On the circumference of a circle there are red and blue points. One may add a red point and change the colour of both its neighbours (to the other colour) or remove a red point and change the colour of both its previous neighbours. Initially there are two red points. Prove that there is no sequence of allowed operations which leads to the configuration consisting of two blue points. (K Kazarnovskiy, Moscow)

Given is a tree $G$ with $2023$ vertices. The longest path in the graph has length $2n$. A vertex is called good if it has degree at most $6$. Find the smallest possible value of $n$ if there doesn't exist a vertex having $6$ good neighbors.

Polynomial $P(x)$ has integer coefficients. Prove, that if polynomials $P(x)$ and $P(P(P(x)))$ have common real root, they also have a common integer root.

Circles ω[size=35]1[/size] and ω[size=35]2[/size] have no common point. Where is outerior tangents a and b, interior tangent c. Lines a, b and c touches circle ω[size=35]1[/size] respectively on points A[size=35]1[/size], B[size=35]1[/size] and C[size=35]1[/size], and circle ω[size=35]2[/size] – respectively on points A[size=35]2[/size], B[size=35]2[/size] and C[size=35]2[/size]. Prove that triangles A[size=35]1[/size]B[size=35]1[/size]C[size=35]1[/size] and A[size=35]2[/size]B[size=35]2[/size]C[size=35]2[/size] area ratio is the same as ratio of ω[size=35]1[/size] and ω[size=35]2[/size] radii.

Let $P(x)=x^4+20x^3+29x^2-666x+2025.$ It is known that $P(x)>0$ for every real $x.$ There is a root $r$ for $P$ in the first quadrant of the complex plane that can be expressed as $r=\frac{1}{2}(a+bi+\sqrt{c+di}),$ where $a,b,c,d$ are integers. Find $a+b+c+d.$

At least how many weighings of a balanced scale are needed to order four stones with distinct weights from the lightest to the heaviest? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 8 $

An (unordered) partition $P$ of a positive integer $n$ is an $n$-tuple of nonnegative integers $P=(x_1,x_2,\cdots,x_n)$ such that $\sum_{k=1}^n kx_k=n$. For positive integer $m\leq n$, and a partition $Q=(y_1,y_2,\cdots,y_m)$ of $m$, $Q$ is called compatible to $P$ if $y_i\leq x_i$ for $i=1,2,\cdots,m$. Let $S(n)$ be the number of partitions $P$ of $n$ such that for each odd $m<n$, $m$ has exactly one partition compatible to $P$ and for each even $m<n$, $m$ has exactly two partitions compatible to $P$. Find $S(2010)$.

In isosceles trapezoid $ABCD$ with $AB < CD$ and $BC = AD$, the angle bisectors of $\angle A$ and $\angle B$ intersect $CD$ at $E$ and $F$ respectively, and intersect each other outside the trapezoid at $G$. Given that $AD = 8$, $EF = 3$, and $EG = 4$, the area of $ABCD$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b$, and $c$, with $a$ and $c$ relatively prime and $b$ squarefree. Find $10000a +100b +c$.

Four segments divide a convex quadrilateral into nine quadrilaterals. The points of intersections of these segments lie on the diagonals of the quadrilateral (see figure). It is known that the quadrilaterals 1, 2, 3, 4 admit inscribed circles. Prove that the quadrilateral 5 also has an inscribed circle. [asy] pair A,B,C,D,E,F,G,H,I,J,K,L; A=(-4.0,4.0);B=(-1.06,4.34);C=(-0.02,4.46);D=(4.14,4.93);E=(3.81,0.85);F=(3.7,-0.42); G=(3.49,-3.05);H=(1.37,-2.88);I=(-1.46,-2.65);J=(-2.91,-2.52);K=(-3.14,-1.03);L=(-3.61,1.64); draw(A--D);draw(D--G);draw(G--J);draw(J--A); draw(A--G);draw(D--J); draw(B--I);draw(C--H);draw(E--L);draw(F--K); pair R,S,T,U,V; R=(-2.52,2.56);S=(1.91,2.58);T=(-0.63,-0.11);U=(-2.37,-1.94);V=(2.38,-2.06); label("1",R,N);label("2",S,N);label("3",T,N);label("4",U,N);label("5",V,N); [/asy] [i]Proposed by Nairi M. Sedrakyan, Armenia[/i]

Let $O$ be the circumcenter of a triangle $ABC$. The projections of points $D$ and $X$ to the sidelines of the triangle lie on lines $\ell $ and $L $ such that $\ell // XO$. Prove that the angles formed by $L$ and by the diagonals of quadrilateral $ABCD$ are equal.

