A point $B$ lies on a chord $AC$ of circle $\omega.$ Segments $AB$ and $BC$ are diameters of circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ respectively. These circles intersect $\omega$ for the second time in points $D$ and $E$ respectively. The rays $O_1D$ and $O_2E$ meet in a point $F,$ and the rays $AD$ and $CE$ do in a point $G.$ Prove that the line $FG$ passes through the midpoint of the segment $AC.$

Prove that if $h_1,h_2,h_3,h_4$ are the altitudes of a tetrahedron and $d_1,d_2,d_3$ the distances between the pairs of opposite edges of the tetrahedron, then $$\frac{1}{h_1^2} +\frac{1}{h_2^2} +\frac{1}{h_3^2} +\frac{1}{h_4^2} =\frac{1}{d_1^2} +\frac{1}{d_2^2} +\frac{1}{d_3^2}.$$

We call the sum of any $k$ of $n$ given numbers (with distinct indices) a $k$-sum. Given $n$, find all $k$ such that, whenever more than half of $k$-sums of numbers $a_{1},a_{2},...,a_{n}$ are positive, the sum $a_{1}+a_{2}+...+a_{n}$ is positive as well.

Determine all polynomials $P(x)$ with real coeffcients such that $(x^3+3x^2+3x+2)P(x-1)=(x^3-3x^2+3x-2)P(x)$.

Let $n$ be a positive integer. Let $S$ be a subset of points on the plane with these conditions: $i)$ There does not exist $n$ lines in the plane such that every element of $S$ be on at least one of them. $ii)$ for all $X \in S$ there exists $n$ lines in the plane such that every element of $S - {X} $ be on at least one of them. Find maximum of $\mid S\mid$. [i]Proposed by Erfan Salavati[/i]

What is the minimum value of \[(x^2+2x+8-4\sqrt{3})\cdot(x^2-6x+16-4\sqrt{3})\] where $x$ is a real number? $ \textbf{(A)}\ 112-64\sqrt{3} \qquad\textbf{(B)}\ 3-\sqrt{3} \qquad\textbf{(C)}\ 8-4\sqrt{3} \\ \textbf{(D)}\ 3\sqrt{3}-4 \qquad\textbf{(E)}\ \text{None of the preceding} $

$M = (a_{i,j} ), \ i, j = 1, 2, 3, 4$, is a square matrix of order four. Given that: [list] [*] [b](i)[/b] for each $i = 1, 2, 3,4$ and for each $k = 5, 6, 7$, \[a_{i,k} = a_{i,k-4};\]\[P_i = a_{1,}i + a_{2,i+1} + a_{3,i+2} + a_{4,i+3};\]\[S_i = a_{4,i }+ a_{3,i+1} + a_{2,i+2} + a_{1,i+3};\]\[L_i = a_{i,1} + a_{i,2} + a_{i,3} + a_{i,4};\]\[C_i = a_{1,i} + a_{2,i} + a_{3,i} + a_{4,i},\] [*][b](ii)[/b] for each $i, j = 1, 2, 3, 4$, $P_i = P_j , S_i = S_j , L_i = L_j , C_i = C_j$, and [*][b](iii)[/b] $a_{1,1} = 0, a_{1,2} = 7, a_{2,1} = 11, a_{2,3} = 2$, and $a_{3,3} = 15$.[/list] find the matrix M.

What is the tens digit of the sum \[\left(1!\right)^2+\left(2!\right)^2+\left(3!\right)^2+\ldots+\left(2018!\right)^2?\] [i]2018 CCA Math Bonanza Individual Round #1[/i]

Define the sequence $(a_n)$ as $a_1 = 1$, $a_{2n} = a_n$ and $a_{2n+1} = a_n + 1$ for all $n\geq 1$. a) Find all positive integers $n$ such that $a_{kn} = a_n$ for all integers $1 \leq k \leq n$. b) Prove that there exist infinitely many positive integers $m$ such that $a_{km} \geq a_m$ for all positive integers $k$.

A parallelogram is inscribed in a quadrilateral, two opposite vertices of which are the midpoints of the opposite sides of the quadrilateral. Determine the area of ​​such a parallelogram if the area of ​​the quadrilateral is equal to $S_o$.

Determine all positive integers that cannot be written as $\frac{a}{b} + \frac{a+1}{b+1}$ where $a$ and $b$ are positive integers.

