In $\vartriangle ABC, \angle A = 66^o$ and $| AB | <| AC |$. The outer bisector in $A$ intersects $BC$ in $D$ and $| BD | = | AB | + | AC |$. Determine the angles of $\vartriangle ABC$.

The convex pentagon $ABCDE$ is cyclic and $AB = BD$. Let point $P$ be the intersection of the diagonals $AC$ and $BE$. Let the straight lines $BC$ and $DE$ intersect at point $Q$. Prove that the straight line $PQ$ is parallel to the diagonal $AD$.

Let $f: Z^+ \to Z^+ \cup \{0\}$ a function that meets the following conditions: a) $f (a b) = f (a) + f (b)$, b) $f (a) = 0$ provided that the digits of the unit of $a$ are $7$, c) $f (10) = 0$. Find $f (2016).$

LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid $A$ dollars and Bernardo had paid $B$ dollars, where $A < B$. How many dollars must LeRoy give to Bernardo so that they share the costs equally? $ \textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B $

Let $\triangle ABC$ be a triangle with $\angle BAC=60^\circ$, $M$ be a point in its interior and $A',\, B',\, C'$ be the orthogonal projections of $M$ on the sides $BC,\, CA,\, AB$. Determine the locus of $M$ when the sum $A'B+B'C+C'A$ is constant. [i]Horea Călin Pop[/i]

A school sent students to compete in an academic olympiad in $11$ differents subjects, each consist of $5$ students. Given that for any $2$ different subjects, there exists a student compete in both subjects. Prove that there exists a student who compete in at least $4$ different subjects.

Prove that for every non-negative integer $m$ there exists a polynomial w with integer coefficients such that $2^m$ is the greatest common divisor of the numbers $$ a_n = 3^n + w(n), n = 0, 1, 2, ....$$

An acute triangle \(ABC\) has circumcircle \(\Gamma\) and circumcentre \(O\). The incentres of \(AOB\) and \(AOC\) are \(I_b\) and \(I_c\) respectively. Let \(M\) be the the point on \(\Gamma\) such that \(MB = MC\) and \(M\) lies on the same side of \(BC\) as \(A\). Prove that the points \(M\), \(A\), \(I_b\), and \(I_c\) are concyclic.

The sequence $(a_n)$ is given by $a_1 = x \in \mathbb{R}$ and $3a_{n+1} = a_n+1$ for $n \geq 1$. Set $A = \sum_{n=1}^\infty \Big[ a_n - \frac{1}{6}\Big]$, $B = \sum_{n=1}^\infty \Big[ a_n + \frac{1}{6}\Big]$. Compute the sum $A+B$ in terms of $x$.

[b]a)[/b] Find all solutions of the equation $p^2+q^2+r^2=pqr$, where $p,q,r$ are positive primes.\\ [b]b)[/b] Show that for every positive integer $N$, there exist three integers $a,b,c\geq N$ with $a^2+b^2+c^2=abc$.

Find the arithmetic mean of all the three-digit palindromes. (Recall that a palindrome is a number that reads the same forward and backward, such as $777$ or $383$.)

Can one find 4004 positive integers such that the sum of any 2003 of them is not divisible by 2003?

The roots of the equation $x^2 + (3a + b)x + a^2 + 2b^2 = 0$ are $x_1$ and $x_2$ with $x_1 \ne x_2$. Determine the values of $a$ and $b$ so that the roots of the equation $ x^2 - 2a(3a + 2b)x + 5a^2b^2 + 4b^4 = 0$ let $x^2_1$ and $x^2_2$.

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

From point $P$ outside a circle draw two tangents to the circle touching at $A$ and $B$. Draw a secant line intersecting the circle at points $C$ and $D$, with $C$ between $P$ and $D$. Choose point $Q$ on the chord $CD$ such that $\angle DAQ=\angle PBC$. Prove that $\angle DBQ=\angle PAC$.

