Two circles $S_1$ and $S_2$ intersect at different points $P,Q$. The arc of $S_1$ lying inside $S_2$ measures $2a$ and the arc of $S_2$ lying inside $S_1$ measures $2b$. Let $T$ be any point on $S_1$. Let $R,S$ be another points of intersection of $S_2$ with $TP$ and $TQ$ respectively. Let $a+2b<\pi$ . Find the locus of the intersection points of $PS$ and $RQ$. S.Shikh

A fair coin is flipped $6$ times. The probability that the coin lands on the same side $3$ flips in a row at some point can be expressed as a common fraction $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.

Let $P$ be the set of all subsets of ${1, 2, ... , n}$. Show that there are $1^n + 2^n + ... + m^n$ functions $f : P \longmapsto {1, 2, ... , m}$ such that $f(A \cap B) = min( f(A), f(B))$ for all $A, B.$

Let $D$ be a closed disc in the complex plane. Prove that for all positive integers $n$, and for all complex numbers $z_1,z_2,\ldots,z_n\in D$ there exists a $z\in D$ such that $z^n = z_1\cdot z_2\cdots z_n$.

The integers from $1$ through $9$ inclusive, are placed in the squares of a $3 \times 3$ grid. Each square contains a different integer. The product of the integers in the first and second rows are $60$ and $96$ respectively. Find the sum of the integers in the third row. [i]Proposed by bissue [/i]

In the plane we have a line $r$ and two points $A$ and $B$ outside the line and in the same half plane. Determine a point $M$ on the line such that the angle of $r$ with $AM$ is double that of $r$ with $BM$. (Consider the smaller angle of two lines of the angles they form).

A unit equilateral triangle is given. Divide each side into three equal parts. Remove the equilateral triangles whose bases are middle one-third segments. Now we have a new polygon. Remove the equilateral triangles whose bases are middle one-third segments of the sides of the polygon. After repeating these steps for infinite times, what is the area of the new shape? $ \textbf{(A)}\ \dfrac {1}{2\sqrt 3} \qquad\textbf{(B)}\ \dfrac {\sqrt 3}{8} \qquad\textbf{(C)}\ \dfrac {\sqrt 3}{10} \qquad\textbf{(D)}\ \dfrac {1}{4\sqrt 3} \qquad\textbf{(E)}\ \text{None of the above} $

Pieces are placed in some squares of an $8 \times 8$ board sothat: a) There is at least one token in any rectangle with sides $2 \times 1$ or $1\times 2$. b) There are at least two neighboring pieces in any rectangle with sides $7\times 1$ or $1\times 7$. Find the smallest number of tokens that can be taken to fulfill with both conditions.

Let be given $ \vartriangle ABC$ with area $ (\vartriangle ABC) = 60$ cm$^2$. Let $R,S $ lie in $BC$ such that $BR = RS = SC$ and $P,Q$ be midpoints of $AB$ and $AC$, respectively. Suppose that $PS$ intersects $QR$ at $T$. Evaluate area $(\vartriangle PQT)$.

$1019$ stones are placed into two non-empty boxes. Each second Alex chooses a box with an even amount of stones and shifts half of these stones into another box. Prove that for each $k$, $1\le k\le1018$, at some moment there will be a box with exactly $k$ stones. [i](O. Izhboldin)[/i]

Three circles $C_1$, $C_2$, $C_3$ in the plane touch each other (in three different points). Connect the common point of $C_1$ and $C_2$ with the other two common points by straight lines. Show that these lines meet $C_3$ in diametrically opposite points.

Two walkers are at the same altitude in a range of mountains. The path joining them is piecewise linear with all its vertices above the two walkers. Can they each walk along the path until they have changed places, so that at all times their altitudes are equal?

Find all positive integers $n$ that have precisely $\sqrt{n + 1}$ natural divisors.

Jared has 3 distinguishable Rolexes. Each day, he selects a subset of his Rolexes and wears them on his arm (the order he wears them does not matter). However, he does not want to wear the same Rolex 2 days in a row. How many ways can he wear his Rolexes during a 6 day period?

