Let $a,b,c,d$ be positive reals such taht $a+b+c+d=1$. Prove that: \[6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.\]

Let $ n$ be natural number. Find the limit value of ${ \lim_{n\to\infty} \frac{1}{n}(\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}}+\cdots\cdots +\frac{n}{\sqrt{n^2+1}}).$

Let $\Omega$ be the circumcircle of an isosceles trapezoid $ABCD$, in which $AD$ is parallel to $BC$. Let $X$ be the reflection point of $D$ with respect to $BC$. Point $Q$ is on the arc $BC$ of $\Omega$ that does not contain $A$. Let $P$ be the intersection of $DQ$ and $BC$. A point $E$ satisfies that $EQ$ is parallel to $PX$, and $EQ$ bisects $\angle BEC$. Prove that $EQ$ also bisects $\angle AEP$. [i]Proposed by Li4.[/i]

Initially, a pair of numbers $(1,1)$ is written on the board. If for some $x$ and $y$ one of the pairs $(x, y-1)$ and $(x+y, y+1)$ is written on the board, then you can add the other one. Similarly for $(x, xy)$ and $(\frac {1} {x}, y)$. Prove that for each pair that appears on the board, its first number will be positive.

$11$ carriers will carry $270$ kg of melons at one step where each melons weighs at most $7$ kg. Each carrier can carry at most $30$ kg in one step. Show that it is possible to carry all the melons at one step whatever a melon weighs.

Compute the number of real numbers $t$ such that \[t = 50 \sin(t - \lfloor t \rfloor).\] Here $\lfloor \cdot\rfloor$ denotes the greatest integer function. [i]Proposed by David Altizio[/i]

Given several squares with the total area $1$. Prove that you can pose them in the square of the area $2$ without any intersections.

For a polynomial $ P$ of degree 2000 with distinct real coefficients let $ M(P)$ be the set of all polynomials that can be produced from $ P$ by permutation of its coefficients. A polynomial $ P$ will be called [b]$ n$-independent[/b] if $ P(n) \equal{} 0$ and we can get from any $ Q \in M(P)$ a polynomial $ Q_1$ such that $ Q_1(n) \equal{} 0$ by interchanging at most one pair of coefficients of $ Q.$ Find all integers $ n$ for which $ n$-independent polynomials exist.

Maisy is at the origin of the coordinate plane. On her first step, she moves $1$ unit up. On her second step, she moves $ 1$ unit to the right. On her third step, she moves $2$ units up. On her fourth step, she moves $2$ units to the right. She repeats this pattern with each odd-numbered step being $ 1$ unit more than the previous step. Given that the point that Maisy lands on after her $21$st step can be written in the form $(x, y)$, find the value of $x + y$. Proposed by Audrey Chun

Consider a sequence of polynomials $P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots$, where $P_0(x) = 2, P_1(x) = x$ and for every $n \geq 1$ the following equality holds: \[P_{n+1}(x) + P_{n-1}(x) = xP_n(x).\] Prove that there exist three real numbers $a, b, c$ such that for all $n \geq 1,$ \[(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.\]

Let $n$ be a positive integer and let $d(n)$ denote the number of ordered pairs of positive integers $(x,y)$ such that $(x+1)^2-xy(2x-xy+2y)+(y+1)^2=n$. Find the smallest positive integer $n$ satisfying $d(n) = 61$. (Patrik Bak, Slovakia)

Find all the finite sequences with $ n$ consecutive natural numbers $ a_1, a_2,..., a_n$, with $ n\geq3$ such that $ a_1\plus{} a_2\plus{}...\plus{} a_n\equal{}2009$.

Let $f(x)$ be a nonconstant real polynomial. The sequence $\{a_i\}_{i=1}^{\infty}$ of real numbers is strictly increasing and unbounded, as $$a_{i+1}<a_i+2020.$$ The integers $\lfloor{|f(a_1)|}\rfloor$ , $\lfloor{|f(a_2)|}\rfloor$ , $\lfloor{|f(a_3)|}\rfloor$ , $\dots$ are written consecutively in such a way that their digits form an infinite sequence of digits $\{s_k\}_{k=1}^{\infty}$ (here $s_k\in\{0, 1, \dots, 9\}$). $\quad$If $n\in\mathbb{N}$ , prove that among the numbers $\overline{s_{n(k-1)+1}s_{n(k-1)+2}\cdots s_{nk}}$ , where $k\in\mathbb{N}$ , all $n$-digit numbers appear.

Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.

Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively. If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$

From $25$ points in a plane, both of whose coordinates are integers of the set $\{-2,-1, 0, 1, 2\}$, some $17$ points are marked. Prove that there are three points on one line, one of them is the midpoint of two others.

A sequence of polynomials $\{f_n\}_{n=0}^{\infty}$ is defined recursively by $f_0(x)=1$, $f_1(x)=1+x$, and \[(k+1)f_{k+1}(x)-(x+1)f_k(x)+(x-k)f_{k-1}(x)=0, \quad k=1,2,\ldots\] Prove that $f_k(k)=2^k$ for all $k\geq 0$.

Ali is hosting a large party. Together with his $n-1$ friends, $n$ people are seated around a circular table in a fixed order. Ali places $n$ apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction). Find all values of $n$ such that after some number of steps, the situation reaches a point where each person has exactly one apple.

Determine all pairs of rational numbers $(x, y)$ such that \[x^{3}+y^{3}= x^{2}+y^{2}.\]

$n$ people sitting at a round table. In the beginning, everyone writes down a positive number $n$ on piece of paper in front of them. From now on, in every minute, they write down the number that they get if they subtract the number of their right-hand neighbour from their own number. They write down the new number and erase the original. Give those number $n$ that there exists an integer $k$ in a way that regardless of the starting numbers, after $k$ minutes, everyone will have a number that is divisible by $n$.

Let $ABC$ be an acute triangle with orthocenter $H$. Segments $AH$ and $CH$ intersect segments $BC$ and $AB$ in points $A_1$ and $C_1$ respectively. The segments $BH$ and $A_1C_1$ meet at point $D$. Let $P$ be the midpoint of the segment $BH$. Let $D'$ be the reflection of the point $D$ in $AC$. Prove that quadrilateral $APCD'$ is cyclic. [i]Proposed by Matko Ljulj.[/i]

a) The triangle $ABC$ was turned around the centre of the circumscribed circle by the angle less than $180$ degrees and thus was obtained the triangle $A_1B_1C_1$. The corresponding segments $[AB]$ and $[A_1B_1]$ intersect in the point $C_2, [BC]$ and $[B_1C_1]$ -- $A_2, [AC]$ and $[A_1C_1]$ -- $B_2$. Prove that the triangle $A_2B_2C_2$ is similar to the triangle $ABC$. b) The quadrangle $ABCD$ was turned around the centre of the circumscribed circle by the angle less than $180$ degrees and thus was obtained the quadrangle $A_1B_1C_1D_1$. Prove that the points of intersection of the corresponding lines ( $(AB$) and $(A_1B_1), (BC)$ and $(B_1C_1), (CD)$ and $(C_1D_1), (DA)$ and $(D_1A_1)$ ) are the vertices of the parallelogram.

Find all $k>0$ for which a strictly decreasing function $g:(0;+\infty)\to(0;+\infty)$ exists such that $g(x)\geq kg(x+g(x))$ for all positive $x$.

[asy] draw((0,1)--(4,1)--(4,2)--(0,2)--cycle); draw((2,0)--(3,0)--(3,3)--(2,3)--cycle); draw((1,1)--(1,2)); label("1",(0.5,1.5)); label("2",(1.5,1.5)); label("32",(2.5,1.5)); label("16",(3.5,1.5)); label("8",(2.5,0.5)); label("6",(2.5,2.5)); [/asy] The image above is a net of a unit cube. Let $n$ be a positive integer, and let $2n$ such cubes are placed to build a $1 \times 2 \times n$ cuboid which is placed on a floor. Let $S$ be the sum of all numbers on the block visible (not facing the floor). Find the minimum value of $n$ such that there exists such cuboid and its placement on the floor so $S > 2011$.