Let $ABC$ be a triangle with $AB=AC$. Also, let $D\in[BC]$ be a point such that $BC>BD>DC>0$, and let $\mathcal{C}_1,\mathcal{C}_2$ be the circumcircles of the triangles $ABD$ and $ADC$ respectively. Let $BB'$ and $CC'$ be diameters in the two circles, and let $M$ be the midpoint of $B'C'$. Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$). [i]Greece[/i]

The game tic-tac is played on a $3$ by $3$ square grid between players $X$ and $O$. They take turns, and on their turn a player writes their symbol onto one empty space of the grid. A player wins if they fill a row or column with three copies of their symbol; a player filling a main diagonal does [i]not[/i] end the game in a win for that player. If the grid is filled without determining the winner, the game is a draw. Assuming player $X$ goes first and the players draw the game, how many possibilities are there for the final state of the grid? $\text{(A) }24\qquad\text{(B) }33\qquad\text{(C) }36\qquad\text{(D) }45\qquad\text{(E) }126$

In a park there are 23 trees $t_0,t_1,\dots,t_{22}$ in a circle and 22 birds $b_1,n_2,\dots,b_{22}.$ Initially, each bird is in a tree. Every minute, the bird $b_i, 1\leqslant i\leqslant 22$ flies from the tree $t_j{}$ to the tree $t_{i+j}$ in clockwise order, indices taken modulo 23. Prove that there exists a moment when at least 6 trees are empty.

Let $\triangle ABC$ have $\angle A=20^{\circ}$ and $\angle C=40^{\circ}$. We've constructed the angle bisector $AL$ ($L\in BC$) and the external angle bisector $CN$ ($N\in AB$). Find $\angle CLN$.

In a general triangle $ ADE$ (as shown) lines $ \overline{EB}$ and $ \overline{EC}$ are drawn. Which of the following angle relations is true? [asy]defaultpen(linewidth(.8pt)); unitsize(2cm); pair A = origin; pair B = (1,0); pair C = (2,0); pair D = (3,0); pair E = (1.25,1.75); draw(A--D--E--cycle); draw(E--B); draw(E--C); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,S); label("$D$",D,SE); label("$E$",E,N); label("$y$",E,3SW + 3S); label("$w$",E,7S + E); label("$b$",E,3SE + 4S + E); label("$x$",A,NE); label("$z$",B,NW); label("$m$",B,NE); label("$n$",C,NW); label("$c$",C,NE); label("$a$",D,NW+W);[/asy] $ \textbf{(A)}\ x \plus{} z \equal{} a \plus{} b\qquad \textbf{(B)}\ y \plus{} z \equal{} a \plus{} b\qquad \textbf{(C)}\ m \plus{} x \equal{} w \plus{} n\qquad \\ \textbf{(D)}\ x \plus{} z \plus{} n \equal{} w \plus{} c \plus{} m\qquad \textbf{(E)}\ x \plus{} y \plus{} n \equal{} a \plus{} b \plus{} m$

The point $D$ on the altitude $AA_1$ of an acute triangle $ABC$ is such that $\angle BDC=90^\circ$; $H$ is the orthocentre of $ABC$. A circle with diameter $AH$ is constructed. Prove that the tangent drawn from $B$ to this circle is equal to $BD$.

If $x^3-3\sqrt3 x^2 +9x - 3\sqrt3 -64=0$ find the value of $x^6-8x^5+13x^4-5x^3+49x^2-137x+2015$ .

The lengths of the diagonals of a rhombus are, in inches, two consecutive integers. The area of the rhombus is $210$ sq. in. Find its perimeter, in inches.

The quadrilateral $ABCD$ is inscribed in a circle. The lines $AB$ and $CD$ meet each other in the point $E$, while the diagonals $AC$ and $BD$ in the point $F$. The circumcircles of the triangles $AFD$ and $BFC$ have a second common point, which is denoted by $H$. Prove that $\angle EHF=90^\circ$.

Let $ABCD$ be a circumscribed quadrilateral and $T=AC\cap BD$. Let $I_1$, $I_2$, $I_3$, $I_4$ the incenters of $\Delta TAB$, $\Delta TBC$, $TCD$, $TDA$, respectively, and $J_1$, $J_2$, $J_3$, $J_4$ the incenters of $\Delta ABC$, $\Delta BCD$, $\Delta CDA$, $\Delta DAB$. Show that $I_1I_2I_3I_4$ is a cyclic quadrilateral and its center is $J_1J_3\cap J_2J_4$