Given an infinite sequence of numbers $a_1, a_2,...$, in which there are no two equal members. Segment $a_i, a_{i+1}, ..., a_{i+m-1}$ of this sequence is called a monotone segment of length $m$, if $a_i < a_{i+1} <...<a_{i+m-1}$ or $a_i > a_{i+1} >... > a_{i+m-1}$. It turned out that for each natural $k$ the term $a_k$ is contained in some monotonic segment of length $k + 1$. Prove that there exists a natural $N$ such that the sequence $a_N , a_{N+1} ,...$ monotonic.

Distinct points $A$ and $B$ are on a semicircle with diameter $MN$ and center $C$. The point $P$ is on $CN$ and $\angle CAP = \angle CBP = 10^{\circ}$. If $\stackrel{\frown}{MA} = 40^{\circ}$, then $\stackrel{\frown}{BN}$ equals [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair C=origin, N=dir(0), B=dir(20), A=dir(135), M=dir(180), P=(3/7)*dir(C--N); draw(M--N^^C--A--P--B--C^^Arc(origin,1,0,180)); markscalefactor=0.03; draw(anglemark(C,A,P)); draw(anglemark(C,B,P)); pair point=C; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, S); label("$M$", M, dir(point--M)); label("$N$", N, dir(point--N)); label("$P$", P, S); label("$40^\circ$", C+(-0.15,0), NW); label("$10^\circ$", B+(0,0.05), W); label("$10^\circ$", A+(0.05,0.02), E);[/asy] $ \textbf{(A)}\ 10^{\circ}\qquad\textbf{(B)}\ 15^{\circ}\qquad\textbf{(C)}\ 20^{\circ}\qquad\textbf{(D)}\ 25^{\circ}\qquad\textbf{(E)}\ 30^{\circ}$

The points $M$ and $N$ are on the side $BC$ of $\Delta ABC$, so that $BM=CN$ and $M$ is between $B$ and $N$. Points $P\in AN$ and $Q\in AM$ are such that $\angle PMC=\angle MAB$ and $\angle QNB=\angle NAC$. Prove that $\angle QBC=\angle PCB$.

Consider an infinite sequence of reals $x_1, x_2, x_3, ...$ such that $x_1 = 1$, $x_2 =\frac{2\sqrt3}{3}$ and with the recursive relationship $$n^2 (x_n - x_{n-1} - x_{n-2}) - n(3x_n + 2x_{n-1} + x_{n-2}) + (x_nx_{n-1}x_{n-2} + 2x_n) = 0.$$ Find $x_{2019}$.

There are $26$ students in the class. They agreed that each of them would either be a liar (liars always lie) or a knight (knights always tell the truth). When they came to the class and sat down for desks, each of them said: “I am sitting next to a liar.” Then some students moved for other desks. After that, everyone says: “ I am sitting next to a knight .” Is this possible? Every time exactly two students sat at any desk.

Prove that if the numbers $p_1, p_2, q_1, q_2$ satisfy the condition $$(q_1 - q_2)^2 + (p_1 - p_2)(p_1q_2 -p_2q_1)<0$$ then the square polynomials $x^2 + p_1x + q_1$ and $x^2 + p_2x + q_2$ have real roots, and between the roots of each there is a root of another one.

The sequence $(z_n)$ of complex numbers satisfies the following properties: [list] [*]$z_1$ and $z_2$ are not real. [*]$z_{n+2}=z_{n+1}^2z_n$ for all integers $n\geq 1$. [*]$\dfrac{z_{n+3}}{z_n^2}$ is real for all integers $n\geq 1$. [*]$\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2$. [/list] Find the product of all possible values of $z_1$.

Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 \equal{} 1,$ and $ a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$

Given a sequence of natural numbers $ (a_i) $, for each natural number $ n $ the sum of the terms of the sequence that are not greater than $ n $ is a number not less than $ n $. Prove that for every natural number $ k $ it is possible to choose from the sequence $ (a_i) $ a finite sequence with the sum of terms equal to $ k $.

For any positive integer $a,$ $\sigma(a)$ denotes the sum of the positive integer divisors of $a.$ Let $n$ be the least positive integer such that $\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a.$ Find the sum of the prime factors in the prime factorization of $n.$

An altitude $AH$ of triangle $ABC$ bisects a median $BM$. Prove that the medians of triangle $ABM$ are sidelengths of a right-angled triangle. by Yu.Blinkov

Let $ABC$ be a triangle. Let $D, E$ be a points on the segment $BC$ such that $BD =DE = EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine $BP:PQ$.

The incircle of $ \triangle A_{0}B_{0}C_{0}$, meets legs $B_{0}C_{0}$, $C_{0}A_{0}$, $A_{0}B_{0}$, respectively on points $A$, $B$, $C$, and the incircle of $ \triangle ABC$, with center $I$, meets legs $BC$, $CA$, $AB$, on points $A_{1}$, $B_{1}$, $C_{1}$, respectively. We write with $ \sigma (ABC)$, and $ \sigma (A_{1}B_{1}C_{1})$ the areas of $ \triangle ABC$, and $ \triangle A_{1}B_{1}C_{1}$ respectively. Prove that if $ \sigma (ABC)=2 \sigma (A_{1}B_{1}C_{1})$, then lines $AA_{0}$, $BB_{0}$, $CC_{0}$ are concurrent.

