Assume that the set of all positive integers is decomposed into $ r$ (disjoint) subsets $ A_1 \cup A_2 \cup \ldots \cup A_r \equal{} \mathbb{N}.$ Prove that one of them, say $ A_i,$ has the following property: There exists a positive $ m$ such that for any $ k$ one can find numbers $ a_1, a_2, \ldots, a_k$ in $ A_i$ with $ 0 < a_{j \plus{} 1} \minus{} a_j \leq m,$ $ (1 \leq j \leq k \minus{} 1)$.

1. Baron Munchhausen was told that some polynomial $P(x)=a_{n} x^{n}+\ldots+a_{1} x+a_{0}$ is such that $P(x)+P(-x)$ has exactly 45 distinct real roots. Baron doesn't know the value of $n$. Nevertheless he claims that he can determine one of the coefficients $a_{n}, \ldots, a_{1}, a_{0}$ (indicating its position and value). Isn't Baron mistaken? Boris Frenkin

In $\triangle ABC$, $CA=1960\sqrt{2}$, $CB=6720$, and $\angle C = 45^{\circ}$. Let $K$, $L$, $M$ lie on $BC$, $CA$, and $AB$ such that $AK \perp BC$, $BL \perp CA$, and $AM=BM$. Let $N$, $O$, $P$ lie on $KL$, $BA$, and $BL$ such that $AN=KN$, $BO=CO$, and $A$ lies on line $NP$. If $H$ is the orthocenter of $\triangle MOP$, compute $HK^2$. [hide="Clarifications"] [list] [*] Without further qualification, ``$XY$'' denotes line $XY$.[/list][/hide] [i]Evan Chen[/i]

The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA, SB, SC, BC, CA, AB,$ or to their extensions. a) Prove that the tetrahedron $SABC$ is regular. b) Prove conversely that for every regular tetrahedron five such spheres exist.

A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive $x$ and $y$ axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly $1989$ minutes? [asy] draw((0,0)--(20,0), EndArrow); draw((0,0)--(0,25), EndArrow); draw((0,0)--(5,0)--(5,5)--(0,5)--(0,10)--(10,10)--(10,0)--(15,0)--(15,15)--(0,15)--(0,20)--(10,20),linewidth(2)); draw((0,20)--(10,20), EndArrow); draw((3.5,.5)--(4,.5)--(4,2), EndArrow); draw((4,3.5)--(4,4)--(2.5,4), EndArrow); draw((2,5.5)--(1,5.5)--(1,7), EndArrow); draw((1,8)--(1,9)--(2.5,9), EndArrow); draw((8,9.5)--(9,9.5)--(9,8), EndArrow); draw((10.5,2)--(10.5,1)--(12,1), EndArrow); draw((13,.5)--(14,.5)--(14,2), EndArrow); draw((14.5,13)--(14.5,14)--(13,14), EndArrow); draw((2,15.5)--(1,15.5)--(1,17), EndArrow); draw((.5,18)--(.5,19)--(2,19), EndArrow); label("x", (21,0), E); label("y", (0,26), N); label("4", (0,20), W); label("3", (0,15), W); label("2", (0,10), W); label("1", (0,5), W); label("0", (0,0), SW); label("1", (5,0), S); label("2", (10,0), S); label("3", (15,0), S); [/asy] $\textbf{(A)}\ (35,44) \qquad\textbf{(B)}\ (36,45) \qquad\textbf{(C)}\ (37,45) \qquad\textbf{(D)}\ (44,35) \qquad\textbf{(E)}\ (45,36)$

There are several coins in a row, and the [i]allowed move[/i] is to remove exactly one coin from the row, which can either be the first or the last. In the initial distribution, there are $n$ coins with not necessarily equal values. Ana and María alternate turns. Ana starts, making two moves, then María makes one move, then Ana makes two moves, and so on until no coins remain: Ana makes two moves and María makes one. (Only in the last move can Ana take one coin if only one coin is left.) Ana's goal is to ensure she takes at least $\dfrac{2}{3}$ of the total value of the coins. Determine if Ana can achieve her goal with certainty if [list=a] [*]$n=2013$ [*]$n=2014$ [/list] If the answer is yes, provide a strategy to achieve it; if the answer is no, give a specific sequence of coins and explain how María prevents Ana from achieving her goal.

