A positive integer $n$ is fixed. Numbers $0$ and $1$ are placed in all cells (exactly one number in any cell) of a $k \times n$ table ($k$ is a number of the rows in the table, $n$ is the number of the columns in it). We call a table nice if the following property is fulfilled: for any partition of the set of the rows of the table into two nonempty subsets $R$[size=75]1[/size] and $R$[size=75]2[/size] there exists a nonempty set $S$ of the columns such that on the intersection of any row from $R$[size=75]1[/size] with the columns from $S$ there are even number of $1's$ while on the intersection of any row from $R$[size=75]2[/size] with the columns from $S$ there are odd number of $1's$. Find the greatest number of $k$ such that there exists at least one nice $k \times n$ table.

In a regular polygon $H$ of $6n+1$ sides ($n$ is a positive integer), we paint $r$ vertices red, and the rest blue. Demonstrate that the number of isosceles triangles that have three of their vertices of the same color does not depend on the way we distribute the colors on the vertices of $H$.

Esmeralda chooses two distinct positive integers \(a\) and \(b\), with \(b > a\), and writes the equation \[ x^2 - ax + b = 0 \] on the board. If the equation has distinct positive integer roots \(c\) and \(d\), with \(d > c\), she writes the equation \[ x^2 - cx + d = 0 \] on the board. She repeats the procedure as long as she obtains distinct positive integer roots. If she writes an equation for which this does not occur, she stops. a) Show that Esmeralda can choose \(a\) and \(b\) such that she will write exactly 2024 equations on the board. b) What is the maximum number of equations she can write knowing that one of the initially chosen numbers is 2024?

Prove that for any integer $n$, there exists a unique polynomial $Q$ with coefficients in $\{0,1,\ldots,9\}$ such that $Q(-2) = Q(-5) = n$.

Prove that there is no positive integer $n>1$ such that $n\mid2^{n} -1.$

Determine all positive integers $N$ for which the sphere \[ x^2+y^2+z^2=N \] has an inscribed regular tetrahedron whose vertices have integer coordinates.

Let $p=101.$ The sum \[\sum_{k=1}^{10}\frac1{\binom pk}\] can be written as a fraction of the form $\dfrac a{p!},$ where $a$ is a positive integer. Compute $a\pmod p.$

There are $2015$ distinct circles in a plane, with radius $1$. Prove that you can select $27$ circles, which form a set $C$, which satisfy the following. For two arbitrary circles in $C$, they intersect with each other or For two arbitrary circles in $C$, they don't intersect with each other.

In a scalene triangle $ABC,$ $I$ is the incenter and $O$ is the circumcenter. The line $IO$ intersects the lines $BC,CA,AB$ at points $D,E,F$ respectively. Let $A_1$ be the intersection of $BE$ and $CF$. The points $B_1$ and $C_1$ are defined similarly. The incircle of $ABC$ is tangent to sides $BC,CA,AB$ at points $X,Y,Z$ respectively. Let the lines $XA_1, YB_1$ and $ZC_1$ intersect $IO$ at points $A_2,B_2,C_2$ respectively. Prove that the circles with diameters $AA_2,BB_2$ and $CC_2$ have a common point.

$ \frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+...+\sqrt{10+\sqrt{99}}}{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+...+\sqrt{10-\sqrt{99}}}=? $

Determine all polynomials P(x) with real coefficients such that [(x + 1)P(x − 1) − (x − 1)P(x)] is a constant polynomial.

Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$: \[ f(3mn+m+n)=4f(m)f(n)+f(m)+f(n). \]

The sequence of real numbers $(a_n)_{n\geq 0}$ is such that $a_0 = 1$, $a_1 = a > 2$ and $\displaystyle a_{n+1} = \left(\left(\frac{a_n}{a_{n-1}}\right)^2 -2\right)a_n$ for every positive integer $n$. Prove that $\displaystyle \sum_{i=0}^k \frac{1}{a_i} < \frac{2+a-\sqrt{a^2-4}}{2}$ for every positive integer $k$.

If $n$ is an integer greater than $7$, prove that ${n \choose 7} - \left[ \frac{n}{7} \right]$ is divisible by $7$.

Prove that there is no function $f:(0,\infty )\rightarrow (0,\infty)$ such that \[f(x+y)\ge f(x)+yf(f(x)) \] for every $x,y\in (0,\infty )$.

Find all positive integer solutions $x,y,z$ such that $1/x +2/y - 3/z=1$

Let $\alpha$, $\beta$ and $\gamma$ be the roots of the polynomial $2023x^3-2023x^2-1$. Find $$\frac{1}{\alpha^3}+\frac{1}{\beta^3}+\frac{1}{\gamma^3}$$. [i]Proposed by Andy Xu[/i]

An ant can move from a vertex of a cube to the opposite side of its diagonal only on the edges and the diagonals of the faces such that it doesn’t trespass a second time through the path made. Find the distance of the maximum journey this ant can make.

A frog makes $ 3$ jumps, each exactly $ 1$ meter long. The directions of the jumps are chosen independently and at random. What is the probability the frog's final position is no more than $ 1$ meter from its starting position? $ \textbf{(A)}\ \frac {1}{6} \qquad \textbf{(B)}\ \frac {1}{5} \qquad \textbf{(C)}\ \frac {1}{4} \qquad \textbf{(D)}\ \frac {1}{3} \qquad \textbf{(E)}\ \frac {1}{2}$

Let $ABCDEF$ be a convex hexagon, and let $P= AB \cap CD$, $Q = CD \cap EF$, $R = EF \cap AB$, $S = BC \cap DE$, $T = DE \cap FA$, $U = FA \cap BC$. Prove that $\frac{PQ}{CD} = \frac{QR}{EF} = \frac{RP}{AB}$ if and only if $\frac{ST}{DE} = \frac{TU}{FA} = \frac{US}{BC}$

Given a circle $\gamma$ and a point $P$ inside it, find the maximum and minimum value of the sum of the lengths of two perpendicular chords of $\gamma$ passing through $P$.

Every unit square of a $n \times n$ board is colored either red or blue so that among all 2 $\times 2$ squares on this board all possible colorings of $2 \times 2$ squares with these two colors are represented (colorings obtained from each other by rotation and reflection are considered different). a) Find the least possible value of $n$. b) For the least possible value of $n$ find the least possible number of red unit squares

Suppose $F$ is a family of finite subsets of $\mathbb{N}$ and for any 2 sets $A,B \in F$ we have $A \cap B \not= \O$. (a) Is it true that there is a finite subset $Y$ of $\mathbb{N}$ such that for any $A,B \in F$ we have $A\cap B\cap Y \not= \O$? (b) Is the above true if we assume that all members of $F$ have the same size?

Given are real numbers $a_1, a_2,...,a_{101}$ from the interval $[-2,10]$ such that their sum is $0$. Prove that the sum of their squares is smaller than $2020$.

Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$. [i]Author: Farzan Barekat, Canada[/i]