Three athletes ran at different constant speeds along a track of length $1$. They started moving at the same time at one end of the track. Having reached one of the ends of the track, the athlete immediately turned around and continued running in the opposite direction. After a while, all three athletes met at the start and finished training. At what maximum $S$ can we knowingly say that at some point the sum of the pairwise distances between athletes was at least $S$? [i]Proposed by A. Golovanov, I. Rubanov[/i]

Let $A_1,B_1,C_1,D_1$ be the midpoints of the sides of a convex quadrilateral $ABCD$ and let $A_2, B_2, C_2, D_2$ be the midpoints of the sides of the quadrilateral $A_1B_1C_1D_1$. If $A_2B_2C_2D_2$ is a rectangle with sides $4$ and $6$, then what is the product of the lengths of the diagonals of $ABCD$ ?

Nils has an $M \times N$ board where $M$ and $N$ are positive integers, and a tile shaped as shown below. What is the smallest number of squares that Nils must color, so that it is impossible to place the tile on the board without covering a colored square? The tile can be freely rotated and mirrored, but it must completely cover four squares. [asy] usepackage("tikz"); label("% \begin{tikzpicture} \draw[step=1cm,color=black] (0,0) grid (2,1); \draw[step=1cm,color=black] (1,1) grid (3,2); \fill [yellow] (0,0) rectangle (2,1); \fill [yellow] (1,1) rectangle (3,2); \draw[step=1cm,color=black] (0,0) grid (2,1); \draw[step=1cm,color=black] (1,1) grid (3,2); \end{tikzpicture} "); [/asy]

Let $a,b,c$ be the side lengths of any triangle $\triangle ABC$ opposite to $A,B$ and $C,$ respectively. Let $x,y,z$ be the length of medians from $A,B$ and $C,$ respectively. If $T$ is the area of $\triangle ABC$, prove that $\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\ge\sqrt{\sqrt{3}T}$

Let $p$ be an arbitrary odd prime and $\sigma(n)$ for $1 \le n \le p-1$ denote the inverse of $n \pmod p$. Show that the number of pairs $(a,b) \in \{1,2,\cdots,p-1\}^2$ with $a<b$ but $\sigma(a) > \sigma(b)$ is at least $$\left \lfloor \left(\frac{p-1}{4}\right)^2 \right \rfloor$$ [i]usjl[/i] Note: Partial credits may be awarded if the $4$ in the statement is replaced with some larger constant

$ABC$ is a triangle. Reflect each vertex in the opposite side to get the triangle $A'B'C'$. Find a necessary and sufficient condition on $ABC$ for $A'B'C'$ to be equilateral.

Find all functions $f:\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}$ for which $xf(xy) + f(-y) = xf(x)$ for all non-zero real numbers $x, y$.

For a positive integer $n$, let $S_n$ be the set of positive integers that do not exceed $n$ and are coprime to $n$. Define $f(n)$ as the smallest positive integer that allows $S_n$ to be partitioned into $f(n)$ disjoint subsets, each forming an arithmetic progression. Prove that there exist infinitely many pairs $(a, b)$ satisfying $a, b > 2025$, $a \mid b$, and $f(a) \nmid f(b)$.

Let $w = w_1 w_2 \dots w_n$ be a word. Define a [i]substring[/i] of $w$ to be a word of the form $w_i w_{i + 1} \dots w_{j - 1} w_j$, for some pair of positive integers $1 \le i \le j \le n$. Show that $w$ has at most $n$ distinct palindromic substrings. For example, $aaaaa$ has $5$ distinct palindromic substrings, and $abcata$ has $5$ ($a$, $b$, $c$, $t$, $ata$).

Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.

Let $ABC$ be a triangle with $AB=20, AC=21,$ and $\angle BAC = 90^{\circ}.$ Suppose $\Gamma_1$ is the unique circle centered at $B$ and passing through $A,$ and $\Gamma_2$ is the unique circle centered at $C$ and passing through $A.$ Points $E$ and $F$ are selected on $\Gamma_1$ and $\Gamma_2,$ respectively, such that $E, A, F$ are collinear in that order. The tangent to $\Gamma_1$ at $E$ and the tangent to $\Gamma_2$ at $F$ intersect at $P$. Given that $PA \bot BC$, compute the area of $PBC$.

