Prove there exist two relatively prime polynomials $P(x),Q(x)$ having integer coefficients and a real number $u>0$ such that if for positive integers $a,b,c,d$ we have: $$|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}$$ $$| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}$$ Then we have : $$bP(\frac{a}{c})=dQ(\frac{a}{c})$$ (Two polynomials are relatively prime if they don't have a common root) Proposed by [i]Navid Safaii[/i] and [i]Alireza Haghi[/i]

Let $ABCD$ be a parallelogram and let $AX$ and $AY$ be the altitudes from $A$ to $CB, CD$, respectively. A line $\ell \perp XY$ bisects $AX$ and meets $AB, BC$ at $K, L$. Similarly, a line $d \perp XY$ bisects $AY$ and meets $DA, DC$ at $P, Q$. Show that the circumcircles of $\triangle BKL$ and $\triangle DPQ$ are tangent to each other.

Let $f:[0,1]\to\mathbb R$ be a continuous function such that $\int^1_0x^mf(x)dx=0$ for $m=0,1,\ldots,1999$. Prove that $f$ has at least $2000$ zeroes on the segment $[0,1]$.

Determine the ordered systems $(x,y,z)$ of positive rational numbers for which $x+\frac{1}{y},y+\frac{1}{z}$ and $z+\frac{1}{x}$ are integers.

Given a scalene triangle $ ABC$. Let $ A'$, $ B'$, $ C'$ be the points where the internal bisectors of the angles $ CAB$, $ ABC$, $ BCA$ meet the sides $ BC$, $ CA$, $ AB$, respectively. Let the line $ BC$ meet the perpendicular bisector of $ AA'$ at $ A''$. Let the line $ CA$ meet the perpendicular bisector of $ BB'$ at $ B'$. Let the line $ AB$ meet the perpendicular bisector of $ CC'$ at $ C''$. Prove that $ A''$, $ B''$ and $ C''$ are collinear.

Evan writes a computer program that randomly rearranges the digits $0,2,4,6$, and $8$ to create a five-digit number with no leading zeroes. If he executes this program once, the probability the program outputs an integer divisible by 4 can be written in the form $\tfrac{m}{n}$ where $m$ and $n$ are positive integers which share no common factors. What is $100m+n$? [i]Proposed by David Altizio[/i]

On the graph of a polynomial with integral coefficients are two points with integral coordinates. Prove that if the distance between these two points is integral, then the segment connecting them is parallel to the $x$-axis. [i](4 points)[/i]

Two boxes contain between them 65 balls of several different sizes. Each ball is white, black, red or yellow. If you take any five balls of the same colour, at least two of them will always be of the same size(radius). Prove that there are at least three ball which lie in the same box have the same colour and have the same size(radius).

Show that: a) There is a sequence of non-zero natural numbers $a_1, a_2, ...$ uniquely determined, so that: $n = \sum _ {d | n} a _ d$ for whatever $n \in N ^ {*}$ . b) There is a sequence of non-zero natural numbers $b_1, b_2, ...$ uniquely determined, so that: $n = \prod _ {d | n} b _ d$ for whatever $n \in N ^ {*}$ . Note: The sum from a), respectively the product from b), are made after all the natural divisors $d$ of the number $n$ , including $1$ and $n$ .

Let $p,q,r$ be primes such that for all positive integer $n$, $$n^{pqr}\equiv n (\mod{pqr})$$ Prove that this happens if and only if $p,q,r$ are pairwise distinct and $LCM(p-1,q-1,r-1)|pqr-1$

Let $a_1,a_2,\ldots,a_n$ ($n\geq 2$) be nonnegative real numbers whose sum is $\frac{n}{2}$. For every $i=1,\ldots,n$ define $$b_i=a_i+a_ia_{i+1}+a_ia_{i+1}a_{i+2}+\cdots+ a_ia_{i+1}\cdots a_{i+n-2}+2a_ia_{i+1}\cdots a_{i+n-1}$$ where $a_{j+n}=a_j$ for every $j$. Prove that $b_i\geq 1$ holds for at least one index $i$.

* Cut out of a $3 \times 3$ square an unfolding of the cube with edge $1$.

Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$. Considering all such sequences $A$, find the greatest value of $m$.

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

Given a circle and $2006$ points lying on this circle. Albatross colors these $2006$ points in $17$ colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common points (not even a common endpoint). Hereby, Frankinfueter intends to draw as many chords as possible, while Albatross is trying to hinder him as much as he can. What is the maximal number of chords Frankinfueter will always be able to draw?

