In the triangle $ABC$ we have $\angle A = 2\angle C$ and $2\angle B = \angle A + \angle C$. The angle bisector of $\angle C$ intersects the segment $AB$ in $E$, let $F$ be the midpoint of $AE$, let $AD$ be the altitude of the triangle $ABC$. The perpendicular bisector of $DF$ intersects $AC$ in $M$. Prove that $AM = CM$.

In the parallelogram $ABCD$, the length of side $AB$ is half the length of side $BC$. The bisector of angle $\angle ABC$ intersects side $AD$ at point $K$ and diagonal $AC$ at point $L$. The bisector of angle $\angle ADC$ intersects the extension of side $AB$ at point $M$, with $B$ between $A$ and $M$. The line $ML$ intersects side $AD$ at point $F$. Calculate the ratio $\frac{AF}{AD}$.

Let $f:[0, \infty) \to \mathbb{R}$ a continuous function, constant on $\mathbb{Z}_{\geq 0}.$ For any $0 \leq a < b < c < d$ which satisfy $f(a)=f(c)$ and $f(b)=f(d)$ we also have $f \left( \frac{a+b}{2} \right) = f \left( \frac{c+d}{2} \right).$ Prove that $f$ is constant.

The infinite decimal notation of the real number $x$ contains all the digits. Let $u_n$ be the number of different $n$-digit segments encountered in $x$ notation. Prove that if for some $n$, $u_n \le (n+8)$, than $x$ is a rational number.

Consider $16$ points arranged as shown, with horizontal and vertical distances of $1$ between consecutive rows and columns. In how many ways can one choose four of these points such that the distance between every two of those four points is strictly greater than $2$? [asy] for (int x = 0; x < 4; ++x) { for (int y = 0; y < 4; ++y) { dot((x, y)); } } [/asy]

Prove that for all positive integers $n$ there exist $n$ distinct, positive rational numbers with sum of their squares equal to $n$. Proposed by Daniyar Aubekerov

The trapezoid $ABCD$, of bases $AB$ and $CD$, is inscribed in a circumference $\Gamma$. Let $X$ a variable point of the arc $AB$ of $\Gamma$ that does not contain $C$ or $D$. We denote $Y$ to the point of intersection of $AB$ and $DX$, and let Z be the point of the segment $CX$ such that $\frac{XZ}{XC}=\frac{AY}{AB}$ . Prove that the measure of $\angle AZX$ does not depend on the choice of $X.$

Let $ABC$ be an equilateral triangle with side length $1$. Say that a point $X$ on side $\overline{BC}$ is [i]balanced[/i] if there exists a point $Y$ on side $\overline{AC}$ and a point $Z$ on side $\overline{AB}$ such that the triangle $XYZ$ is a right isosceles triangle with $XY = XZ$. Find with proof the length of the set of all balanced points on side $\overline{BC}$.

Find all pairs $(m, n)$ of positive integers such that $m^2 + 2n^2 = 3(m + 2n)$

There are $n$ cities, $2$ airline companies in a country. Between any two cities, there is exactly one $2$-way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of $n$.

One bisector of a given triangle is parallel to one sideline of its Nagel triangle. Prove that one of two remaining bisectors is parallel to another sideline of the Nagel triangle. Proposed by:L.Emelyanov

Let $a_1 \leq a_2 \leq \cdots$ be a non-decreasing sequence of positive integers. A positive integer $n$ is called [i]good[/i] if there is an index $i$ such that $n=\dfrac{i}{a_i}$. Prove that if $2013$ is [i]good[/i], then so is $20$.

$\textbf{Juan's Old Stamping Grounds}$ Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, 6 cents each, Peru 4 cents each, and Spain 5 cents each. (Brazil and Peru are South American countries and France and Spain are in Europe.) [asy] /* AMC8 2002 #8, 9, 10 Problem */ size(3inch, 1.5inch); for ( int y = 0; y <= 5; ++y ) { draw((0,y)--(18,y)); } draw((0,0)--(0,5)); draw((6,0)--(6,5)); draw((9,0)--(9,5)); draw((12,0)--(12,5)); draw((15,0)--(15,5)); draw((18,0)--(18,5)); draw(scale(0.8)*"50s", (7.5,4.5)); draw(scale(0.8)*"4", (7.5,3.5)); draw(scale(0.8)*"8", (7.5,2.5)); draw(scale(0.8)*"6", (7.5,1.5)); draw(scale(0.8)*"3", (7.5,0.5)); draw(scale(0.8)*"60s", (10.5,4.5)); draw(scale(0.8)*"7", (10.5,3.5)); draw(scale(0.8)*"4", (10.5,2.5)); draw(scale(0.8)*"4", (10.5,1.5)); draw(scale(0.8)*"9", (10.5,0.5)); draw(scale(0.8)*"70s", (13.5,4.5)); draw(scale(0.8)*"12", (13.5,3.5)); draw(scale(0.8)*"12", (13.5,2.5)); draw(scale(0.8)*"6", (13.5,1.5)); draw(scale(0.8)*"13", (13.5,0.5)); draw(scale(0.8)*"80s", (16.5,4.5)); draw(scale(0.8)*"8", (16.5,3.5)); draw(scale(0.8)*"15", (16.5,2.5)); draw(scale(0.8)*"10", (16.5,1.5)); draw(scale(0.8)*"9", (16.5,0.5)); label(scale(0.8)*"Country", (3,4.5)); label(scale(0.8)*"Brazil", (3,3.5)); label(scale(0.8)*"France", (3,2.5)); label(scale(0.8)*"Peru", (3,1.5)); label(scale(0.8)*"Spain", (3,0.5)); label(scale(0.9)*"Juan's Stamp Collection", (9,0), S); label(scale(0.9)*"Number of Stamps by Decade", (9,5), N); [/asy] The average price of his '70s stamps is closest to $\text{(A)}\ 3.5 \text{ cents} \qquad \text{(B)}\ 4 \text{ cents} \qquad \text{(C)}\ 4.5 \text{ cents} \qquad \text{(D)}\ 5 \text{ cents} \qquad \text{(E)}\ 5.5 \text{ cents}$

