Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos k \theta$ and $ \cos[(k \plus{} 1)\theta ]$ are both rational for some positive integer $ k$, show that $ \theta \equal{} \pi/6$.

Katie begins juggling five balls. After every second elapses, there is a chance she will drop a ball. If she is currently juggling $ k$ balls, this probability is $ \frac{k}{10}$. Find the expected number of seconds until she has dropped all the balls.

$27$ students in a school take French. $32$ students in a school take Spanish. $5$ students take both courses. How many of these students in total take only $1$ language course?

Let $ABCD$ be a tetrahedron and let $H_{a},H_{b},H_{c},H_{d}$ be the orthocenters of triangles $BCD,CDA,DAB,ABC$, respectively. Prove that lines $AH_{a},BH_{b},CH_{c}, DH_{d}$ are concurrent if and only if $AB^2 + CD^2 = AC^2 + BD^2 = AD^2 + BC^2$

Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca\leq 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\leq \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)\] [i]Proposed by Dzianis Pirshtuk, Belarus[/i]

Determine the perfect squares $ \overline{aabcd} $ of five digits such that $ \overline{dcbaa} $ is a perfect square of five digits.

Find all possible integer values of $\frac{m^2+n^2}{mn}$ where m and n are integers.

The number of maps $f$ from $1,2,3$ into the set $1,2,3,4,5$ such that $f(i)\le f(j)$ whenever $i\le j$ is $\textbf{(A)}~60$ $\textbf{(B)}~50$ $\textbf{(C)}~35$ $\textbf{(D)}~30$

Let $S_P$ be the set of all polynomials $P$ with complex coefficients, such that $P(x^2) = P(x)P(x-1)$ for all complex numbers $x$. Suppose $P_0$ is the polynomial in $S_P$ of maximal degree such that $P_0(1) \mid 2016$. Find $P_0(10)$.

Let $ABC$ be a triangle, where $AB = AC$ and $BC = 12$. Let $D$ be the midpoint of $BC$. Let $E$ be a point in $AC$ such that $DE \perp AC$. Let $F$ be a point in $AB$ such that $EF \parallel BC$. If $EC = 4$, determine the length of $EF$.

For a positive integer $n$, denote $rad(n)$ as product of prime divisors of $n$. And also $rad(1)=1$. Define the sequence $\{a_i\}_{i=1}^{\infty}$ in this way: $a_1 \in \mathbb N$ and for every $n \in \mathbb N$, $a_{n+1}=a_n+rad(a_n)$. Prove that for every $N \in \mathbb N$, there exist $N$ consecutive terms of this sequence which are in an arithmetic progression.

Find all ordered triplets of positive integers $(a,\ b,\ c)$ such that $2^a+3^b+1=6^c$.

Given that $\tan A=\frac{12}{5}$, $\cos B=-\frac{3}{5}$ and that $A$ and $B$ are in the same quadrant, find the value of $\cos (A-B)$. $ \textbf{(A) }-\frac{63}{65}\qquad\textbf{(B) }-\frac{64}{65}\qquad\textbf{(C) }\frac{63}{65}\qquad\textbf{(D) }\frac{64}{65}\qquad\textbf{(E) }\frac{65}{63} $

How many positive integers $n<10^6$ are there such that $n$ is equal to twice of square of an integer and is equal to three times of cube of an integer? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of the above} $

On the sides of triangle $ABC$, isosceles right-angled triangles $AUB, CVB$, and $AWC$ are placed. These three triangles have their right angles at vertices $U, V$ , and $W$, respectively. Triangle $AUB$ lies completely inside triangle $ABC$ and triangles $CVB$ and $AWC$ lie completely outside $ABC$. See the figure. Prove that quadrilateral $UVCW$ is a parallelogram. [asy] import markers; unitsize(1.5 cm); pair A, B, C, U, V, W; A = (0,0); B = (2,0); C = (1.7,2.5); U = (B + rotate(90,A)*(B))/2; V = (B + rotate(90,C)*(B))/2; W = (C + rotate(90,A)*(C))/2; draw(A--B--C--cycle); draw(A--W, StickIntervalMarker(1,1,size=2mm)); draw(C--W, StickIntervalMarker(1,1,size=2mm)); draw(B--V, StickIntervalMarker(1,2,size=2mm)); draw(C--V, StickIntervalMarker(1,2,size=2mm)); draw(A--U, StickIntervalMarker(1,3,size=2mm)); draw(B--U, StickIntervalMarker(1,3,size=2mm)); draw(rightanglemark(A,U,B,5)); draw(rightanglemark(B,V,C,5)); draw(rightanglemark(A,W,C,5)); dot("$A$", A, S); dot("$B$", B, S); dot("$C$", C, N); dot("$U$", U, NE); dot("$V$", V, NE); dot("$W$", W, NW); [/asy]

A set $S$ of $n-1$ natural numbers is given ($n\ge 3$). There exist at least at least two elements in this set whose difference is not divisible by $n$. Prove that it is possible to choose a non-empty subset of $S$ so that the sum of its elements is divisible by $n$.