Ankit, Box, and Clark are taking the tiebreakers for the geometry round, consisting of three problems. Problem $k$ takes each $k$ minutes to solve. If for any given problem there is a $\frac13$ chance for each contestant to solve that problem first, what is the probability that Ankit solves a problem first?

[b]Q[/b]. $ABC$ is an acute - angled triangle with $\odot(ABC)$ and $\Omega$ as the circumcircle and incircle respectively. Let $D, E, F$ to be the respective intouch points on $\overline{BC}, \overline{CA}$ and $\overline{AB}$. Circle $\gamma_A$ is drawn internally tangent to sides $\overline{AC}, \overline{AB}$ and $\odot(ABC)$ at $X, Y$ and $Z$ respectively. Another circle $(\omega)$ is constructed tangent to $\overline{BC}$ at $\mathcal{T}_1$ and internally tangent to $\odot(ABC)$ at $\mathcal{T}_2$. A tangent is drawn from $A$ such that it touches $\omega$ at $W$ and meets $BC$ at $V$, with $V$ lying inside $\odot(ABC)$. Now if $\overline{EF}$ meets $\odot(BC)$ at $\mathcal{X}_1$ and $\mathcal{X}_2$, opposite to vertex $B$ and $C$ respectively, where $\odot(BC)$ denotes the circle with $BC$ as diameter, prove that the set of lines $\{\overline{B\mathcal{X}_1}, \overline{ZS}, \overline{C\mathcal{X}_2}, \overline{DU}, \overline{YX}, \overline{\mathcal{T}_1W} \}$ are concurrent where $S$ is the mid-point of $\widehat{BC}$ containing $A$ and $U$ is the anti-pode of $D$ with respect to $\Omega$. If the line joining that concurrency point and $A$ meets $\odot(ABC)$ at $N\not = A$ prove that $\overline{AD}, \overline{ZN}$ and $\gamma_A$ pass through a common point. [i] Proposed by srijonrick[/i]

Inside the inscribed quadrilateral $ABCD$ there is a point $K$, the distances from which to the sides $ABCD$ are proportional to these sides. Prove that $K$ is the intersection point of the diagonals of $ABCD$.

Let $x, y$ be distinct positive real numbers satisfying $$\frac{1}{\sqrt{x + y} -\sqrt{x - y}}+\frac{1}{\sqrt{x + y} +\sqrt{x - y}} =\frac{x}{\sqrt{y^3}}.$$ If $\frac{x}{y} =\frac{a+\sqrt{b}}{c}$ for positive integers $a, b, c$ with $gcd (a, c) = 1$, find $a + b + c$.

Find all pairs $(p,n)$ of integers so that $p$ is a prime and there exists $x,y\not\equiv0\pmod p$ with $$x^2+y^2\equiv n\pmod p.$$

Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

Find all pairs $ (p, q)$ of prime numbers such that $ p$ divides $ 5^q \plus{} 1$ and $ q$ divides $ 5^p \plus{} 1$.

\[\int_1^\infty \frac{\lfloor x^2\rfloor}{x^5}\mathrm dx\] [i]Proposed by Robert Trosten[/i]

If $n\heartsuit m=n^3m^2$, what is $\frac{2\heartsuit 4}{4\heartsuit 2}$? $\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4$

Let $P_n \ (n=3,4,5,6,7)$ be the set of positive integers $n^k+n^l+n^m$, where $k,l,m$ are positive integers. Find $n$ such that: i) In the set $P_n$ there are infinitely many squares. ii) In the set $P_n$ there are no squares.

Prove that if $n$ is a natural number such that both $3n+1$ and $4n+1$ are squares, then $n$ is divisible by $56$.

In the triangle $ABC$ the incircle $J$ touches the sides $BC$, $CA$, $AB$ in $D$, $E$, $F$, respectively. The segments $(BE)$ and $(CF)$ intersect $J$ in $G,H$. If $B$ and $C$ are fixed points, find the loci of points $A, D, E, F, G, H$ if $GH \parallel BC$ and the loci of the same points if $BCHG$ is an inscriptible quadrilateral.

[b] Problem 6.[/b] Let $m\geq 5$ and $n$ are given natural numbers, and $M$ is regular $2n+1$-gon. Find the number of the convex $m$-gons with vertices among the vertices of $M$, who have at least one acute angle. [i]Alexandar Ivanov[/i]

Determine all triples $(a,b,c)$ of positive integers satisfying the conditions $$\gcd(a,20) = b$$ $$\gcd(b,15) = c$$ $$\gcd(a,c) = 5$$ (Richard Henner)