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.

Let $ABCDA_1B_1C_1D_1$ be a cube and $P$ a variable point on the side $[AB]$. The perpendicular plane on $AB$ which passes through $P$ intersects the line $AC'$ in $Q$. Let $M$ and $N$ be the midpoints of the segments $A'P$ and $BQ$ respectively. a) Prove that the lines $MN$ and $BC'$ are perpendicular if and only if $P$ is the midpoint of $AB$. b) Find the minimal value of the angle between the lines $MN$ and $BC'$.

Let $O$ be the center of a circle. Let $OU,OV$ be perpendicular radii of the circle. The chord $PQ$ passes through the midpoint $M$ of $UV$. Let $W$ be a point such that $PM = PW$, where $U, V,M,W$ are collinear. Let $R$ be a point such that $PR = MQ$, where $R$ lies on the line $PW$. Prove that $MR = UV$. [u]Alternative version:[/u] A circle $S$ is given with center $O$ and radius $r$. Let $M$ be a point whose distance from $O$ is $\frac{r}{\sqrt{2}}$. Let $PMQ$ be a chord of $S$. The point $N$ is defined by $\overrightarrow{PN} =\overrightarrow{MQ}$. Let $R$ be the reflection of $N$ by the line through $P$ that is parallel to $OM$. Prove that $MR =\sqrt{2}r$.

Consider triangle $ABC$ with $AB<AC<BC$, inscribed in triangle $\Gamma_1$ and the circles $\Gamma_2 (B,AC)$ and $\Gamma_2 (C,AB)$. A common point of circle $\Gamma_2$ and $\Gamma_3$ is point $E$, a common point of circle $\Gamma_1$ and $\Gamma_3$ is point $F$ and a common point of circle $\Gamma_1$ and $\Gamma_2$ is point $G$, where the points $E,F,G$ lie on the same semiplane defined by line $BC$, that point $A$ doesn't lie in. Prove that circumcenter of triangle $EFG$ lies on circle $\Gamma_1$. Note: By notation $\Gamma (K,R)$, we mean random circle $\Gamma$ has center $K$ and radius $R$.

Let $a$ be negative constant. Find the value of $a$ and $f(x)$ such that $\int_{\frac{a}{2}}^{\frac{t}{2}} f(x)dx=t^2+3t-4$ holds for any real numbers $t$.

Given the set of $ n$ numbers; $ n > 1$, of which one is $ 1 \minus{} \frac {1}{n}$ and all the others are $ 1.$ The arithmetic mean of the $ n$ numbers is: $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ n \minus{} \frac {1}{n} \qquad \textbf{(C)}\ n \minus{} \frac {1}{n^2} \qquad \textbf{(D)}\ 1 \minus{} \frac {1}{n^2} \qquad \textbf{(E)}\ 1 \minus{} \frac {1}{n} \minus{} \frac {1}{n^2}$

Natural numbers from $1$ to $400$ are divided in $100$ disjoint sets. Prove that one of the sets contains three numbers which are lengths of a non-degenerate triangle's sides.

Find all functions $f:\mathbb{R} \to \mathbb{R}$, such that for any $x, y \in \mathbb{R}$ holds the following: $$f(x)f(yf(x)) + yf(xy) = xf(xy) + y^2f(x)$$ [i]Proposed by Mykhailo Shtandenko[/i]

Let points $P$ and $Q$ be isogonally conjugated with respect to a triangle $ABC$. The line $PQ$ meets the circumcircle of $ABC$ at point $X$. The reflection of $BC$ about $PQ$ meets $AX$ at point $E$. Prove that $A, P, Q, E$ are concyclic.

Find all pairs of $p,q$ prime numbers that satisfy the equation $$p(p^4+p^2+10q)=q(q^2+3)$$

If $ a$ and $ b$ are positive and $ a\not\equal{} 1,\,b\not\equal{} 1$, then the value of $ b^{\log_b{a}}$ is: $ \textbf{(A)}\ \text{dependent upon }{b} \qquad \textbf{(B)}\ \text{dependent upon }{a}\qquad \textbf{(C)}\ \text{dependent upon }{a}\text{ and }{b}\qquad \textbf{(D)}\ \text{zero}\qquad \textbf{(E)}\ \text{one}$

Acute scalene triangle $\triangle{}ABC$ has circumcenter $O$ and orthocenter $H$. Points $X$ and $Y$, distinct from $B$ and $C$, lie on the circumcircle of $\triangle{}ABC$ such that $\angle{}BXH=\angle{}CYH=90^{\circ{}}$. Show that if lines $XY$, $AH$, and $BC$ are concurrent, then $OH$ is parallel to $BC$.