The incircle of the triangle $ ABC $ touches the sides of $ AB, BC, CA $ at points $ C_0, A_0, B_0 $, respectively. Let the point $ M $ be the midpoint of the segment connecting the vertex $ C_0 $ with the intersection point of the altitudes of the triangle $ A_0B_0C_0 $, point $ N $ be the midpoint of the arc $ ACB $ of the circumscribed circle of the triangle $ ABC $. Prove that line $ MN $ passes through the center of incircle of triangle $ ABC $.

Define $ x\otimes y \equal{} x^3 \minus{} y$. What is $ h\otimes (h\otimes h)$? $ \textbf{(A) } \minus{} h\qquad \textbf{(B) } 0\qquad \textbf{(C) } h\qquad \textbf{(D) } 2h\qquad \textbf{(E) } h^3$

Given an isocleos triangle $ABC$ with equal sides $AB=AC$ and incenter $I$.Let $\Gamma_1$be the circle centered at $A$ with radius $AB$,$\Gamma_2$ be the circle centered at $I$ with radius $BI$.A circle $\Gamma_3$ passing through $B,I$ intersects $\Gamma_1$,$\Gamma_2$ again at $P,Q$ (different from $B$) respectively.Let $R$ be the intersection of $PI$ and $BQ$.Show that $BR \perp CR$.

Jack writes whole numbers starting from $ 1$ and skips all numbers that contain either a $2$ or $9$. What is the $100$th number that Jack writes down?

Let $a$ be a real number, such that $a\ne 0, a\ne 1, a\ne -1$ and $m,n,p,q$ be natural numbers . Prove that if $a^m+a^n=a^p+a^q$ and $a^{3m}+a^{3n}=a^{3p}+a^{3q}$ , then $m \cdot n = p \cdot q$.

When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$ Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$ 30 points

For a positive integer $n$, $(a_0, a_1, \cdots , a_n)$ is a $n+1$-tuple with integer entries. For all $k=0, 1, \cdots , n$, we denote $b_k$ as the number of $k$s in $(a_0, a_1, \cdots ,a_n)$. For all $k = 0,1, \cdots , n$, we denote $c_k$ as the number of $k$s in $(b_0, b_1, \cdots ,b_n)$. Find all $(a_0, a_1, \cdots ,a_n)$ which satisfies $a_0 = c_0$, $a_1=c_1$, $\cdots$, $a_n=c_n$.

Show that the coefficients $a_1 , a_2 , a_3 ,\ldots$ in the expansion $$\frac{1}{4}\left(1+x-\frac{1}{\sqrt{1-6x+x^{2}}}\right) =a_{1} x+ a_2 x^2 + a_3 x^3 +\ldots$$ are positive integers.

A right-triangulated prism made of benzene sits on a table. The hypotenuse makes an angle of $30^\circ$ with the horizontal table. An incoming ray of light hits the hypotenuse horizontally, and leaves the prism from the vertical leg at an acute angle of $ \gamma $ with respect to the vertical leg. Find $\gamma$, in degrees, to the nearest integer. The index of refraction of benzene is $1.50$. [i](Proposed by Ahaan Rungta)[/i]

$16$ white and $4$ red balls that are not identical are distributed randomly into $4$ boxes which contain at most $5$ balls. What is the probability that each box contains exactly $1$ red ball? $ \textbf{(A)}\ \dfrac{5}{64} \qquad\textbf{(B)}\ \dfrac{1}{8} \qquad\textbf{(C)}\ \dfrac{4^4}{\binom{16}{4}} \qquad\textbf{(D)}\ \dfrac{5^4}{\binom{20}{4}} \qquad\textbf{(E)}\ \dfrac{3}{32} $

In the parallelogram $CMNP$ extend the bisectors of angles $MCN$ and $PCN$ and intersect with extensions of sides PN and $MN$ at points $A$ and $B$, respectively. Prove that the bisector of the original angle $C$ of the the parallelogram is perpendicular to $AB$. [img]https://cdn.artofproblemsolving.com/attachments/f/3/fde8ef133758e06b1faf8bdd815056173f9233.png[/img]

Determine the area of the region defined by [i]x[/i]²+[i]y[/i]²≤[i]π[/i]² and [i]y[/i] ≥ sin [i]x[/i].

The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

In every $1\times1$ cell of a rectangle board a natural number is written. In one step it is allowed the numbers written in every cell of arbitrary chosen row, to be doubled, or the numbers written in the cells of the arbitrary chosen column to be decreased by 1. Will after final number of steps all the numbers on the board be $0$?