Given two concentric circles and a pair of parallel lines. Find the locus of the fourth vertices of all rectangles with three vertices on the concentric circles, two vertices on one circle and the third on the other and with sides parallel to the given lines.

Real numbers $a, b, c$ are not equal $0$ and are solution of the system: $\begin{cases} a^2 + a = b^2 \\ b^2 + b = c^2 \\ c^2 +c = a^2 \end{cases}$ Prove that $(a - b)(b - c)(c - a) = 1$.

Let $A$ be one of the two points where the circles whose centers are the points $M$ and $N$ intersect. The tangents in $A$ to such circles intersect them again in $B$ and $C$, respectively. Let $P$ be a point such that the quadrilateral $AMPN$ is a parallelogram. Show that $P$ is the circumcenter of triangle $ABC$.

Let $l$ be the tangent line at the point $P(s,\ t)$ on a circle $C:x^2+y^2=1$. Denote by $m$ the line passing through the point $(1,\ 0)$, parallel to $l$. Let the line $m$ intersects the circle $C$ at $P'$ other than the point $(1,\ 0)$. Note : if $m$ is the line $x=1$, then $P'$ is considered as $(1,\ 0)$. Call $T$ the operation such that the point $P'(s',\ t')$ is obtained from the point $P(s,\ t)$ on $C$. (1) Express $s',\ t'$ as the polynomials of $s$ and $t$ respectively. (2) Let $P_n$ be the point obtained by $n$ operations of $T$ for $P$. For $P\left(\frac{\sqrt{3}}{2},\ \frac{1}{2}\right)$, plot the points $P_1,\ P_2$ and $P_3$. (3) For a positive integer $n$, find the number of $P$ such that $P_n=P$.

Let $x_1=y_1=x_2=y_2=1$, then for $n\geq 3$ let $x_n=x_{n-1}y_{n-2}+x_{n-2}y_{n-1}$ and $y_n=y_{n-1}y_{n-2}-x_{n-1}x_{n-2}$. What are the last two digits of $|x_{2012}|?$

Prove that for all positive integers $n$ we can find a permutation of {$1,2,...,n$} such that the average of two numbers doesn't appear in-between them.For example {$1,3,2,4$}works,but {$1,4,2,3$} doesn't because $2$ is between $1$ and $3$.

Let $z$ be a complex number satisfying $12\lvert z\rvert^2 = 2 \lvert z+2 \rvert ^2+\lvert z^2+1\rvert ^2+31.$ What is the value of $z+\frac{6}{z}?$ $\textbf{(A) }-2\qquad\textbf{(B) }-1\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }1\qquad\textbf{(E) }4$

What is the largest possible number of subsets of the set $\{1, 2, \dots , 2n+1\}$ such that the intersection of any two subsets consists of one or several consecutive integers?

A polyhedron has $7n$ faces. Show that there exist $n + 1$ of the polyhedron's faces that all have the same number of edges.

Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

Ninety-nine points are given on one of the diagonals of a unit square. Prove that there is at most one vertex of the square such that the average squared distance from a given point to the vertex is less than or equal to $1/2$.

Define a sequence of integers $a_0=m, a_1=n$ and $a_{k+1}=4a_k-5a_{k-1}$ for all $k \ge 1$. Suppose $p>5$ is a prime with $p \equiv 1 \pmod{4}$. Prove that it is possible to choose $m,n$ such that $p \nmid a_k$ for any $k \ge 0$.

Let $\Gamma $ be the circumcircle of triangle $ABC$. A circle $\Omega$ is tangent to the line segment $AB$ and is tangent to $\Gamma$ at a point lying on the same side of the line $AB$ as $C$. The angle bisector of $\angle BCA$ intersects $\Omega$ at two different points $P$ and $Q$. Prove that $\angle ABP = \angle QBC$.

In a triangle $ABC$, let $I$ be its incenter; $Q$ the point at which the incircle touches the line $AC$; $E$ the midpoint of $AC$ and $K$ the orthocenter of triangle $BIC$. Prove that the line $KQ$ is perpendicular to the line $IE$.

Prove that the number $n^n-n$ is divisible by $24$ for any odd integer $n$.

Let $0\le x_{i,j} \le 1$, where $i=1,2, \ldots m$ and $j=1,2, \ldots n$. Prove the inequality \[ \prod_{j=1}^n\left(1-\prod_{i=1}^mx_{i,j} \right)+ \prod_{i=1}^m\left(1-\prod_{j=1}^n(1-x_{i,j}) \right) \ge 1 \]

Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. How many real numbers $x$ satisfy the equation $x^2 + 10{,}000\lfloor x \rfloor = 10{,}000x$? $\textbf{(A) } 197 \qquad \textbf{(B) } 198 \qquad \textbf{(C) } 199 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 201$

In acute $\vartriangle ABC$, let points $D$, $E,$ and $F$ be the feet of the altitudes of the triangle from $A$, $B$,and $C$, respectively. The area of $\vartriangle AEF$ is $1$, the area of $\vartriangle CDE$ is $2$, and the area of $\vartriangle BF D$ is $2 -\sqrt3$. What is the area of $\vartriangle DEF$?

Let $T$ be a non-isosceles triangle and $n \ge 4$ an integer . Prove that you can divide $T$ in $n$ triangles and draw in each of them an inner bisector so that those $n$ bisectors are parallel.

In a convex quadrilateral $ ABCD, $ let $ A’,B’ $ be the orthogonal projections to $ CD $ of $ A, $ respectively, $ B. $ [b]a)[/b] Assuming that $ BB’\le AA’ $ and that the perimeter of $ ABCD $ is $ (AB+CD)\cdot BB’, $ is $ ABCD $ necessarily a trapezoid? [b]b)[/b] The same question with the addition that $ \angle BAD $ is obtuse.

Let $S = \{1, 2, \dots, 9\}.$ Compute the number of functions $f : S \rightarrow S$ such that, for all $s \in S, f(f(f(s))) =s$ and $f(s) - s$ is not divisible by $3$.

For a positive integer $n$, the equation $a^2 + b^2 + c^2 + n = abc$ is given in the positive integers. Prove that: 1. There does not exist a solution $(a, b, c)$ for $n = 2017$. 2. For $n = 2016$, $a$ is divisible by $3$ for all solutions $(a, b, c)$. 3. There are infinitely many solutions $(a, b, c)$ for $n = 2016$.

Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is: $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 96 $

In the figure $ \overline{AB} \equal{} \overline{AC}$, angle $ BAD \equal{} 30^{\circ}$, and $ \overline{AE} \equal{} \overline{AD}$. [asy]unitsize(20); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair A=(3,3),B=(0,0),C=(6,0),D=(2,0),E=(5,1); draw(A--B--C--cycle); draw(A--D--E); label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); label("$D$",D,S); label("$E$",E,NE);[/asy]Then angle $ CDE$ equals: $ \textbf{(A)}\ 7\frac {1}{2}^{\circ} \qquad\textbf{(B)}\ 10^{\circ} \qquad\textbf{(C)}\ 12\frac {1}{2}^{\circ} \qquad\textbf{(D)}\ 15^{\circ} \qquad\textbf{(E)}\ 20^{\circ}$

In the acute-angled triangle $ABC$, the altitudes $BD$ and $CE$ are drawn. Let $F$ and $G$ be the points of the line $ED$ such that $BF$ and $CG$ are perpendicular to $ED$. Prove that $EF = DG$.