We have three circles $w_1$, $w_2$ and $\Gamma$ at the same side of line $l$ such that $w_1$ and $w_2$ are tangent to $l$ at $K$ and $L$ and to $\Gamma$ at $M$ and $N$, respectively. We know that $w_1$ and $w_2$ do not intersect and they are not in the same size. A circle passing through $K$ and $L$ intersect $\Gamma$ at $A$ and $B$. Let $R$ and $S$ be the reflections of $M$ and $N$ with respect to $l$. Prove that $A, B, R, S$ are concyclic.

Restore a triangle $ABC$ by the Nagel point, the vertex $B$ and the foot of the altitude from this vertex.

If at least one of the integers $ a,b $ is not divisible by $ 3, $ then the polynom $ X^2-abX+a^2+b^2 $ is irreducible over the integers. [i]Ion Cucurezeanu[/i]

On each horizontal line in the figure below, the five large dots indicate the populations of cities $A$, $B$, $C$, $D$ and $E$ in the year indicated. Which city had the greatest percentage increase in population from 1970 to 1980? [asy] size(300); defaultpen(linewidth(0.7)+fontsize(10)); pair A=(5,0), B=(7,0), C=(10,0), D=(13,0), E=(16,0); pair F=(4,3), G=(5,3), H=(7,3), I=(10,3), J=(12,3); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); dot(I); dot(J); draw((0,0)--(18,0)^^(0,3)--(18,3)); draw((0,0)--(0,.5)^^(5,0)--(5,.5)^^(10,0)--(10,.5)^^(15,0)--(15,.5)); draw((0,3)--(0,2.5)^^(5,3)--(5,2.5)^^(10,3)--(10,2.5)^^(15,3)--(15,2.5)); draw((1,0)--(1,.2)^^(2,0)--(2,.2)^^(3,0)--(3,.2)^^(4,0)--(4,.2)^^(6,0)--(6,.2)^^(7,0)--(7,.2)^^(8,0)--(8,.2)^^(9,0)--(9,.2)^^(10,0)--(10,.2)^^(11,0)--(11,.2)^^(12,0)--(12,.2)^^(13,0)--(13,.2)^^(14,0)--(14,.2)^^(16,0)--(16,.2)^^(17,0)--(17,.2)^^(18,0)--(18,.2)); draw((1,3)--(1,2.8)^^(2,3)--(2,2.8)^^(3,3)--(3,2.8)^^(4,3)--(4,2.8)^^(6,3)--(6,2.8)^^(7,3)--(7,2.8)^^(8,3)--(8,2.8)^^(9,3)--(9,2.8)^^(10,3)--(10,2.8)^^(11,3)--(11,2.8)^^(12,3)--(12,2.8)^^(13,3)--(13,2.8)^^(14,3)--(14,2.8)^^(16,3)--(16,2.8)^^(17,3)--(17,2.8)^^(18,3)--(18,2.8)); label("A", A, S); label("B", B, S); label("C", C, S); label("D", D, S); label("E", E, S); label("A", F, N); label("B", G, N); label("C", H, N); label("D", I, N); label("E", J, N); label("1970", (0,3), W); label("1980", (0,0), W); label("0", (0,1.5)); label("50", (5,1.5)); label("100", (10,1.5)); label("150", (15,1.5)); label("Population", (21,2)); label("in thousands", (21.4,1));[/asy] $ \textbf{(A)}\ A\qquad\textbf{(B)}\ B\qquad\textbf{(C)}\ C\qquad\textbf{(D)}\ D\qquad\textbf{(E)}\ E $

The function is given $f(x) = \frac{2x^3 -6x^2 + 13x + 10}{2x^2 - 9x}$. Determine all positive integers $x$ for which $f(x)$ is an integer

Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds: (i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$ (ii) some plane contains exactly three points from $E.$

