It is a fact that every set of 2016 consecutive integers can be partitioned in two sets with the following four properties: (i) The sets have the same number of elements. (ii) The sums of the elements of the sets are equal. (iii) The sums of the squares of the elements of the sets are equal. (iv) The sums of the cubes of the elements of the sets are equal. Let $S =\{n + 1; n + 2;$ [b]. . .[/b] $; n + k\}$ be a set of $k$ consecutive integers. (a) Determine the smallest value of $k$ such that property (i) holds for $S$. (b) Determine the smallest value of $k$ such that properties (i) and (ii) hold for $S$. (c) Show that properties (i), (ii) and (iii) hold for $S$ when $k = 8$. (d) Show that properties (i), (ii), (iii) and (iv) hold for $S$ when $k = 16$.

$a$ and $b$ are fixed real numbers. With $x_n$ we denote the sum of the digits of $an+b$ in the decimal number system. Prove that the sequence $x_n$ contains an infinite constant subsequence.

On the plane are chosen $n \ge 3$ points, not all on the same line. Drawing all lines passing through two of these points one obtains k different lines. Prove that $k \ge n$.

Let $\{a_{n}\}_{n \geq 0}$ and $\{b_{n}\}_{n \geq 0}$ be two sequences of integer numbers such that: i. $a_{0}=0$, $b_{0}=8$. ii. For every $n \geq 0$, $a_{n+2}=2a_{n+1}-a_{n}+2$, $b_{n+2}=2b_{n+1}-b_{n}$. iii. $a_{n}^{2}+b_{n}^{2}$ is a perfect square for every $n \geq 0$. Find at least two values of the pair $(a_{1992},\, b_{1992})$.

Let $ABC$ be a right-angled triangle, $I$ its incenter and $B_0, A_0$ points of tangency of the incircle with the legs $AC$ and $BC$ respectively. Let the perpendicular dropped to $AI$ from $A_0$ and the perpendicular dropped to $BI$ from $B_0$ meet at point $P$. Prove that the lines $CP$ and $AB$ are perpendicular.

Prove that if the cross-section of a cube cut by a plane is a pentagon, as shown in the figure below, then there are two adjacent sides of the pentagon such that the sum of the lengths of those two sides is greater than the sum of the lengths of the other three sides. (For ease of grading, please use the names of the points from the figure below in your solution.) [asy] import three; defaultpen(linewidth(0.8)); currentprojection=orthographic(1,3/5,1/2); draw(unitcube, white, thick(), nolight); draw(O--(1,0,0)^^O--(0,1,0)^^O--(0,0,1), linetype("4 4")+linewidth(0.7)); triple A=(1/3, 1, 1), B=(2/3, 1, 0), C=(1, 1/2, 0), D=(1, 0, 1/2), E=(2/3, 0, 1); draw(E--A--B^^C--D); draw(B--C^^D--E, linetype("4 4")+linewidth(0.7)); label("$A$", A, dir(85)); label("$B$", B, SE); label("$C$", C, S); label("$D$", D, W); label("$E$", E, NW);[/asy]

Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by \[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \] Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality \[ n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \] and establish the cases of equality. [i]Proposed by Finbarr Holland, Ireland[/i]

Let $A, B, C, D$ be points chosen on a circle, in that order. Line $BD$ is reflected over lines $AB$ and $DA$ to obtain lines $\ell_1$ and $\ell_2$ respectively. If lines $\ell_1$, $\ell_2$, and $AC$ meet at a common point and if $AB = 4$, $BC = 3$, $CD = 2$, compute the length $DA$.

Find all triplets of matrices $A,B,C\in\mathcal{M}_2(\mathbb{R})$ which satisfy \begin{align*} A=BC-CB \\ B=CA-AC \\ C=AB-BA \end{align*} [i]Proposed by David Anghel[/i]

Find all real values of $\alpha$, for which there exists one and only one function $f: \mathbb{R} \mapsto \mathbb{R}$ and satisfying the equation \[ f(x^2 + y + f(y)) = (f(x))^2 + \alpha \cdot y \] for all $x, y \in \mathbb{R}$.

Find the value of $(1+2)(1+2^2)(1+2^4)(1+2^8)...(1+2^{2048})$.

If ${a, b}$ and $c$ are positive real numbers, prove that \begin{align*} a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 &\ge{ abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right)}. \end{align*} [i](Montenegro).[/i]

There are $10$ consecutive 30-digit numbers. We write the biggest divisor for every number ( divisor is not equal number). Prove that some written numbers ends with same digit.

