty) rectangle of cells in the grid. Find the... | 83 |
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