A positive integer $d$ and a permutation of positive integers $a_1,a_2,a_3,\dots$ is given such that for all indices $i\geq 10^{100}$, $|a_{i+1}-a_{i}|\leq 2d$ holds. Prove that there exists infinity many indices $j$ such that $|a_j -j|< d$.

The sequence $a_1, a_2, a_3, ...$ satisfies that $a_1 = 3$, $a_2 = -1$, $a_na_{n-2} +a_{n-1} = 2$ for all $n \ge 3$. Calculate $a_1 + a_2+ ... + a_{99}$.

It is known that the graph of the function $y = f(x)$ after a rotation of $45^o$ around a certain point turns into the graph of the function $y = x^3 + ax^2 + 19x + 97$. At what $a$ is this possible?

A $\textit{cubic sequence}$ is a sequence of integers given by $a_n =n^3 + bn^2 + cn + d$, where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative integers. $\textbf{(a)}$ Show that there exists a cubic sequence such that the only terms of the sequence which are squares of integers are $a_{2015}$ and $a_{2016}$. $\textbf{(b)}$ Determine the possible values of $a_{2015} \cdot a_{2016}$ for a cubic sequence satisfying the condition in part $\textbf{(a)}$.

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers.

$H$ is the orthocenter of an acute triangle $ABC$, and let $M$ be the midpoint of $BC$. Suppose $(AH)$ meets $AB$ and $AC$ at $D,E$ respectively. $AH$ meets $DE$ at $P$, and the line through $H$ perpendicular to $AH$ meets $DM$ at $Q$. Prove that $P,Q,B$ are collinear.

Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$. [i]Proposed by Evan Chen, Taiwan[/i]

Given a $15\times 15$ chessboard. We draw a closed broken line without self-intersections such that every edge of the broken line is a segment joining the centers of two adjacent cells of the chessboard. If this broken line is symmetric with respect to a diagonal of the chessboard, then show that the length of the broken line is $\leq 200$.

Mary's top book shelf holds five books with the following widths, in centimeters: $ 6$, $ \frac12$, $ 1$, $ 2.5$, and $ 10$. What is the average book width, in centimeters? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Consider a line $ \ell $ in the plane and let $ B_1, B_2, B_3 $ be different points in $ \ell$. Let $ A $ be a point that is not in $ \ell$. Show that there is $ P, Q $ in $ {B_1, B_2, B_3} $ with $ P \ne Q $ so that the distance from $ A $ to $ \ell$ is greater than the distance from $ P $ to the line that passes through $ A $ and $ Q $.

Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

Find all pairs of integers $(a,b)$ such that $n|( a^n + b^{n+1})$ for all positive integer $n$

For positive is true $$\frac{3}{abc} \geq a+b+c$$ Prove $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq a+b+c$$

Two circles $\omega$ and $\gamma$ have radii $3$ and $4$ respectively, and their centers are $10$ units apart. Let $x$ be the shortest possible distance between a point on $\omega$ and a point on $\gamma$ , and let$ y$ be the longest possible distance between a point on $\omega$ and a point on $\gamma$ . Find the product $xy$.

Find all triplets natural numbers $(a, b, c)$ satisfied \[GCD(a, b) + LCM(a,b) = 2021^c\] with $|a - b|$ and $(a+b)^2 + 4$ are both prime number

In a circle with a radius $R$ a convex hexagon is inscribed. The diagonals $AD$ and $BE$,$BE$ and $CF$,$CF$ and $AD$ of the hexagon intersect at the points $M$,$N$ and$K$, respectively. Let $r_1,r_2,r_3,r_4,r_5,r_6$ be the radii of circles inscribed in triangles $ ABM,BCN,CDK,DEM,EFN,AFK$ respectively. Prove that.$$r_1+r_2+r_3+r_4+r_5+r_6\leq R\sqrt{3}$$ .

The squares of a squared paper are enumerated as shown on the picture. \[\begin{array}{|c|c|c|c|c|c} \ddots &&&&&\\ \hline 10&\ddots&&&&\\ \hline 6&9&\ddots&&&\\ \hline 3&5&8&12&\ddots&\\ \hline 1&2&4&7&11&\ddots\\ \hline \end{array}\] Devise a polynomial $p(m, n)$ in two variables such that for any $m, n \in \mathbb{N}$ the number written in the square with coordinates $(m, n)$ is equal to $p(m, n)$.