Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[ \angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.

Consider the infinite, strictly increasing sequence of positive integer $(a_n)$ such that i. All terms of sequences are pairwise coprime. ii. The sum $\frac{1}{\sqrt{a_1a_2}} +\frac{1}{\sqrt{a_2a_3}}+ \frac{1}{\sqrt{a_3a_4}} + ..$ is unbounded. Prove that this sequence contains infinitely many primes.

Suppose $a$, $b$, and $c$ are positive integers such that \[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\] Which of the following statements are necessarily true? I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$. II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both. III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$. $\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}$

On a party, there are 6 boys and a number of girls. Two of the girls know exactly four boys each and the remaining girls know exactly two boys each. None of the boys know more than three girls. (We assume that if $ A$ knows $ B$, then $ B$ will also know $ A$). Then, the greatest possible number of girls on the party is $ \text{(A)}\ 6 \qquad \text{(B)}\ 7 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ \text{10 or more}$

Let $ S$ be a set of all points of a plane whose coordinates are integers. Find the smallest positive integer $ k$ for which there exists a 60-element subset of set $ S$ with the following condition satisfied for any two elements $ A,B$ of the subset there exists a point $ C$ contained in $ S$ such that the area of triangle $ ABC$ is equal to k .

Let $n$, $(n \geq3)$ be a positive integer and the polynomial $f(x)=(1+x) \cdot (1+2x) \cdot (1+3x) \cdot ... \cdot (1+nx)$ $= a_0+a_1 \cdot x+a_2 \cdot x^2+a_3 \cdot x^3+...+a_n \cdot x^n$. Show that the number $a_3$ divides the number $k=C^2_{n+1} \cdot (2 \cdot C^2_n \cdot C^2_{n+1}-3 \cdot a_2).$

Find all integer triples $(p,m,n)$ that satisfy: $p^m-n^3=27$ where $p$ is a prime number.

Let $n$ be a positive integer, and let $1=d_1<d_2<\dots < d_k=n$ denote all positive divisors of $n$, If the following conditions are satisfied: $$ 2d_2+d_4+d_5=d_7$$ $$ d_3 d_6 d_7=n$$ $$ (d_6+d_7)^2=n+1$$ find all possible values of $n$.

Suppose that a function $f : R^+ \to R^+$ satisfies $$f(f(x))+x = f(2x).$$ Prove that $f(x) \ge x$ for all $x >0$

Pasha and Vova play the game crossing out the cells of the $3\times 101$ board by turns. At the start, the central cell is crossed out. By one move the player chooses the diagonal (there can be $1, 2$ or $3$ cells in the diagonal) and crosses out cells of this diagonal which are still uncrossed. At least one new cell must be crossed out by any player's move. Pasha begins, the one who can not make any move loses. Who has a winning strategy?

Consider the transformation $$T(x,y,z) = (\sin y + \sin z - \sin x,\sin z + \sin x - \sin y,\sin x +\sin y -\sin z).$$ Determine all the points $(x,y,z) \in [0,1]^3$ such that $T^n(x,y,z) \in [0,1]^3,$ for every $n \geq 1$.

Let $ ABC$ be a triangle. Points $ A',B',C'$ are on the sides $ BC, CA, AB,$ respectively such that we have \[ \overline{A'B'} \equal{} \overline{B'C'} \equal{} \overline{C'A'}\] and \[ \overline{AB'} \equal{} \overline{BC'} \equal{} \overline{CA'}.\] Prove that triangle $ ABC$ is equilateral.

The value of $\frac{(487,000)(12,027,300)+(9,621,001)(487,000)}{(19,367)(.05)}$ is closest to $\text{(A)}\ 10,000,000 \qquad \text{(B)}\ 100,000,000 \qquad \text{(C)}\ 1,000,000,000 \\ \text{(D)}\ 10,000,000,000 \qquad \text{(E)}\ 100,000,000,000$

Prove that if two segments of a tetrahedron, going from the ends of some edge to the centers of the inscribed circles of opposite faces, intersect, then the segments issued from the ends of the crossing with it edges to the centers of the inscribed circles of the other two faces, also intersect.

Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. Let $m$ be a positive integer, $m\geq 3$. For every integer $i$ with $1\leq i\leq m$, let \[S_{m,i}=\left\{\left\lfloor\dfrac{2^m-1}{2^{i-1}}n-2^{m-i}+1\right\rfloor\,:\,n=1,2,3,\ldots\right\}.\] For example, for $m=3$, \begin{align*}S_{3,1}&=\{\lfloor 7n-3\rfloor\,:\,n=1,2,3,\ldots\} \\&=\{4,11,18,\ldots\}, \\S_{3,2}&=\left\{\left\lfloor\dfrac72n-1\right\rfloor\,:\,n=1,2,3,\ldots\right\} \\&=\{2,6,9,\ldots\}, \\S_{3,3}&=\left\{\left\lfloor\dfrac74n\right\rfloor\,:\,n=1,2,3,\ldots\right\} \\&=\{1,3,5,\ldots\}.\end{align*} Prove that for all $m\geq 3$, each positive integer occurs in exactly one of the sets $S_{m,i}$.

Find all function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for all $a,b\in\mathbb{N}$ , $(f(a)+b) f(a+f(b))=(a+f(b))^2$

Determine the maximum value of $x^2 y^2 z^2 w$ for $\{x,y,z,w\}\in\mathbb{R}^{+} \cup\{0\}$ and $2x+xy+z+yzw=1$.

$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.

Carl is given three distinct non-parallel lines $\ell_1, \ell_2, \ell_3$ and a circle $\omega$ in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line $\ell$ and a point $P$, constructs a new line passing through $P$ parallel to $\ell$. (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle $\omega$ whose sides are parallel to $\ell_1,\ell_2,\ell_3$ in some order. [i]Proposed by Vincent Huang[/i]

A spacecraft landed on an asteroid. It is known that the asteroid is either a ball or a cube. The rover started its route at the landing site and finished it at the point symmetric to the landing site with respect to the center of the asteroid. On its way, the rover transmitted its spatial coordinates to the spacecraft on the landing site so that the trajectory of the rover movement was known. Can it happen that this information is not suffcient to determine whether the asteroid is a ball or a cube?

In the number $74982.1035$ the value of the ''place'' occupied by the digit 9 is how many times as great as the value of the ''place'' occupied by the digit 3? $\textbf{(A)}\ 1,000 \qquad \textbf{(B)}\ 10,000 \qquad \textbf{(C)}\ 100,000 \qquad \textbf{(D)}\ 1,000,000 \qquad \textbf{(E)}\ 10,000,000$

In each unit cell of a finite set of cells of an infinite checkered board, an integer is written so that the sum of the numbers in each row, as well as in each column, is divided by $2002$. Prove that every number $\alpha$ can be replaced by a certain number $\alpha'$ , divisible by $2002$ so that $|\alpha-\alpha'| <2002$ and the sum of the numbers in all rows, and in all columns will not change.

Find the limit \[\lim_{n\to \infty}\frac{\sqrt[n+1]{(2n+3)(2n+4)\ldots (3n+3)}}{n+1}\]

The game involves two players $A$ and $B$. Player $A$ sets the value of one of the coefficients $a, b$ or $c$ of the polynomial $$x^3 + ax^2 + bx + c.$$ Player $B$ indicates the value of any of the two remaining coefficients . Player $A$ then sets the value of the last coefficients. Is there a strategy for player A such that no matter how player $B$ plays, the equation $$x^3 + ax^2 + bx + c = 0$$ to have three different (real) solutions?

Given a triangle $ABC$ with $BC = 5$, $AC = 7$, and $AB = 8$, find the side length of the largest equilateral triangle $P QR$ such that $A, B, C$ lie on $QR$, $RP$, $P Q$ respectively.