Prove that for any positive integers $a_1, a_2, \ldots , a_n$ the following inequality holds true: \[\left\lfloor\frac{a_1^2}{a_2}\right\rfloor+\left\lfloor\frac{a_2^2}{a_3}\right\rfloor+\cdots+\left\lfloor\frac{a_n^2}{a_1}\right\rfloor\geqslant a_1+a_2+\cdots+a_n.\] [i]Maxim Didin[/i]

Circles $\Omega$ and $\omega$ are internally tangent at point $C$. Chord $AB$ of $\Omega$ is tangent to $\omega$ at $E$, where $E$ is the midpoint of $AB$. Another circle, $\omega_1$ is tangent to $\Omega, \omega,$ and $AB$ at $D,Z,$ and $F$ respectively. Rays $CD$ and $AB$ meet at $P$. If $M$ is the midpoint of major arc $AB$, show that $\tan \angle ZEP = \tfrac{PE}{CM}$. [i]Ray Li.[/i]

Let $m$ and $n$ are fixed natural numbers and $Oxy$ is a coordinate system in the plane. Find the total count of all possible situations of $n+m-1$ points $P_1(x_1,y_1),P_2(x_2,y_2),\ldots,P_{n+m-1}(x_{n+m-1},y_{n+m-1})$ in the plane for which the following conditions are satisfied: (i) The numbers $x_i$ and $y_i~(i=1,2,\ldots,n+m-1)$ are integers and $1\le x_i\le n,1\le y_i\le m$. (ii) Every one of the numbers $1,2,\ldots,n$ can be found in the sequence $x_1,x_2,\ldots,x_{n+m-1}$ and every one of the numbers $1,2,\ldots,m$ can be found in the sequence $y_1,y_2,\ldots,y_{n+m-1}$. (iii) For every $i=1,2,\ldots,n+m-2$ the line $P_iP_{i+1}$ is parallel to one of the coordinate axes. [i](Ivan Gochev, Hristo Minchev)[/i]

Determine all real numbers $\alpha$ with the property that all numbers in the sequence $\cos\alpha,\cos2\alpha,\cos2^2\alpha,\ldots,\cos2^n\alpha,\ldots$ are negative.

Prove: for all positive integers $m, n$ $\frac 1m + \frac 1{m+1} + \dotsb + \frac 1 {m+n} $ is not an integer.

In $\triangle ABC,$ $AB=6, AC=8, BC=10,$ and $D$ is the midpoint of $\overline{BC}.$ What is the sum of the radii of the circles inscribed in $\triangle ADB$ and $\triangle ADC?$ $\textbf{(A)} \sqrt{5} \qquad \textbf{(B)} \frac{11}{4}\qquad \textbf{(C)} 2\sqrt{2} \qquad \textbf{(D)} \frac{17}{6} \qquad \textbf{(E)} 3$

Let $a$ be an integer and $p$ a prime number that divides both $5a-1$ and $a-10$. Show that $p$ also divides $a-3$.

Prove that equation $a^2 - b^2=ab - 1$ has infinitely many solutions, if $a,b$ are positive integers

Someone wants to unscrew a square nut with side $a$, with a wrench whose hole has the form of a regular hexagon with side $b$. What condition should the lengths $a$ and $b$ meet to make this possible?

a) There are 8 teams in a Quidditch tournament. Each team plays every other team once without draws. Prove that there exist teams $A,B,C,D$ such that pairs of teams $A,B$ and $C,D$ won the same number of games in total. b) There are 25 teams in a Quidditch tournament. Each team plays every other team once without draws. Prove that there exist teams $A,B,C,D,E,F$ such that pairs of teams $A,B,$ $~$ $C,D$ and $E,F$ won the same number of games in total.

Compute the maximum integer value of $k$ such that $2^k$ divides $3^{2n+3}+40n-27$ for any positive integer $n$.