We have $n$ countries. Each country have $m$ persons who live in that country ($n>m>1$). We divide $m \cdot n$ persons into $n$ groups each with $m$ members such that there don't exist two persons in any groups who come from one country. Prove that one can choose $n$ people into one class such that they come from different groups and different countries.

In the fall of $1996$, a total of $800$ students participated in an annual school clean-up day. The organizers of the event expect that in each of the years $1997$, $1998$, and $1999$, participation will increase by $50 \%$ over the previous year. The number of participants the organizers will expect in the fall of $1999$ is $\text{(A)}\ 1200 \qquad \text{(B)}\ 1500 \qquad \text{(C)}\ 2000 \qquad \text{(D)}\ 2400 \qquad \text{(E)}\ 2700$

We have a collection of $1720$ balls, half of which are black and half of which are white, aligned in a straight line. Our task is to make the balls alternating in color along the line. The following greedy algorithm accomplishes that task for $2n$ balls: \begin{tabular}{l} 1: \textbf{FOR} $i$ \textbf{IN} $[2,3,\dots,2n]$ \\ 2: $\quad$ \textbf{IF} balls $i-1$ and $i$ have the same color: \\ 3: $\quad\quad$ $j\gets$ smallest index greater than $i$ for which balls $i-1$ and $j$ have different colors \\ 4: $\quad\quad$ swap balls $i$ and $j$ \end{tabular} Given a configuration $C$ of our $1720$ balls, let $\hat{\sigma}(C)$ denote the number of swaps the greedy algorithm takes, and let $\sigma(C)$ denote the minimum number of swaps actually necessary to perform the task. Find the maximum value over all configurations $C$ of $\hat{\sigma}(C)-\sigma(C)$.

If a,b,c positive numbers and such that $a+\sqrt{b+\sqrt{c}}=c+\sqrt{b+\sqrt{a}}$. Prove that if $a\neq c$ then $40ac<1$.

Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle? $ \textbf{(A)}\ \frac{1}{36} \qquad \textbf{(B)}\ \frac{1}{24} \qquad \textbf{(C)}\ \frac{1}{18} \qquad \textbf{(D)}\ \frac{1}{12} \qquad \textbf{(E)}\ \frac{1}{9}$

Suppose that the real numbers $a_{1},a_{2},...,a_{2002}$ satisfying $\frac{a_{1}}{2}+\frac{a_{2}}{3}+...+\frac{a_{2002}}{2003}=\frac{4}{3}$ $\frac{a_{1}}{3}+\frac{a_{2}}{4}+...+\frac{a_{2002}}{2004}=\frac{4}{5}$ $...$ $\frac{a_{1}}{2003}+\frac{a_{2}}{2004}+...+\frac{a_{2002}}{4004}=\frac{4}{4005}$ Evaluate the sum $\frac{a_{1}}{3}+\frac{a_{2}}{5}+...+\frac{a_{2002}}{4005}$.

The $ABC$ triangle is given, point $I_a$ is the center of an exscribed circle touching the side $BC$ , the point $M$ is the midpoint of the side $BC$, the point $W$ is the intersection point of the bisector of the angle $A$ of the triangle $ABC$ with the circumscribed circle around him. Prove that the area of the triangle $I_aBC$ is calculated by the formula $S_{ (I_aBC)} = MW \cdot p$, where $p$ is the semiperimeter of the triangle $ABC$. (Mykola Moroz)

If $ABCDEF$ is a convex cyclic hexagon, then its diagonals $AD$, $BE$, $CF$ are concurrent if and only if $\frac{AB}{BC}\cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$. [i]Alternative version.[/i] Let $ABCDEF$ be a hexagon inscribed in a circle. Then, the lines $AD$, $BE$, $CF$ are concurrent if and only if $AB\cdot CD\cdot EF=BC\cdot DE\cdot FA$.

Let $a$ be a permutation on $\{0,1,\ldots ,2015\}$ and $b,c$ are also permutations on $\{1,2,\ldots ,2015\}$. For all $x\in \{1,2,\ldots ,2015\}$, the following conditions are satisfied: (i) $a(x)-a(x-1)\neq 1$,\\ (ii) if $b(x)\neq x$, then $c(x)=x$,\\ Prove that the number of $a$'s is equal to the number of ordered pairs of $(b,c)$.