Sequences $x_1,x_2,\dots,$ and $y_1,y_2,\dots,$ are defined with $x_1=\dfrac{1}{8}$, $y_1=\dfrac{1}{10}$ and $x_{n+1}=x_n+x_n^2$, $y_{n+1}=y_n+y_n^2$. Prove that $x_m\neq y_n$ for all $m,n\in\mathbb{Z}^{+}$. [I]Proposed by A. Golovanov[/i]

Let $ABCD$ be a parallelogram. $E$ and $F$ are on $BC, CD$ respectively such that the triangles $ABE$ and $BCF$ have the same area. Let $BD$ intersect $AE, AF$ at $M, N$ respectively. Prove there exists a triangle whose side lengths are $BM, MN, ND$.

Find all integers $x,y$ such that $2x+3y=185$ and $xy>x+y$.

There are $1999$ people participating in an exhibition. Among any $50$ people there are two who don't know each other. Prove that there are $41$ people, each of whom knows at most $1958$ people.

Let $a,b$ be two integers. Prove that a) $13 \mid 2a+3b$ if and only if $13 \mid 2b-3a$; b) If $13 \mid a^2+b^2$ then $13 \mid (2a+3b)(2b+3a)$. [i]Mircea Fianu[/i]

A large square region is paved with $n^2$ gray square tiles, each measuring $s$ inches on a side. A border $d$ inches wide surrounds each tile. The figure below shows the case for $n = 3$. When $n = 24$, the $576$ gray tiles cover $64\%$ of the area of the large square region. What is the ratio $\frac{d}{s}$ for this larger value of $n$? [asy] draw((0,0)--(13,0)--(13,13)--(0,13)--cycle); filldraw((1,1)--(4,1)--(4,4)--(1,4)--cycle, mediumgray); filldraw((1,5)--(4,5)--(4,8)--(1,8)--cycle, mediumgray); filldraw((1,9)--(4,9)--(4,12)--(1,12)--cycle, mediumgray); filldraw((5,1)--(8,1)--(8,4)--(5,4)--cycle, mediumgray); filldraw((5,5)--(8,5)--(8,8)--(5,8)--cycle, mediumgray); filldraw((5,9)--(8,9)--(8,12)--(5,12)--cycle, mediumgray); filldraw((9,1)--(12,1)--(12,4)--(9,4)--cycle, mediumgray); filldraw((9,5)--(12,5)--(12,8)--(9,8)--cycle, mediumgray); filldraw((9,9)--(12,9)--(12,12)--(9,12)--cycle, mediumgray); [/asy] $\textbf{(A) }\frac6{25} \qquad \textbf{(B) }\frac14 \qquad \textbf{(C) }\frac9{25} \qquad \textbf{(D) }\frac7{16} \qquad \textbf{(E) }\frac9{16}$

Let $ABCD$ be a regular tetrahedron with side length $1$. Let $X$ be the point in the triangle $BCD$ such that $[XBC]=2[XBD]=4[XCD]$, where $[\overline{\omega}]$ denotes the area of figure $\overline{\omega}$. Let $Y$ lie on segment $AX$ such that $2AY=YX$. Let $M$ be the midpoint of $BD$. Let $Z$ be a point on segment $AM$ such that the lines $YZ$ and $BC$ intersect at some point. Find $\frac{AZ}{ZM}$.

Find the number of ordered triples of integers $(a,b,c)$, each between $1$ and $64$, such that \[ a^2 + b^2 \equiv c^2\pmod{64}. \]

In an isosceles triangle $ABC$ with $AB = AC$ and $\angle A = 30^o$, points $L$ and $M$ lie on the sides $AB$ and $AC$, respectively such that $AL = CM$. Point $K$ lies on $AB$ such that $\angle AMK = 45^o$. If $\angle LMC = 75^o$, prove that $KM +ML = BC$. [i]Proposed by Mahdi Etesamifard - Iran[/i]

Given any four distinct positive real numbers, show that one can choose three numbers $A,B,C$ from among them, such that all three quadratic equations \begin{eqnarray*} Bx^2 + x + C &=& 0\\ Cx^2 + x + A &=& 0 \\ Ax^2 + x +B &=& 0 \end{eqnarray*} have only real roots, or all three equations have only imaginary roots.