Let $\mathbb N = B_1\cup\cdots \cup B_q$ be a partition of the set $\mathbb N$ of all positive integers and let an integer $l \in \mathbb N$ be given. Prove that there exist a set $X \subset \mathbb N$ of cardinality $l$, an infinite set $T \subset \mathbb N$, and an integer $k$ with $1 \leq k \leq q$ such that for any $t \in T$ and any finite set $Y \subset X$, the sum $t+ \sum_{y \in Y} y$ belongs to $B_k.$

Let $ ABC$ be a triangle with $ AB \equal{} 5$, $ BC \equal{} 4$ and $ AC \equal{} 3$. Let $ \mathcal P$ and $ \mathcal Q$ be squares inside $ ABC$ with disjoint interiors such that they both have one side lying on $ AB$. Also, the two squares each have an edge lying on a common line perpendicular to $ AB$, and $ \mathcal P$ has one vertex on $ AC$ and $ \mathcal Q$ has one vertex on $ BC$. Determine the minimum value of the sum of the areas of the two squares. [asy]import olympiad; import math; import graph; unitsize(1.5cm); pair A, B, C; A = origin; B = A + 5 * right; C = (9/5, 12/5); pair X = .7 * A + .3 * B; pair Xa = X + dir(135); pair Xb = X + dir(45); pair Ya = extension(X, Xa, A, C); pair Yb = extension(X, Xb, B, C); pair Oa = (X + Ya)/2; pair Ob = (X + Yb)/2; pair Ya1 = (X.x, Ya.y); pair Ya2 = (Ya.x, X.y); pair Yb1 = (Yb.x, X.y); pair Yb2 = (X.x, Yb.y); draw(A--B--C--cycle); draw(Ya--Ya1--X--Ya2--cycle); draw(Yb--Yb1--X--Yb2--cycle); label("$A$", A, W); label("$B$", B, E); label("$C$", C, N); label("$\mathcal P$", Oa, origin); label("$\mathcal Q$", Ob, origin);[/asy]

Find the number of sets $A$ containing $9$ positive integers with the following property: for any positive integer $n\le 500$, there exists a subset $B\subset A$ such that $\sum_{b\in B}{b}=n$. [i]Bogdan Enescu & Dan Ismailescu[/i]

The diagonals $AC,BD$ of the quadrilateral $ABCD$ intersect at $E$. Let $M,N$ be the midpoints of $AB,CD$ respectively. Let the perpendicular bisectors of the segments $AB,CD$ meet at $F$. Suppose that $EF$ meets $BC,AD$ at $P,Q$ respectively. If $MF\cdot CD=NF\cdot AB$ and $DQ\cdot BP=AQ\cdot CP$, prove that $PQ\perp BC$.

Alphonse and Beryl play a game starting with a blank blackboard. Alphonse goes first and the two players alternate turns. On Alphonse's first turn, he writes the integer $10^{2011}$ on the blackboard. On each subsequent turn, each player can do exactly one of the following two things: [b](i)[/b] replace any number $x$ that is currently on the blackboard with two integers a and b greater than $1$ such that $x = ab,$ or [b](ii)[/b] erase one or two copies of a number $y$ that appears at least twice on the blackboard. Thus, there may be many numbers on the board at any time. The first player who cannot do either of these things loses. Determine which player has a winning strategy and explain the strategy.

For positive integers $a$ and $b$, let $M(a,b)$ denote their greatest common divisor. Determine all pairs of positive integers $(m,n)$ such that for any two positive integers $x$ and $y$ such that $x\mid m$ and $y\mid n$, $$M(x+y,mn)>1.$$ [i]Proposed by Ivan Novak[/i]

Let $x, y$ be distinct positive integers. Show that the number $$\frac{(x + y)^2}{x^3 + xy^2- x^2y -y^3}$$ is not an integer.