$f(n,k)$ is defined by (1) $f(n,0) = f(n,n) = 1$ and (2) $f(n,k) = f(n-1,k-1) + f(n-1,k)$ for $0 < k < n$. How many times do we need to use (2) to find $f(3991,1993)$?

For how many ordered triplets $(a,b,c)$ of positive integers less than $10$ is the product $a\times b\times c$ divisible by $20$?

Rectangular sheet of paper $ABCD$ is folded as shown in the figure. Find the rato $DK: AB$, given that $C_1$ is the midpoint of $AD$. [img]https://3.bp.blogspot.com/-9EkSdxpGnPU/W6dWD82CxwI/AAAAAAAAJHw/iTkEOejlm9U6Dbu427vUJwKMfEOOVn0WwCK4BGAYYCw/s400/Yasinsky%2B2017%2BVIII-IX%2Bp1.png[/img]

Let $ABC$ be a triangle with $\angle BAC = 90^o$. Angle bisector of the $\angle CBA$ intersects the segment $(AB)$ at point $E$. If there exists $D \in (CE)$ so that $\angle DAC = \angle BDE =x^o$ , calculate $x$.

Determine all functions $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ such that \[ f(x,y) + f(y,z) + f(z,x) = \max \{ x,y,z \} - \min \{ x,y,z \} \] for every $x,y,z \in \mathbb{R}$ and there exists some real $a$ such that $f(x,a) = f(a,x) $ for every $x \in \mathbb{R}$.

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

Let $ x $ be a real number such that $ x^3 $ and $ x^2+x $ are rational numbers. Prove that $ x $ is rational.

Evaluate $\displaystyle\lim_{x\to 1}x^{\dfrac{x}{\sin(1-x)}}$.

Find all functions $f : Z^+ \to Z^+$ satisfying $f (1) = 2, f (2) \ne 4$, and max $\{f (m) + f (n), m + n\} |$ min $\{2m + 2n, f (m + n) + 1\}$ for all $m, n \in Z^+$.

Let $S$ be the set $\{-1, 1\}^n$, that is, $n$-tuples such that each coordinate is either $-1$ or $1$. For \[s = (s_1, s_2, \ldots, s_n), t = (t_1, t_2, \ldots, t_n) \in \{-1, 1\}^n,\] define $s \odot t = (s_1t_1, s_2t_2, \ldots, s_nt_n)$. Let $c$ be a positive constant, let $f : S \to \{-1, 1\}$ be a function such that there are at least $(1-c) \cdot 2^{2n}$ pairs $(s, t)$ with $s, t \in S$ such that $f(s \odot t) = f(s)f(t)$. Show that there exists a function $f'$ such that $f'(s \odot t) = f'(s)f'(t)$ for all $s, t \in S$ and $f(s) = f'(s)$ for at least $(1-10c) \cdot 2^n$ values of $s \in S$.

Let $a,b,c$ be positive real numbers. Prove that $$\frac{a}{\sqrt{(2a+b)(2a+c)}} +\frac{b}{\sqrt{(2b+c)(2b+a)}} +\frac{c}{\sqrt{(2c+a)(2c+b)}} \le 1 $$

A sequence is defined as follows $a_1=a_2=a_3=1$, and, for all positive integers $n$, $a_{n+3}=a_{n+2}+a_{n+1}+a_n$. Given that $a_{28}=6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\displaystyle \sum^{28}_{k=1} a_k$ is divided by 1000.

Seven squares are arranged to form a rectangle as shown below. The side length of the smallest square is $3$ cm. What is the perimeter in centimeters of the rectangle formed by the $7$ squares? [asy] draw((0,0)--(57,0)--(57,63)--(0,63)--cycle); draw((12,27)--(12,39)); draw((24,27)--(24,63)); draw((27,0)--(27,30)); draw((0,27)--(27,27)); draw((24,30)--(57,30)); draw((0,39)--(24,39)); [/asy]

There are $1988$ towns and $4000$ roads in a certain country (each road connects two towns) . Prove that there is a closed path passing through no more than $20$ towns. (A. Razborov , Moscow)

$f(x),g(x),h(x)$ are square trinomials with discriminant, that equals $2$. And $f(x)+g(x),f(x)+h(x),g(x)+h(x)$ are square trinomials with discriminant, that equals $1$. Prove,that $f(x)+g(x)+h(x)$ has not roots.