seq int64 1 10 | id stringclasses 10
values | problem stringclasses 10
values | answer int64 50 57.4k |
|---|---|---|---|
3 | 0e644e | Let $ABC$ be an acute-angled triangle with integer side lengths and $AB<AC$. Points $D$ and $E$ lie on segments $BC$ and $AC$, respectively, such that $AD=AE=AB$. Line $DE$ intersects $AB$ at $X$. Circles $BXD$ and $CED$ intersect for the second time at $Y \neq D$. Suppose that $Y$ lies on line $AD$. There is a unique ... | 336 |
6 | 26de63 | Define a function $f \colon \mathbb{Z}_{\geq 1} \to \mathbb{Z}_{\geq 1}$ by
\begin{equation*}
f(n) = \sum_{i = 1}^n \sum_{j = 1}^n j^{1024} \left\lfloor\frac1j + \frac{n-i}{n}\right\rfloor.
\end{equation*}
Let $M=2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13$ and let $N = f{\left(M^{15}\right)} - f{\left(M^{15}-1\rig... | 32,951 |
5 | 424ee000 | A tournament is held with $2^{20}$ runners each of which has a different running speed. In each race, two runners compete against each other with the faster runner always winning the race. The competition consists of $20$ rounds with each runner starting with a score of $0$. In each round, the runners are paired in suc... | 21,818 |
8 | 42d360 | On a blackboard, Ken starts off by writing a positive integer $n$ and then applies the following move until he first reaches $1$. Given that the number on the board is $m$, he chooses a base $b$, where $2 \leq b \leq m$, and considers the unique base-$b$ representation of $m$,
\begin{equation*}
m = \sum_{k = 0}^\in... | 32,193 |
7 | 641659 | Let $ABC$ be a triangle with $AB \neq AC$, circumcircle $\Omega$, and incircle $\omega$. Let the contact points of $\omega$ with $BC$, $CA$, and $AB$ be $D$, $E$, and $F$, respectively. Let the circumcircle of $AFE$ meet $\Omega$ at $K$ and let the reflection of $K$ in $EF$ be $K'$. Let $N$ denote the foot of the perpe... | 57,447 |
10 | 86e8e5 | Let $n \geq 6$ be a positive integer. We call a positive integer $n$-Norwegian if it has three distinct positive divisors whose sum is equal to $n$. Let $f(n)$ denote the smallest $n$-Norwegian positive integer. Let $M=3^{2025!}$ and for a non-negative integer $c$ define
\begin{equation*}
g(c)=\frac{1}{2025!}\left... | 8,687 |
1 | 92ba6a | Alice and Bob are each holding some integer number of sweets. Alice says to Bob: ``If we each added the number of sweets we're holding to our (positive integer) age, my answer would be double yours. If we took the product, then my answer would be four times yours.'' Bob replies: ``Why don't you give me five of your swe... | 50 |
4 | 9c1c5f | Let $f \colon \mathbb{Z}_{\geq 1} \to \mathbb{Z}_{\geq 1}$ be a function such that for all positive integers $m$ and $n$,
\begin{equation*}
f(m) + f(n) = f(m + n + mn).
\end{equation*}
Across all functions $f$ such that $f(n) \leq 1000$ for all $n \leq 1000$, how many different values can $f(2024)$ take? | 580 |
2 | a295e9 | A $500 \times 500$ square is divided into $k$ rectangles, each having integer side lengths. Given that no two of these rectangles have the same perimeter, the largest possible value of $k$ is $\mathcal{K}$. What is the remainder when $k$ is divided by $10^{5}$? | 520 |
9 | dd7f5e | Let $\mathcal{F}$ be the set of functions $\alpha \colon \mathbb{Z}\to \mathbb{Z}$ for which there are only finitely many $n \in \mathbb{Z}$ such that $\alpha(n) \neq 0$.
For two functions $\alpha$ and $\beta$ in $\mathcal{F}$, define their product $\alpha\star\beta$ to be $\sum\limits_{n\in\mathbb{Z}} \alpha(n)\cdot... | 160 |
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