Albert and Brita play a game with a bar of $19$ adjacent squares. Initially, there is a button on the middle square of the bar. At every turn Albert mentions one positive integer less than $5$, and Brita moves button a number of squares in the direction of her choice - while doing so however, Brita must not move the button more than twice in one direction order. Prove that Albert can choose the numbers so that by the $19$th turn, Brita to be forced to move the button out of the bar.

Let $a, b, c$ be pairwise coprime natural numbers. A positive integer $n$ is said to be [i]stubborn[/i] if it cannot be written in the form $n = bcx+cay+abz$, for some $x, y, z \in\mathbb{ N}.$ Determine the number of stubborn numbers.

Triangle $ABC$ has circumcircle $(O,R)$, and orthocenter $H$. The symmedians through $A,B,C$ meet the perpendicular bisectors of $BC,CA,AB$ at $D,E, F$ respectively. Let $M,N, P$ be the perpendicular projections of H on the line $OD,OE,OF.$ Prove that $$\frac{OH^2}{R^2} =\frac{\overline{OM}}{\overline{OD}}+\frac{\overline{ON}}{\overline{OE}} +\frac{\overline{OP}}{\overline{OF}}$$ Đỗ Thanh Sơn

A function $f$ is defined on the positive integers by \[\left\{\begin{array}{rcl}f(1) &=& 1, \\ f(3) &=& 3, \\ f(2n) &=& f(n), \\ f(4n+1) &=& 2f(2n+1)-f(n), \\ f(4n+3) &=& 3f(2n+1)-2f(n), \end{array}\right.\] for all positive integers $n$. Determine the number of positive integers $n$, less than or equal to 1988, for which $f(n) = n$.

Prove that for every natural $n$ the following inequality is valid $$|\sin 1| + |\sin 2| + |\sin (3n-1)| + |\sin 3n| > \frac{8n}{5}$$

Find the 1000th number (from small to large) that is coprime to $105$.

The polynomial $f(x)=1-x+x^2-x^3+\cdots-x^{19}+x^{20}$ is written into the form $g(y)=a_0+a_1y+a_2y^2+\cdots+a_{20}y^{20}$, where $y=x-4$, then $a_0+a_1+\cdots+a_{20}=$________.

There is a circle $c$ centered about the origin of radius $ 1$. There are circles $c_1$,$ . . .$ ,$c_6$, each of radius $r_1$, such that each circle is completely inside c and is tangent to it, and $c_2$ is tangent to $c_1$, $c_3$ is tangent to $c_2$, . . ., and $c_1$ is tangent to $c_6$. There is a circle $d$ which is tangent to $c$, $c_1$, and $c_2$, but does not intersect any of these circles. What is the radius of circle $d$? Express your answer in the form $\frac{a+b\sqrt{c}}{d}$ , where $a,b,c,d$ are integers, $d$ is positive and as small as possible, and $c$ is squarefree.

Find a four-digit number that is perfect square and such that the first two digits are the same and the last two as well.

Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]

Let $a,b,c,$ and $d$ be real numbers such that at least one of $c$ and $d$ is non-zero. Let $ f:\mathbb{R}\to\mathbb{R}$ be a function defined as $f(x)=\frac{ax+b}{cx+d}$. Suppose that for all $x\in\mathbb{R}$, we have $f(x) \neq x$. Prove that if there exists some real number $a$ for which $f^{1387}(a)=a$, then for all $x$ in the domain of $f^{1387}$, we have $f^{1387}(x)=x$. Notice that in this problem, \[f^{1387}(x)=\underbrace{f(f(\cdots(f(x)))\cdots)}_{\text{1387 times}}.\] [i]Hint[/i]. Prove that for every function $g(x)=\frac{sx+t}{ux+v}$, if the equation $g(x)=x$ has more than $2$ roots, then $g(x)=x$ for all $x\in\mathbb{R}-\left\{\frac{-v}{u}\right\}$.

Find, with proof, the number of positive integers whose base-$n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\pm 1$ from some digit further to the left. (Your answer should be an explicit function of $n$ in simplest form.)