Find all triples $(p,M, z)$ of integers, where $p$ is prime, $m$ is positive and $z$ is negative, that satisfy the equation $$p^3 + pm + 2zm = m^2 + pz + z^2$$

function $f:\mathbb{N}^{\star} \times \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star}$ ($\mathbb{N}^{\star}=\mathbb{N}\cup \{0\}$)with these conditon: 1- $f(0,x)=x+1$ 2- $f(x+1,0)=f(x,1)$ 3- $f(x+1,y+1)=f(x,f(x+1,y))$(romania 1997) find $f(3,1997)$

There are three circles $w_1, w_2, w_3$. Let $w_{i+1} \cap w_{i+2} = A_i, B_i$, where $A_i$ lies insides of $w_i$. Let $\gamma$ be the circle that is inside $w_1,w_2,w_3$ and tangent to the three said circles at $T_1, T_2, T_3$. Let $T_iA_{i+1}T_{i+2}$'s circumcircle and $T_iA_{i+2}T_{i+1}$'s circumcircle meet at $S_i$. Prove that the circumcircles of $A_iB_iS_i$ meet at two points. ($1 \le i \le 3$, indices taken modulo $3$) If one of $A_i,B_i,S_i$ are collinear - the intersections of the other two circles lie on this line. Prove this as well.

Let $P$ and $Q$ be fixed points in the Euclidean plane. Consider another point $O_0$. Define $O_{i+1}$ as the center of the unique circle passing through $O_i$, $P$ and $Q$. (Assume that $O_i,P,Q$ are never collinear.) How many possible positions of $O_0$ satisfy that $O_{2021}=O_{0}$? [i]Proposed by Fei Peng[/i]

Let $ABC$ be a triangle with $\angle A=60^{\circ}$ and $AB\neq AC$, $I$-incenter, $O$-circumcenter. Prove that perpendicular bisector of $AI$, line $OI$ and line $BC$ have a common point.

For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers. (a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$. (b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.

Show that there are only finitely many solutions to $1/a + 1/b + 1/c = 1/1983$ in positive integers.

Given a nonequilateral triangle $ABC$, the vertices listed counterclockwise, find the locus of the centroids of the equilateral triangles $A'B'C'$ (the vertices listed counterclockwise) for which the triples of points $A,B', C'; A',B, C';$ and $A',B', C$ are collinear. [i]Proposed by Poland.[/i]

Let $f : (0, \infty) \to \mathbb{R}$ is differentiable such that $\lim \limits_{x \to \infty} f(x)=2019$ Then which of the following is correct? [list=1] [*] $\lim \limits_{x \to \infty} f'(x)$ always exists but not necessarily zero. [*] $\lim \limits_{x \to \infty} f'(x)$ always exists and is equal to zero. [*] $\lim \limits_{x \to \infty} f'(x)$ may not exist. [*] $\lim \limits_{x \to \infty} f'(x)$ exists if $f$ is twice differentiable. [/list]

Find all pairs of integers $(m, n)$ which satisfy the equation \[(2n^2+5m-5n-mn)^2=m^3n\]

Let $ABCD$ be a cyclic quadrilateral, and $M$, $N$ be the midpoints of arcs $AB$ and $CD$ respectively. Prove that $MN$ bisects the segment between the incenters of triangles $ABC$ and $ADC$.

Let $O$ be the circumcenter of triangle $ABC$. Suppose the perpendicular bisectors of $\overline{OB}$ and $\overline{OC}$ intersect lines $AB$ and $AC$ at $D\neq A$ and $E\neq A$, respectively. Determine the maximum possible number of distinct intersection points between line $BC$ and the circumcircle of $\triangle ADE$. [i]Andrew Wen[/i]

In the triangle $ABC$ it is known that$\angle A = 75^o, \angle C = 45^o$. On the ray $BC$ beyond the point $C$ the point $T$ is taken so that $BC = CT$. Let $M$ be the midpoint of the segment $AT$. Find the measure of the $\angle BMC$. (Anton Trygub)

Which of the following is the correct order of the fractions $\frac{15}{11}, \frac{19}{15}$, and $\frac{17}{13}$, from least to greatest? $\textbf{(A) } \frac{15}{11} < \frac{17}{13} < \frac{19}{15} \qquad\textbf{(B) } \frac{15}{11} < \frac{19}{15} < \frac{17}{13} \qquad\textbf{(C) } \frac{17}{13} < \frac{19}{15} < \frac{15}{11} \newline\newline \qquad\textbf{(D) } \frac{19}{15} < \frac{15}{11} < \frac{17}{13} \qquad\textbf{(E) } \frac{19}{15} < \frac{17}{13} < \frac{15}{11}$