How many pairs $(m,n)$ of integers satisfy the equation $m+n=mn$? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }\text{more than }4$

For a set $A$ of integers, define $f(A)=\{x^2+xy+y^2: x,y\in A\}$. Is there a constant $c$ such that for all positive integers $n$, there exists a set $A$ of size $n$ such that $|f(A)|\le cn$? [i]David Yang.[/i]

$5$ people stand in a line facing one direction. In every round, the person at the front moves randomly to any position in the line, including the front or the end. Suppose that $\frac{m}{n}$ is the expected number of rounds needed for the last person of the initial line to appear at the front of the line, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?

Let $M$ be a set of points in the Cartesian plane, and let $\left(S\right)$ be a set of segments (whose endpoints not necessarily have to belong to $M$) such that one can walk from any point of $M$ to any other point of $M$ by travelling along segments which are in $\left(S\right)$. Find the smallest total length of the segments of $\left(S\right)$ in the cases [b]a.)[/b] $M = \left\{\left(-1,0\right),\left(0,0\right),\left(1,0\right),\left(0,-1\right),\left(0,1\right)\right\}$. [b]b.)[/b] $M = \left\{\left(-1,-1\right),\left(-1,0\right),\left(-1,1\right),\left(0,-1\right),\left(0,0\right),\left(0,1\right),\left(1,-1\right),\left(1,0\right),\left(1,1\right)\right\}$. In other words, find the Steiner trees of the set $M$ in the above two cases.

The last digit of the number A = $7^{2011}$ is ?

Let $a, b$ be two nonnegative real numbers and $n$ a positive integer. Prove that \[\left(1-2^{-n}\right)\left|a^{2^n}-b^{2^n}\right|\ge\sqrt{ab}\left|a^{2^n-1}-b^{2^n-1}\right|.\]

Given $ 0< c< 1$, we define $f(x) = \begin{cases} \frac{x}{c} & x \in [0,c] \\ \frac{1-x}{1-c} & x \in [c, 1] \end{cases} $ Let $f^{n}(x)=f(f(...f(x)))$ . Show that for each positive integer $n$, $f^{n}$ has a non-zero finite nunber of fixed points which aren't fixed points of $f^k$ for $k< n$.

The decimal number $13^{101}$ is given. It is instead written as a ternary number. What are the two last digits of this ternary number?

A large gathering of people stand in a triangular array with $2020$ rows, such that the first row has $1$ person, the second row has $2$ people, and so on. Every day, the people in each row infect all of the people adjacent to them in their own row. Additionally, the people at the ends of each row infect the people at the ends of the rows in front of and behind them that are at the same side of the row as they are. Given that two people are chosen at random to be infected with COVID at the beginning of day 1, what is the earliest possible day that the last uninfected person will be infected with COVID? [i]Proposed by Richard Chen[/i]

How many triangles appear in the diagram below? [asy] import graph; size(4.4cm); real labelscalefactor = 0.5; pen dotstyle = black; draw((-2,5)--(-2,1)); draw((-2,5)--(2,5)); draw((2,5)--(2,1)); draw((-2,1)--(2,1)); draw((0,5)--(0,1)); draw((-2,3)--(2,3)); draw((-1,5)--(-1,1)); draw((1,5)--(1,1)); draw((-2,2)--(2,2)); draw((-2,4)--(2,4)); draw((1,5)--(-2,2)); draw((-2,2)--(-1,1)); draw((-1,1)--(2,4)); draw((2,4)--(1,5)); draw((-1,5)--(-2,4)); draw((-2,4)--(1,1)); draw((1,1)--(2,2)); draw((2,2)--(-1,5)); [/asy]

Real numbers $x$ and $y$ satisfy the following equations: \begin{align*} x &= \log_{10} (10^{y-1}+1)-1 \\ y &= \log_{10} (10^x+1)-1. \end{align*} Find $10^{x-y}.$

$ABC$ is an acute triangle with $AB> AC$. $\Gamma_B$ is a circle that passes through $A,B$ and is tangent to $AC$ on $A$. Define similar for $ \Gamma_C$. Let $D$ be the intersection $\Gamma_B$ and $\Gamma_C$ and $M$ be the midpoint of $BC$. $AM$ cuts $\Gamma_C$ at $E$. Let $O$ be the center of the circumscibed circle of the triangle $ABC$. Prove that the circumscibed circle of the triangle $ODE$ is tangent to $\Gamma_B$.