If all edges of a non-planar quadrilateral tangent the faces of a sphere, prove that all of the points of tangency belong to a plane.

A pile of n pebbles is placed in a vertical column. This configuration is modified according to the following rules. A pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right. (If there are no pebbles to the right, think of this as a column with 0 pebbles.) At each stage, choose a pebble from among those that can be moved (if there are any) and place it at the top of the column to its right. If no pebbles can be moved, the configuration is called a final configuration. For each n, show that, no matter what choices are made at each stage, the final configuration obtained is unique. Describe that configuration in terms of n.

Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals. [i]Proposed by Paisan Nakmahachalasint, Thailand[/i]

$2013$ smurfs are sitting at a large round table. Each of them has two tickets. on each card represents a number from $\{1, 2, . . ., 2013\}$ such that each of the numbers from this set occurs exactly twice. Every smurf takes the card every minute with the smaller of the two numbers, it smurfs on to its left neighbor and receives a card from his right neighbor. Show that there will come a time when a smurf has two cards with the same number.

What number is one third of the way from $ \frac14$ to $ \frac34$? $ \textbf{(A)}\ \frac{1}{3} \qquad \textbf{(B)}\ \frac{5}{12} \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{7}{12} \qquad \textbf{(E)}\ \frac{2}{3}$

[b]p1.[/b] Solve the equation $\log_x (x + 2) = 2$. [b]p2.[/b] Solve the inequality: $0.5^{|x|} > 0.5^{x^2}$. [b]p3.[/b] The integers from 1 to 2015 are written on the blackboard. Two randomly chosen numbers are erased and replaced by their difference giving a sequence with one less number. This process is repeated until there is only one number remaining. Is the remaining number even or odd? Justify your answer. [b]p4.[/b] Four circles are constructed with the sides of a convex quadrilateral as the diameters. Does there exist a point inside the quadrilateral that is not inside the circles? Justify your answer. [b]p5.[/b] Prove that for any finite sequence of digits there exists an integer the square of which begins with that sequence. [b]p6.[/b] The distance from the point $P$ to two vertices $A$ and $B$ of an equilateral triangle are $|P A| = 2$ and $|P B| = 3$. Find the greatest possible value of $|P C|$. PS. You should use hide for answers.

Suppose $a$, $b$, and $c$ are positive integers such that \[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\] Which of the following statements are necessarily true? I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$. II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both. III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$. $\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}$

For every positive integer $n$, denote by $D_n$ the number of permutations $(x_1, \dots, x_n)$ of $(1,2,\dots, n)$ such that $x_j\neq j$ for every $1\le j\le n$. For $1\le k\le \frac{n}{2}$, denote by $\Delta (n,k)$ the number of permutations $(x_1,\dots, x_n)$ of $(1,2,\dots, n)$ such that $x_i=k+i$ for every $1\le i\le k$ and $x_j\neq j$ for every $1\le j\le n$. Prove that $$\Delta (n,k)=\sum_{i=0}^{k=1} \binom{k-1}{i} \frac{D_{(n+1)-(k+i)}}{n-(k+i)}$$ (Proposed by Combinatorics; Ferdowsi University of Mashhad, Iran; Mirzavaziri)

Find the smallest natural number for which there exist that many natural numbers such that the sum of the squares of their squares is equal to $ 1998. $ [i]Gheorghe Iurea[/i]

Inside a triangle ABC, consider points $D, E$ such that $\angle ABD =\angle DBE=\angle EBC$ and $\angle ACD=\angle DC E=\angle ECB$. Calculate angles $\angle BDE$, $\angle B EC$, $\angle D E C$ in terms of the angle of the triangle $ABC$.

Let $m_1, m_2, \cdots ,m_k$ be positive numbers with the sum $S$. Prove that \[\displaystyle\sum_{i=1}^k\left(m_i +\frac{1}{m_i}\right)^2 \ge k\left(\frac{k}{S}+\frac{S}{k}\right)^2\]