reference.csv

id,problem
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 such triangle with minimal perimeter. This triangle has side lengths $a=BC$, $b=CA$, and $c=AB$. Find the remainder when $abc$ is divided by $10^{5}$."
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 $N = f{\left(M^{15}\right)} - f{\left(M^{15}-1\right)}$. Let $k$ be the largest non-negative integer such that $2^k$ divides $N$. What is the remainder when $2^k$ is divided by $5^7$?"
424e18,"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 such a way that in each pair, both runners have the same score at the beginning of the round. The winner of each race in the $i^{\text{th}}$ round receives $2^{20-i}$ points and the loser gets no points.

At the end of the tournament, we rank the competitors according to their scores. Let $N$ denote the number of possible orderings of the competitors at the end of the tournament. Let $k$ be the largest positive integer such that $10^k$ divides $N$. What is the remainder when $k$ is divided by $10^{5}$?"
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}^\infty a_k \cdot b^k
\end{equation*}
where $a_k$ are non-negative integers and $0 \leq a_k < b$ for each $k$. Ken then erases $m$ on the blackboard and replaces it with $\sum\limits_{k = 0}^\infty a_k$.

Across all choices of $1 \leq n \leq 10^{10^5}$, the largest possible number of moves Ken could make is $M$. What is the remainder when $M$ is divided by $10^{5}$?"
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 perpendicular from $D$ to $EF$. The circle tangent to line $BN$ and passing through $B$ and $K$ intersects $BC$ again at $T \neq B$. 
    
Let sequence $(F_n)_{n \geq 0}$ be defined by $F_0 = 0$, $F_1 = 1$ and for $n \geq 2$, $F_n = F_{n-1} + F_{n-2}$. Call $ABC$ $n$\emph{-tastic} if $BD = F_n$, $CD = F_{n+1}$, and $KNK'B$ is cyclic. Across all $n$-tastic triangles, let $a_n$ denote the maximum possible value of $\frac{CT \cdot NB}{BT \cdot NE}$. Let $\alpha$ denote the smallest real number such that for all sufficiently large $n$, $a_{2n} < \alpha$. Given that $\alpha = p + \sqrt{q}$ for rationals $p$ and $q$, what is the remainder when $\left\lfloor p^{q^p} \right\rfloor$ is divided by $99991$?"
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\lfloor \frac{2025! f(M+c)}{M}\right\rfloor.
\end{equation*}
We can write 
\begin{equation*}
    g(0)+g(4M)+g(1848374)+g(10162574)+g(265710644)+g(44636594)=\frac{p}{q}
\end{equation*}
where $p$ and $q$ are coprime positive integers. What is the remainder when $p+q$ is divided by $99991$?"
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 sweets because then both our sum and product would be equal.'' What is the product of Alice and Bob's ages?"
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?"
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 $10^{5}$?"
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 $\sum\limits_{n\in\mathbb{Z}} \alpha(n)\cdot \beta(n)$. Also, for $n\in\mathbb{Z}$, define a shift operator $S_n \colon \mathcal{F}\to \mathcal{F}$ by $S_n(\alpha)(t)=\alpha(t+n)$ for all $t \in \mathbb{Z}$.

A function $\alpha \in \mathcal{F}$ is called \emph{shifty} if 
\begin{itemize}
    \item $\alpha(m)=0$ for all integers $m<0$ and $m>8$ and
    \item There exists $\beta \in \mathcal{F}$ and integers $k \neq l$ such that for all $n \in \mathbb{Z}$
    \begin{equation*}
        S_n(\alpha)\star\beta =
        \begin{cases}
            1 & n \in \{k,l\} \\
            0 & n \not \in \{k,l\}
        \end{cases}
        \; .
    \end{equation*}
\end{itemize}
How many shifty functions are there in $\mathcal{F}$?"