Let $X$ be a set of $100$ elements. Find the smallest possible $n$ satisfying the following condition: Given a sequence of $n$ subsets of $X$, $A_1,A_2,\ldots,A_n$, there exists $1 \leq i < j < k \leq n$ such that $$A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k.$$

For a positive integer $n$, define a function $ f_n (x) $ at an interval $ [ 0, n+1 ] $ as \[ f_n (x) = ( \sum_{i=1} ^ {n} | x-i | )^2 - \sum_{i=1} ^{n} (x-i)^2 . \] Let $ a_n $ be the minimum value of $f_n (x) $. Find the value of \[ \sum_{n=1}^{11} (-1)^{n+1} a_n . \]

Consider two distinct positive integers $a$ and $b$ having integer arithmetic, geometric and harmonic means. Find the minimal value of $|a-b|$. [i]Mircea Fianu[/i]

The numbers $1, 2,..., 2020$ are written on the blackboard. Venus and Serena play the following game. First, Venus connects by a line segment two numbers such that one of them divides the other. Then Serena connects by a line segment two numbers which has not been connected and such that one of them divides the other. Then Venus again and they continue until there is a triangle with one vertex in $2020$, i.e. $2020$ is connected to two numbers that are connected with each other. The girl that has drawn the last line segment (completed the triangle) is the winner. Which of the girls has a winning strategy? (Tomáš Bárta, Czech Republic)

Find all relatively prime integers $a,b$ such that $$a^2+a=b^3+b$$

Call a k-digit positive integer a [i]hyperprime[/i] if all its segments consisting of $ 1, 2, ..., k$ consecutive digits are prime. Find all hyperprimes.

Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die? $ \textbf{(A)}\ \dfrac16\qquad\textbf{(B)}\ \dfrac{13}{72}\qquad\textbf{(C)}\ \dfrac7{36}\qquad\textbf{(D)}\ \dfrac5{24}\qquad\textbf{(E)}\ \dfrac29 $

Tom painted round fence which consists of $2n \ge6$ sections in such way that every section is painted in one of four colours. Then he repeats the following while it is possible: he chooses three neighbouring sections of distinct colours and repaints them into the fourth colour. For which $n$ Tom can repaint the fence in such way infinitely many times?

Let $ABC$ be a triangle and let $P$ be a point on $BC$. Points $M$ and $N$ lie on $AB$ and $AC$, respectively such that $MN$ is not parallel to $BC$ and $AMP N$ is a parallelogram. Line $MN$ meets the circumcircle of $ABC$ at $R$ and $S$. Prove that the circumcircle of triangle $RP S$ is tangent to $BC$.

It is known that a $n$-vertex contains within itself a polyhedron $M$ with a center of symmetry at some point $Q$ and is itself contained in a polyhedron homothetic to $M$ with a homothety center at a point $Q$ and coefficient $k$. Find the smallest value of $k$ if a) $n = 4$, b) $n = 5$.

Find the number of three-digit positive integers which have three distinct digits where the sum of the digits is an even number such as 925 and 824.

Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$ [i]Proposed by Sergei Berlov, Ilya Bogdanov, Russia[/i]

The permutation $\sigma$ consisting of four words $A,B,C,D$ has $f_{AB}(\sigma)$, the sum of the number of $B$ placed rightside of every $A$. We can define $f_{BC}(\sigma)$,$f_{CD}(\sigma)$,$f_{DA}(\sigma)$ as the same way too. For example, $\sigma=ACBDBACDCBAD$, $f_{AB}(\sigma)=3+1+0=4$, $f_{BC}(\sigma)=4$,$f_{CD}(\sigma)=6$, $f_{DA}(\sigma)=3$ Find the maximal value of $f_{AB}(\sigma)+f_{BC}(\sigma)+f_{CD}(\sigma)+f_{DA}(\sigma)$, when $\sigma$ consists of $2020$ letters for each $A,B,C,D$

Given is a triangle $ABC$. Let the points $P$ and $Q$ be on the sides $AB, AC$, respectively, so that $AP=AQ$, and $PQ$ passes through the incenter $I$. Let $(BPI)$ meet $(CQI)$ at $M$, $PM$ meets $BI$ at $D$ and $QM$ meets $CI$ at $E$. Prove that the line $MI$ passes through the midpoint of $DE$.

Prove that the equation $x^2+y^2+z^2+t^2=2^{2004}$, where $0 \leq x \leq y \leq z \leq t$, has exactly $2$ solutions in $\mathbb Z$. [i]Mihai Baluna[/i]

Three squares $ABB_1B_2,BCC_1C_2,CAA_1A_2$ are constructed in the exterior of a triangle $ABC$. In the exterior of these squares, another three squares $A_1B_2B_3B_4,B_1C_2C_3C_4,C_1A_2A_3A_4$ are constructed. Prove that the area of a triangle with sides $C_3A_4,A_3B_4,B_3C_4$ is $16$ times the area of $\triangle ABC$.

