Let $\omega_1$ and $\omega_2$ be two circles that intersect at two points: $A$ and $B$. Let $C$ and $E$ be on $\omega_1$, and $D$ and $F$ be on $\omega_2$ such that $CD$ and $EF$ meet at $B$ and the three lines $CE$, $DF$, and $AB$ concur at a point $P$ that is closer to $B$ than $A$. Let $\Omega$ denote the circumcircle of $\triangle DEF$. Now, let the line through $A$ perpendicular to $AB$ hit $EB$ at $G$, $GD$ hit $\Omega$ at $J$, and $DA$ hit $\Omega$ again at $I$. A point $Q$ on $IE$ satisfies that $CQ=JQ$. If $QJ=36$, $EI=21$, and $CI=16$, then the radius of $\Omega$ can be written as $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers, $b$ is not divisible by the square of a prime, and $\gcd(a, c) = 1$. Find $a+b+c$. [i]Proposed by Kevin Zhao[/i]

Find all positive integer $k$s for which such $f$ exists and unique: $f(mn)=f(n)f(m)$ for $n, m \in \mathbb{Z^+}$ $f^{n^k}(n)=n$ for all $n \in \mathbb{Z^+}$ for which $f^x (n)$ means the n times operation of function $f$(i.e. $f(f(...f(n))...)$)

Let $a, b$, and $c$ be integers that satisfy $2a + 3b = 52$, $3b + c = 41$, and $bc = 60$. Find $a + b + c$

Let $n\ge 2$ be an integer. Thibaud the Tiger lays $n$ $2\times 2$ overlapping squares out on a table, such that the centers of the squares are equally spaced along the line $y=x$ from $(0,0)$ to $(1,1)$ (including the two endpoints). For example, for $n=4$ the resulting figure is shown below, and it covers a total area of $\frac{23}{3}$. [asy] fill((0,0)--(2,0)--(2,.333333333333)--(0.333333333333,0.333333333333)--(0.333333333333,2)--(0,2)--cycle, lightgrey); fill((0.333333333333,0.333333333333)--(2.333333333333,0.333333333333)--(2.333333333333,.6666666666666)--(0.666666666666,0.666666666666666)--(0.66666666666,2.33333333333)--(.333333333333,2.3333333333333)--cycle, lightgrey); fill((0.6666666666666,.6666666666666)--(2.6666666666666,.6666666666)--(2.6666666666666,.6666666666666)--(2.6666666666666,1)--(1,1)--(1,2.6666666666666)--(0.6666666666666,2.6666666666666)--cycle, lightgrey); fill((1,1)--(3,1)--(3,3)--(1,3)--cycle, lightgrey); draw((0.33333333333333,2)--(2,2)--(2,0.333333333333), dashed+grey+linewidth(0.4)); draw((0.66666666666666,2.3333333333333)--(2.3333333333333,2.3333333333333)--(2.3333333333333,0.66666666666), dashed+grey+linewidth(0.4)); draw((1,2.666666666666)--(2.666666666666,2.666666666666)--(2.666666666666,1), dashed+grey+linewidth(0.4)); draw((0,0)--(2,0)--(2,.333333333333)--(0.333333333333,0.333333333333)--(0.333333333333,2)--(0,2)--(0,0),linewidth(0.4)); draw((0.333333333333,0.333333333333)--(2.333333333333,0.333333333333)--(2.333333333333,.6666666666666)--(0.666666666666,0.666666666666666)--(0.66666666666,2.33333333333)--(.333333333333,2.3333333333333)--(0.333333333333,.333333333333),linewidth(0.4)); draw((0.6666666666666,.6666666666666)--(2.6666666666666,.6666666666)--(2.6666666666666,.6666666666666)--(2.6666666666666,1)--(1,1)--(1,2.6666666666666)--(0.6666666666666,2.6666666666666)--(0.6666666666666,0.6666666666666),linewidth(0.4)); draw((1,1)--(3,1)--(3,3)--(1,3)--cycle,linewidth(0.4)); [/asy] Find, with proof, the minimum $n$ such that the figure covers an area of at least $\sqrt{63}$.

Let $x_0 = x_{101} = 0$. The numbers $x_1, x_2,...,x_{100}$ are chosen at random from the interval $[0, 1]$ uniformly and independently. Compute the probability that $2x_i \ge x_{i-1} + x_{i+1}$ for all $i = 1, 2,..., 100.$

Real numbers $x,y,z$ satisfy the inequalities $$x^2\le y+z,\qquad y^2\le z+x\qquad z^2\le x+y.$$Find the minimum and maximum possible values of $z$.

Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

Does there exist positive integer $n$ such that sum of all digits of number $n(4n+1)$ is equal to $2017$

Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$. $\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$

Let $a, b, c, d$ be natural numbers such that $a + b + c + d = 2018$. Find the minimum value of the expression: $$E = (a-b)^2 + 2(a-c)^2 + 3(a-d)^2+4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2.$$

Prove that you can pick $15$ distinct positive integers between $1$ and $2023$, such that each one of them and the sum between some of them is never a perfect square, nor a perfect cube or any other greater perfect power.

The subsets $A_1,A_2,\ldots ,A_{2000}$ of a finite set $M$ satisfy $|A_i|>\frac{2}{3}|M|$ for each $i=1,2,\ldots ,2000$. Prove that there exists $m\in M$ which belongs to at least $1334$ of the subsets $A_i$.

Let $X$ and $Y$ be two finite subsets of the half-open interval $[0, 1)$ such that $0 \in X \cap Y$ and $x + y = 1$ for no $x \in X$ and no $y \in Y$. Prove that the set $\{x + y - \lfloor x + y \rfloor : x \in X \textrm{ and } y \in Y\}$ has at least $|X| + |Y| - 1$ elements. [i]***[/i]

The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation.

On a circle there are $2n+1$ points, dividing it into equal arcs ($n\ge 2$). Two players take turns to erase one point. If after one player's turn, it turned out that all the triangles formed by the remaining points on the circle were obtuse, then the player wins and the game ends. Who has a winning strategy: the starting player or his opponent?

A sequence of real numbers $\{a_n\}_{n = 1}^\infty (n=1,2,...)$ has the following property: \begin{align*} 6a_n+5a_{n-2}=20+11a_{n-1}\ (\text{for }n\geq3). \end{align*} The first two elements are $a_1=0, a_2=1$. Find the integer closest to $a_{2011}$.

Points $A'$ and $B'$ lie inside the parallelogram $ABCD$ and points $C'$ and $D'$ lie outside of it, so that all sides of 8-gon $AA'BB'CC'DD'$ are equal. Prove that $A'$, $B'$, $C'$, $D'$ are concyclic.

If for the coefficients of equation $x^3+ax^2+bx+c=0$ whose roots are all real, holds, $a^2= 2(b+1)$ then $|a-c|\le 2$. Prove it.

Let $M$ be the midpoint of side $BC$ of an equilateral triangle $ABC$. The point $D$ is on $CA$ extended such that $A$ is between $D$ and $C$. The point $E$ is on $AB$ extended such that $B$ is between $A$ and $E$, and $|MD| = |ME|$. The point $F$ is the intersection of $MD$ and $AB$. Prove that $\angle BFM = \angle BME$.

Let $f_1, f_2, \ldots$ be continuous real functions on the real line. Is it true that if the series $\sum_{n=1}^{\infty} f_n(x)$ is divergent for every $x$, then this holds also true for any typical choice of the signs in the sum (i.e. the set of those $\{ \epsilon _n\}_{n=1}^{\infty} \in \{ +1, -1\}^{\mathbb{N}}$ sequences, for which there series $\sum_{n=1}^{\infty} \epsilon_nf_n(x)$ is convergent at least at one point $x$, forms a subset of first category within the set $\{+1,-1\}^{\mathbb{N}} $)? (translated by L. Erdős)

Let $ABC$ be a an acute triangle. Points $A_1, B_1$ and $C_1$ are respectively the projections of the vertices $A, B$ and $C$ on the opposite sides of the triangle, the point $H$ is the orthocenter of the triangle, and the point $P$ is the middle of the segment $[AH]$. The lines $BH$ and $A_1C_1$, $P B_1$ and $AB$ intersect respectively at the points $M$ and $N$. Prove that the lines $MN$ and $BC$ are perpendicular.

A lattice point is a point on the coordinate plane with integer coefficients. Prove or disprove : there exists a finite set $S$ of lattice points such that for every line $l$ in the plane with slope $0,1,-1$, or undefined, either $l$ and $S$ intersect at exactly $2022$ points, or they do not intersect.