We have two lists of numbers: One initially containing 1,6,11,...,96, and the other initially containing 4,9,14,...,99. In every turn, we erase two numbers from one of the lists, and write $\frac{1}{3}$ of their sum (not necessarily an integer) in the other list. We continue this process until there are no possible moves. [list=a] [*] Prove that at the end of the process, there is exactly one number in each list. [*] Prove that those two numbers are [u]not[/u] equal. [/list]

Consider increasing integer sequences with elements from $1,\ldots,10^6$. Such a sequence is [i]Adriatic[/i] if its first element equals 1 and if every element is at least twice the preceding element. A sequence is [i]Tyrrhenian[/i] if its final element equals $10^6$ and if every element is strictly greater than the sum of all preceding elements. Decide whether the number of Adriatic sequences is smaller than, equal to, or greater than the number of Tyrrhenian sequences. (Proposed by Gerhard Woeginger, Austria)

Nine fair coins are flipped independently and placed in the cells of a $3$ by $3$ square grid. Let $p$ be the probability that no row has all its coins showing heads and no column has all its coins showing tails. If $p=\frac ab$ for relatively prime positive integers $a$ and $b$, compute $100a+b$.

Let $(b_n)_{n \ge 0}=\sum_{k=0}^{n} (a_0+kd)$ for positive integers $a_0$ and $d$. We consider all such sequences containing an element $b_i$ which equals $2010$. Determine the greatest possible value of $i$ and for this value the integers $a_0$ and $d$. [i](41th Austrian Mathematical Olympiad, regional competition, problem 4)[/i]

Let $O_1$ be the center of a semicircle $\omega_1$ with diameter $AB$ and let $O_2$ be the center of a circle $\omega_2$ inscribed in $\omega_1$ and which is tangent to $AB$ at $O_1$. Let $O_3$ be a point on $AB$ that is the center of a semicircle $\omega_3$ which is tangent to both $\omega_1$ and $\omega_2$. Let $P$ be the intersection of the line through $O_3$ perpendicular to $AB$ and the line through $O_2$ parallel to $AB$. Show that $P$ is the center of a circle $\Gamma$ tangent to all of $\omega_1, \omega_2$ and $\omega_3$.

$ABCD$ is a rectangle with $AB = 20$ and $BC = 3$. A circle with radius $5$, centered at the midpoint of $DC$, meets the rectangle at four points: $W, X, Y$ , and $Z$. Find the area of quadrilateral $WXY Z$.

In the following figure, the largest square is divided into two squares and three rectangles, as shown: The area of each smaller square is equal to $a$ and the area of each small rectangle is equal to $b$. If $a+b=24$ and the root square of $a$ is a natural number, find all possible values for the area of the largest square. [img]https://cdn.artofproblemsolving.com/attachments/f/a/0b424d9c293889b24d9f31b1531bed5081081f.png[/img]

Let $a,b$ be positive integers. Prove that the equation $x^2+(a+b)^2x+4ab=1$ has rational solutions if and only if $a=b$. [i]Mihai Opincariu[/i]

50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine. [i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]

Simplify $ \left(\frac {1 \cdot 2 \cdot 4 \plus{} 2 \cdot 4 \cdot 8 \plus{} \cdots \plus{} n \cdot 2n \cdot 4n}{1 \cdot 3 \cdot 9 \plus{} 2 \cdot 6 \cdot 18 \plus{} \cdots \plus{} n \cdot 3n \cdot 9n}\right)^{\frac {1}{3}}$

Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]

Let $ABC$ be an acute triangle with orthocenter $H$. Let $D$ and $E$ be feet of the altitudes from $B$ and $C$ respectively. Let $M$ be the midpoint of segment $AH$ and $F$ be the intersection point of $AH$ and $DE$. Furthermore, let $P$ and $Q$ be the points inside triangle $ADE$ so that $P$ is an intersection of $CM$ and the circumcircle of $DFH$, and $Q$ is an intersection of $BM$ and the circumcircle of $EFH$. Prove that the intersection of lines $DQ$ and $EP$ lies on segment $AH$.