Shreyas has a rectangular piece of paper $ABCD$ such that $AB = 20$ and $AD = 21$. Given that Shreyas can make exactly one straight-line cut to split the paper into two pieces, compute the maximum total perimeter of the two pieces

For a positive integer $n$ let $\sigma (n)$ be the sum of all its positive divisors. Find all positive integers $n$ such that the number $\frac{\sigma (n)}{n + 1}$ is an integer.

Circle $B$ has radius $6\sqrt{7}$. Circle $A$, centered at point $C$, has radius $\sqrt{7}$ and is contained in $B$. Let $L$ be the locus of centers $C$ such that there exists a point $D$ on the boundary of $B$ with the following property: if the tangents from $D$ to circle $A$ intersect circle $B$ again at $X$ and $Y$, then $XY$ is also tangent to $A$. Find the area contained by the boundary of $L$.

Is it possible to number the $8$ vertices of a cube from $1$ to $8$ in such a way that the value of the sum on every edge is different?

For $n\in\mathbb{N}$, prove that $2^n$ can begin with any sequence of digits. Hint: $\log 2$ is irrational number.

Let $f:\mathbb C\to\mathbb C$ be a holomorphic function with the property that $|f(z)|=1$ for all $z\in\mathbb C$ such that $|z|=1$. Prove that there exists a $\theta\in\mathbb R$ and a $k\in\{0,1,2,\ldots\}$ such that $$f(z)=e^{i\theta}z^k$$for all $z\in\mathbb C$.

A set $ S$ of points in space satisfies the property that all pairwise distances between points in $ S$ are distinct. Given that all points in $ S$ have integer coordinates $ (x,y,z)$ where $ 1 \leq x,y, z \leq n,$ show that the number of points in $ S$ is less than $ \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).$ [i]Dan Schwarz, Romania[/i]

Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ with $ x,y \in \mathbb{R}$ such that \[ f(x \minus{} f(y)) \equal{} f(x\plus{}y) \plus{} f(y)\]

Consider the integral $Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}$. [b](a)[/b] Show that the integral $Z(j)=\int^{\infty}_{-\infty} dx e^{-x^{2}+jx}$, where $j$ is not a function of $x$, is $Z(j)=e^{j^{2}/4a} Z(0)$. [b](b)[/b] Show that $$\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n},$$ where $(2n-1)!!$ is defined as $(2n-1)(2n-3)\times\cdots\times3\times 1$. [b](c)[/b] What is the number of ways to form $n$ pairs from $2n$ distinct objects? Interpret the previous part of the problem in term of this answer.

Let $Q(x) = a_{2023}x^{2023}+a_{2022}x^{2022}+\dots+a_{1}x+a_{0} \in \mathbb{Z}[x]$ be a polynomial with integer coefficients. For an odd prime number $p$ we define the polynomial $Q_{p}(x) = a_{2023}^{p-2}x^{2023}+a_{2022}^{p-2}x^{2022}+\dots+a_{1}^{p-2}x+a_{0}^{p-2}.$ Assume that there exist infinitely primes $p$ such that $$\frac{Q_{p}(x)-Q(x)}{p}$$ is an integer for all $x \in \mathbb{Z}$. Determine the largest possible value of $Q(2023)$ over all such polynomials $Q$. [i]Authored by Nikola Velov[/i]

A sequence $a_n$ is defined by $\int_{a_n}^{a_{n+1}} (1+|\sin x|)dx=(n+1)^2\ (n=1,\ 2,\ \cdots),\ a_1=0$. Find $\lim_{n\to\infty} \frac{a_n}{n^3}$.

Is it possible to put $1987$ points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a non-degenerate triangle with rational area? [i](IMO Problem 5)[/i] [i]Proposed by Germany, DR[/i]

Several students are competing in a series of three races. A student earns $5$ points for winning a race, $3$ points for finishing second and $1$ point for finishing third. There are no ties. What is the smallest number of points that a student must earn in the three races to be guaranteed of earning more points than any other student? $\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11 \qquad \text{(D)}\ 13 \qquad \text{(E)}\ 15$

A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

Let $y = x^2 + ax + b$ be a parabola that cuts the coordinate axes at three distinct points. Show that the circle passing through these three points also passes through $(0,1)$.