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. [asy]unitsize(.3cm); defaultpen(linewidth(.8pt)); path c=Circle((0,2),1); filldraw(Circle((0,0),3),grey,black); filldraw(Circle((0,0),1),white,black); filldraw(c,white,black); filldraw(rotate(60)*c,white,black); filldraw(rotate(120)*c,white,black); filldraw(rotate(180)*c,white,black); filldraw(rotate(240)*c,white,black); filldraw(rotate(300)*c,white,black);[/asy]$ \textbf{(A)}\ \pi \qquad \textbf{(B)}\ 1.5\pi \qquad \textbf{(C)}\ 2\pi \qquad \textbf{(D)}\ 3\pi \qquad \textbf{(E)}\ 3.5\pi$

Three circular arcs $\gamma_1, \gamma_2,$ and $\gamma_3$ connect the points $A$ and $C.$ These arcs lie in the same half-plane defined by line $AC$ in such a way that arc $\gamma_2$ lies between the arcs $\gamma_1$ and $\gamma_3.$ Point $B$ lies on the segment $AC.$ Let $h_1, h_2$, and $h_3$ be three rays starting at $B,$ lying in the same half-plane, $h_2$ being between $h_1$ and $h_3.$ For $i, j = 1, 2, 3,$ denote by $V_{ij}$ the point of intersection of $h_i$ and $\gamma_j$ (see the Figure below). Denote by $\widehat{V_{ij}V_{kj}}\widehat{V_{kl}V_{il}}$ the curved quadrilateral, whose sides are the segments $V_{ij}V_{il},$ $V_{kj}V_{kl}$ and arcs $V_{ij}V_{kj}$ and $V_{il}V_{kl}.$ We say that this quadrilateral is $circumscribed$ if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals $\widehat{V_{11}V_{21}}\widehat{V_{22}V_{12}}, \widehat{V_{12}V_{22}}\widehat{V_{23}V_{13}},\widehat{V_{21}V_{31}}\widehat{V_{32}V_{22}}$ are circumscribed, then the curved quadrilateral $\widehat{V_{22}V_{32}}\widehat{V_{33}V_{23}}$ is circumscribed, too. [i]Proposed by Géza Kós, Hungary[/i] [asy] pathpen=black; size(400); pair A=(0,0), B=(4,0), C=(10,0); draw(L(A,C,0.3)); MP("A",A); MP("B",B); MP("C",C); pair X=(5,-7); path G1=D(arc(X,C,A)); pair Y=(5,7), Z=(9,6); draw(Z--B--Y); struct T {pair C;real r;}; T f(pair X, pair B, pair Y, pair Z) { pair S=unit(Y-B)+unit(Z-B); real s=abs(sin(angle((Y-B)/(Z-B))/2)); real t=10, r=abs(X-A); pair Q; for(int k=0;k<30;++k) { Q=B+t*S; t-=(abs(X-Q)-r)/abs(S)-s*t; } T T=new T; T.C=Q; T.r=s*t*abs(S); return T; } void g(pair Q, real r) { real t=0; for(int k=0;k<30;++k) { X=(5,t); t+=(abs(X-Q)+r-abs(X-A)); } } pair Z1=(1.07,6); draw(B--Z1); T T=f(X,B,Y,Z1); draw(CR(T.C,T.r)); T T=f(X,B,Y,Z); draw(CR(T.C,T.r)); g(T.C,T.r); path G2=D(arc(X,C,A)); T T=f(X,B,Y,Z1); draw(CR(T.C,T.r)); T=f(X,B,Y,Z); draw(CR(T.C,T.r)); g(T.C,T.r); path G3=D(arc(X,C,A)); pen p=black+fontsize(8); MC("\gamma_1",G1,0.85,p); MC("\gamma_2",G2,0.85,NNW,p); MC("\gamma_3",G3,0.85,WNW,p); MC("h_1",B--Z1,0.95,E,p); MC("h_2",B--Y,0.95,E,p); MC("h_3",B--Z,0.95,E,p); path[] G={G1,G2,G3}; path[] H={B--Z1,B--Y,B--Z}; pair[][] al={{S+SSW,S+SSW,3*S},{SE,NE,NW},{2*SSE,2*SSE,2*E}}; for(int i=0;i<3;++i) for(int j=0;j<3;++j) MP("V_{"+string(i+1)+string(j+1)+"}",IP(H[i],G[j]),al[i][j],fontsize(8));[/asy]

Find all triples of real numbers $(a, b, c)$ satisfying $$a+b+c=14, \quad a^2+b^2+c^2=84,\quad a^3+b^3+c^3=584.$$