How many zeros does $101^{100} - 1$ end with?

Determine all natural numbers $m$ and $n$ such that \[ n \cdot (n + 1) = 3^m + s(n) + 1182, \] where $s(n)$ represents the sum of the digits of the natural number $n$.

A cat is trying to catch a mouse in the non-negative quadrant \[N=\{(x_1,x_2)\in \mathbb{R}^2: x_1,x_2\geq 0\}.\] At time $t=0$ the cat is at $(1,1)$ and the mouse is at $(0,0)$. The cat moves with speed $\sqrt{2}$ such that the position $c(t)=(c_1(t),c_2(t))$ is continuous, and differentiable except at finitely many points; while the mouse moves with speed $1$ such that its position $m(t)=(m_1(t),m_2(t))$ is also continuous, and differentiable except at finitely many points. Thus $c(0)=(1,1)$ and $m(0)=(0,0)$; $c(t)$ and $m(t)$ are continuous functions of $t$ such that $c(t),m(t)\in N$ for all $t\geq 0$; the derivatives $c'(t)=(c'_1(t),c'_2(t))$ and $m'(t)=(m'_1(t),m'_2(t))$ each exist for all but finitely many $t$ and \[(c'_1(t)^2+(c'_2(t))^2=2 \qquad (m'_1(t)^2+(m'_2(t))^2=1,\] whenever the respective derivative exists. At each time $t$ the cat knows both the mouse's position $m(t)$ and velocity $m'(t)$. Show that, no matter how the mouse moves, the cat can catch it by time $t=1$; that is, show that the cat can move such that $c(\tau)=m(\tau)$ for some $\tau\in[0,1]$.

In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is 2 5 that of triangle AOB, determine a : b. [asy] pair A = (0,3); pair B = (0,0); pair C = (3,0); pair D = (0,1.5); pair E = (0.35,0); pair F = (1.2,1.8); pair J = (0.17,0); pair Y = (0.17,0.75); pair Z = (1.6,0.2); draw(A--B); draw(B--C); draw(C--A); draw(D--F--Z--E--D); draw("$O$", B, dir(180)); draw("$B$", A, dir(45)); draw("$A$", C, dir(45)); draw("$Q$", E, dir(45)); draw("$P$", D, dir(45)); draw("$R$", Z, dir(45)); draw("$S$", F, dir(45)); draw("$a$", Y, dir(210)); draw("$b$", J, dir(100)); [/asy]

Given positive integers $m,n(2\le m\le n)$, let $a_1,a_2,\ldots ,a_m$ be a permutation of any $m$ pairwise distinct numbers taken from $1,2,\ldots ,n$. If there exist $k\in\{1,2,\ldots ,m\}$ such that $a_k+k$ is odd, or there exist positive integers $k,l(1\le k<l\le m)$ such that $a_k>a_l$, then call $a_1,a_2,\ldots ,a_m$ a [i]good[/i] sequence. Find the number of good sequences.

Let $Q$ be a polynomial \[Q(x)=a_0+a_1x+\cdots+a_nx^n,\] where $a_0,\ldots,a_n$ are nonnegative integers. Given that $Q(1)=4$ and $Q(5)=152$, find $Q(6)$.

Convex quadrilateral $ABCD$ is inscribed in circle $w.$Rays $AB$ and $DC$ intersect at $K.\ L$ is chosen on the diagonal $BD$ so that $\angle BAC= \angle DAL.\ M$ is chosen on the segment $KL$ so that $CM \mid\mid BD.$ Prove that line $BM$ touches $w.$ [i](Kungozhin M.)[/i]

A cube is formed from $n^3$ ($n \ge 2$) unit cubes, each painted white on five randomly selected sides. This cube is dipped into paint remover and broken into the original unit cubes. What is the expected number of these unit cubes with exactly four sides painted white?

Let be a natural number $ n\ge 4 $ and $ n $ nonnegative numbers $ a,b,\ldots ,c. $ Prove that $$ \prod_{\text{cyc} } (a+b+c)^2 \ge 2^n\prod_{\text{cyc} } (a+b)^2, $$ and tell in which circumstances equality happens.

Let be given a positive integer $n \ge 3$. Consider integers $a_1,a_2,...,a_n >1$ with the product equals to $A$ such that: for each $k \in \{1, 2,..., n\}$ then the remainder when $\frac{A}{a_k}$ divided by $a_k$ are all equal to $r$. Prove that $r \le n- 2$

Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ \sqrt2 \qquad \textbf{(C)}\ \sqrt3 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 2\sqrt2$

Let $AD, BE$ and $CF$ be the medians of triangle $ABC$. The points $X$ and $Y$ are the reflections of $F$ about $AD$ and $BE$, respectively. Prove that the circumcircles of triangles $BEX$ and $ADY$ are concentric.