Let $n$ be a positive integer. For any positive integer $j$ and positive real number $r$, define $f_j(r)$ and $g_j(r)$ by \[f_j(r) = \min (jr, n) + \min\left(\frac{j}{r}, n\right), \text{ and } g_j(r) = \min (\lceil jr\rceil, n) + \min \left(\left\lceil\frac{j}{r}\right\rceil, n\right),\] where $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Prove that \[\sum_{j=1}^n f_j(r)\leq n^2+n\leq \sum_{j=1}^n g_j(r)\] for all positive real numbers $r$.

Let n be an integer greater than 1 and let $x_1, x_2, . . . , x_n$ be real numbers such that $|x_1| + |x_2| + ... + |x_n| = 1$ and $x_1 + x_2 + ... + x_n = 0$ Prove that $\left| \frac{x_1}{1}+\frac{x_2}{2}+\cdots+\frac{x_n}{n} \right| \leq \frac{1}{2} \left(1-\frac{1}{n}\right)$

Prove or disprove that there exists a positive real number $u$ such that $\lfloor u^n \rfloor -n$ is an even integer for all positive integer $n$.

Let $r_1, r_2, ..., r_{2021}$ be the not necessarily real and not necessarily distinct roots of $x^{2022} + 2021x = 2022$. Let $S_i = r_i^{2021}+2022r_i$ for all $1 \le i \le 2021$. Find $\left|\sum^{2021}_{i=1} S_i \right| = |S_1 +S_2 +...+S_{2021}|$.

For a finite set $ X$ of positive integers, let $ \Sigma(X) \equal{} \sum_{x \in X} \arctan \frac{1}{x}.$ Given a finite set $ S$ of positive integers for which $ \Sigma(S) < \frac{\pi}{2},$ show that there exists at least one finite set $ T$ of positive integers for which $ S \subset T$ and $ \Sigma(S) \equal{} \frac{\pi}{2}.$ [i]Kevin Buzzard, United Kingdom[/i]

Let $ABC$ a triangle be given with $BC = a$, $CA = b$, $AB = c$ ($a \neq b \neq c \neq a$). In plane ($ABC$) take the points $A'$, $B'$, $C'$ such that: [b]I.[/b] The pairs of points $A$ and $A'$, $B$ and $B'$, $C$ and $C'$ either all lie in one side either all lie in different sides under the lines $BC$, $CA$, $AB$ respectively; [b]II.[/b] Triangles $A'BC$, $B'CA$, $C'AB$ are similar isosceles triangles. Find the value of angle $A'BC$ as function of $a, b, c$ such that lengths $AA', BB', CC'$ are not sides of an triangle. (The word "triangle" must be understood in its ordinary meaning: its vertices are not collinear.)

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

Ali has $100$ cards with numbers $1,2,\ldots,100$. Ali and Amin play a game together. In each step, first Ali chooses a card from the remaining cards and Amin decides to pick that card for himself or throw it away. In the case that he picks the card, he can't pick the next card chosen by Amin, and he has to throw it away. This action repeats until when there is no remaining card for Ali. Amin wants to pick cards in a way that the sum of the number of his cards is maximized and Ali wants to choose cards in a way that the sum of the number of Amin's cards is minimized. Find the most value of $k$ such that Amin can play in a way that is sure the sum of the number of his cards will be at least equal to $k$.

Let $f:(0,\infty) \rightarrow (0,\infty)$ be a function such that \[ 10\cdot \frac{x+y}{xy}=f(x)\cdot f(y)-f(xy)-90 \] for every $x,y \in (0,\infty)$. What is $f(\frac 1{11})$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 21 \qquad\textbf{(D)}\ 31 \qquad\textbf{(E)}\ \text{There is more than one solution} $

A circle with center $O$ and a point $A$ in this circle are given. Let $P_{B}$ is the intersection point of $[AB]$ and the internal bisector of $\angle AOB$ where $B$ is a point on the circle such that $B$ doesn't lie on the line $OA$, Find the locus of $P_{B}$ as $B$ varies.