A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$. [i]Proposed by Aaron Lin[/i]

Let $y_1$ and $y_2$ be two linearly independent solutions of the equation $$y''+P(x)y'+Q(x)=0.$$ Find the differential equation satisfied by the product $z=y_1 y_2$.

A friendly football match lasts 90 minutes. In this problem, we consider one of the teams, coached by Sir Alex, which plays with 11 players at all times. a) Sir Alex wants for each of his players to play the same integer number of minutes, but each player has to play less than 60 minutes in total. What is the minimum number of players required? b) For the number of players found in a), what is the minimum number of substitutions required, so that each player plays the same number of minutes? [i]Remark:[/i] Substitutions can only take place after a positive integer number of minutes, and players who have come off earlier can return to the game as many times as needed. There is no limit to the number of substitutions allowed. Proposed by Athanasios Kontogeorgis and Demetres Christofides.

Show that there exist infinitely many integers that cannot be written as the difference between two perfect squares.

A group of $4046$ friends will play a videogame tournament. For that, $2023$ of them will go to one room which the computers are labeled with $a_1,a_2,\dots,a_{2023}$ and the other $2023$ friends go to another room which the computers are labeled with $b_1,b_2,\dots,b_{2023}$. The player of computer $a_i$ [b]always[/b] challenges the players of computer $b_i,b_{i+2},b_{i+3},b_{i+4}$(the player doesn't challenge $b_{i+1}$). After the first round, inside both rooms, the players may switch the computers. After the reordering, all the players realize that they are challenging the same players of the first round. Prove that if [b]one[/b] player has [b]not[/b] switched his computer, then all the players have not switched their computers.

Given an integer $a_1$($a_1 \neq -1$), find a real number sequence $\{ a_n \}$($a_i \neq 0, i=1,2,\cdots,5$) such that $x_1,x_2,\cdots,x_5$ and $y_1,y_2,\cdots,y_5$ satisfy $b_{i1}x_1+b_{i2}x_2+\cdots +b_{i5}x_5=2y_i$, $i=1,2,3,4,5$, then $x_1y_1+x_2y_2+\cdots+x_5y_5=0$, where $b_{ij}=\prod_{1 \leq k \leq i} (1+ja_k)$.

Find all integers $n\ge1$ such that $2^n-1$ has exactly $n$ positive integer divisors. [i]Proposed by Ankan Bhattacharya [/i]

In the eight-term sequence $A,B,C,D,E,F,G,H$, the value of $C$ is 5 and the sum of any three consecutive terms is 30. What is $A+H$? $\textbf{(A)}\,17 \qquad\textbf{(B)}\,18 \qquad\textbf{(C)}\,25 \qquad\textbf{(D)}\,26 \qquad\textbf{(E)}\,43$

For all real numbers $a$ and $b$, let \[a\Join b=\dfrac{a+b}{a-b}.\] Compute $1008\Join 1007$. [i] Proposed by David Altizio [/i]

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Which number is bigger $\sqrt[2012]{2012!}$ or $\sqrt[2013]{2013!}$.

Given a positive integer $n$, consider a triangular array with entries $a_{ij}$ where $i$ ranges from $1$ to $n$ and $j$ ranges from $1$ to $n-i+1$. The entries of the array are all either $0$ or $1$, and, for all $i > 1$ and any associated $j$ , $a_{ij}$ is $0$ if $a_{i-1,j} = a_{i-1,j+1}$, and $a_{ij}$ is $1$ otherwise. Let $S$ denote the set of binary sequences of length $n$, and define a map $f \colon S \to S$ via $f \colon (a_{11}, a_{12},\cdots ,a_{1n}) \to (a_{n1}, a_{n-1,2}, \cdots , a_{1n})$. Determine the number of fixed points of $f$.

Determine, with proof, the smallest positive integer $n$ with the following property: For every choice of $n$ integers, there exist at least two whose sum or difference is divisible by $2009$.

Does there exist a pair $ (f; g)$ of strictly monotonic functions, both from $ \mathbb{N}$ to $ \mathbb{N}$, such that \[ f(g(g(n))) < g(f(n))\] for every $ n \in\mathbb{N}$?

A sleeping rabbit lies in the interior of a convex $2024$-gon. A hunter picks three vertices of the polygon and he lays a trap which covers the interior and the boundary of the triangular region determined by them. Determine the minimum number of times he needs to do this to guarantee that the rabbit will be trapped. [i]Proposed by Anant Mudgal and Rohan Goyal[/i]

Let $S = \{1, 2, 3, . . . , 64\}.$ Compute the number of ways to partition $S$ into $16$ arithmetic sequences such that each arithmetic sequence has length $4$ and common difference $1, 4,$ or $16.$