Let points $A,B,$ and $C$ lie on a line such that $AB=1, BC=1,$ and $AC=2.$ Let $C_1$ be the circle centered at $A$ passing through $B,$ and let $C_2$ be the circle centered at $A$ passing through $C.$ Find the area of the region outside $C_1,$ but inside $C_2.$

If $s$ and $d$ are positive integers such that $\frac{1}{s} + \frac{1}{2s} + \frac{1}{3s} = \frac{1}{d^2 - 2d},$ then the smallest possible value of $s + d$ is $ \mathrm{(A) \ } 6 \qquad \mathrm{(B) \ } 8 \qquad \mathrm {(C) \ } 10 \qquad \mathrm{(D) \ } 50 \qquad \mathrm{(E) \ } 96$

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

Let $ ABCDA'B'C'D'$ be a cube. On the sides $ (A'D')$, $ (A'B')$ and $ (A'A)$ we consider the points $ M_1$, $ N_1$ and $ P_1$ respectively. On the sides $ (CB)$, $ (CD)$ and $ (CC')$ we consider the points $ M_2$, $ N_2$ and $ P_2$ respectively. Let $ d_1$ be the distance between the lines $ M_1N_1$ and $ M_2N_2$, $ d_2$ be the distance between the lines $ N_1P_1$ and $ N_2P_2$, and $ d_3$ be the distance between the lines $ P_1M_1$ and $ P_2M_2$. Suppose that the distances $ d_1$, $ d_2$ and $ d_3$ are pairwise distinct. Prove that the lines $ M_1M_2$, $ N_1N_2$ and $ P_1P_2$ are concurrent.

For any nonempty, finite set $B$ and real $x$, define $$d_B(x) = \min_{b\in B} |x-b|$$ (1) Given positive integer $m$. Find the smallest real number $\lambda$ (possibly depending on $m$) such that for any positive integer $n$ and any reals $x_1,\cdots,x_n \in [0,1]$, there exists an $m$-element set $B$ of real numbers satisfying $$d_B(x_1)+\cdots+d_B(x_n) \le \lambda n$$ (2) Given positive integer $m$ and positive real $\epsilon$. Prove that there exists a positive integer $n$ and nonnegative reals $x_1,\cdots,x_n$, satisfying for any $m$-element set $B$ of real numbers, we have $$d_B(x_1)+\cdots+d_B(x_n) > (1-\epsilon)(x_1+\cdots+x_n)$$

Let $(x_{n})_{n=1}^{+\infty}$ be a sequence defined recursively with $x_{n+1} = x_{n}(x_{n}-2)$ and $x_{1} = \frac{7}{2}$. Let $x_{2021} = \frac{a}{b}$, where $a,b \in \mathbb{N}$ are coprime. Show that if $p$ is a prime divisor of $a$, then either $3|p-1$ or $p=3$. [i]Authored by Nikola Velov[/i]

Let $k$ and $d$ be positive integers. Prove that there exists a positive integer $N$ such that for every odd integer $n>N$, the digits in the base-$2n$ representation of $n^k$ are all greater than $d$.

We have an acute-angled triangle $ABC$, and $AA',BB'$ are its altitudes. A point $D$ is chosen on the arc $ACB$ of the circumcircle of $ABC$. If $P=AA'\cap BD,Q=BB'\cap AD$, show that the midpoint of $PQ$ lies on $A'B'$.

A group of 6 math students is staying at a mathematical hotel to participate in a math tournament that will take place in the city in the coming days. This group, composed of 3 women and 3 men, was assigned rooms in a specific way by the hotel administration: in separate rooms and alternating between genders, specifically: woman, man, woman, man, woman, man, occupying the last 6 rooms in a corridor numbered from 101 to 110. \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline M & H & M & H & M & H & & & & \\ \hline 110 & 109 & 108 & 107 & 106 & 105 & 104 & 103 & 102 & 101 \\ \hline \end{tabular} Against the hotel's rules, the group devised the following game: A valid room exchange occurs when two students in consecutive rooms move to two empty rooms, such that the difference between their new room numbers and their original ones is the same. For example, if the students in rooms 105 and 106 move to rooms 101 and 102, this would be a valid exchange since both numbers decreased by 4 units. Determine if, following these rules, the students can manage to have rooms 101, 102, and 103 occupied by men and rooms 104, 105, and 106 occupied by women in just 3 valid exchanges.