For how many (not necessarily positive) integer values of $n$ is the value of $4000\cdot \left(\tfrac{2}{5}\right)^n$ an integer? $ \textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }6 \qquad \textbf{(D) }8 \qquad \textbf{(E) }9 \qquad $

Let $l$ be the tangent line at the point $P(s,\ t)$ on a circle $C:x^2+y^2=1$. Denote by $m$ the line passing through the point $(1,\ 0)$, parallel to $l$. Let the line $m$ intersects the circle $C$ at $P'$ other than the point $(1,\ 0)$. Note : if $m$ is the line $x=1$, then $P'$ is considered as $(1,\ 0)$. Call $T$ the operation such that the point $P'(s',\ t')$ is obtained from the point $P(s,\ t)$ on $C$. (1) Express $s',\ t'$ as the polynomials of $s$ and $t$ respectively. (2) Let $P_n$ be the point obtained by $n$ operations of $T$ for $P$. For $P\left(\frac{\sqrt{3}}{2},\ \frac{1}{2}\right)$, plot the points $P_1,\ P_2$ and $P_3$. (3) For a positive integer $n$, find the number of $P$ such that $P_n=P$.

$A$ is the set of points of a convex $n$-gon on a plane. The distinct pairwise distances between any $2$ points in $A$ arranged in descending order is $d_1>d_2>...>d_m>0$. Let the number of unordered pairs of points in $A$ such that their distance is $d_i$ be exactly $\mu _i$, for $i=1, 2,..., m$. Prove: For any positive integer $k\leq m$, $\mu _1+\mu _2+...+\mu _k\leq (3k-1)n$.

For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/AB=4/5$, the point $Y$ on the segment $(CX)$ such that $CY = 2YX$ and, if possible, the point $Z$ on the ray ($CA$ such that $\widehat{CXZ} = 180 - \widehat{ABC}$. We denote by $\Sigma$ the set of all triangles $T$ for which $\widehat{XYZ} = 45$. Prove that all triangles from $\Sigma$ are similar and find the measure of their smallest angle.

Let $f(n)$ and $g(n)$ be functions satisfying $$f(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}$$and $$g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}$$for positive integers $n$. Find the least positive integer $n$ such that $\tfrac{f(n)}{g(n)} = \tfrac{4}{7}$.

What is the sum of real roots of $(x\sqrt{x})^x = x^{x\sqrt{x}}$? $ \textbf{(A)}\ \frac{18}{7} \qquad\textbf{(B)}\ \frac{71}{4} \qquad\textbf{(C)}\ \frac{9}{4} \qquad\textbf{(D)}\ \frac{24}{19} \qquad\textbf{(E)}\ \frac{13}{4} $

What’s the smallest integer $n>1$ such that $p \mid \left(n^{p-1}-1\right)$ for all integers $2 \leq p \leq 10?$ [i]Individual #6[/i]

Let $S_n = 1 + 2 + ,,, + n$. Define $$T_n =\frac{S_2}{S_2- 1}\cdot \frac{S_3}{S_3 - 1}\cdot ... \cdot \frac{S_n}{S_n - 1}.$$ Find $T_{2015}.$

Let $a$, $b$ and $c$ be positive real numbers such that $a^2+b^2+c^2=3$. Prove the following inequality: $\frac{a}{3c(a^2-ab+b^2)} + \frac{b}{3a(b^2-bc+c^2)} + \frac{c}{3b(c^2-ca+a^2)} \leq \frac{1}{abc}$

We call two circles in the space fighting if they are intersected or they are clipsed. Find a good necessary and sufficient condition for four distinct points $A,B,A',B'$ such that each circle passing through $A,B$ and each circle passing through $A',B'$ are fighting circles. [i]proposed by Ali Khezeli[/i]

Let $p$ be an odd prime and $a_1, a_2,...,a_p$ be integers. Prove that the following two conditions are equivalent: 1) There exists a polynomial $P(x)$ with degree $\leq \frac{p-1}{2}$ such that $P(i) \equiv a_i \pmod p$ for all $1 \leq i \leq p$ 2) For any natural $d \leq \frac{p-1}{2}$, $$ \sum_{i=1}^p (a_{i+d} - a_i )^2 \equiv 0 \pmod p$$ where indices are taken $\pmod p$

For $n$ measured in degrees, let $T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) +\cos^2(30^\circ +n)$. Evaluate $$ 4\sum^{30}_{n=1} n \cdot T(n).$$

Let $m>1$ be a natural number and $N=m^{2017}+1$. On a blackboard, left to right, are written the following numbers: \[N, N-m, N-2m,\dots, 2m+1,m+1, 1.\] On each move, we erase the most left number, written on the board, and all its divisors (if any). This procces continues till all numbers are deleted. Which numbers will be deleted on the last move.

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 $\angle DBQ=\angle PAC$.

A quarter of an checkered plane is given, infinite to the right and up. All its rows and columns are numbered starting from $0$. All cells with coordinates $(2n, n)$, were cut out from this figure, starting from $n = 1$. In each of the remaining cells they wrote a number, the number of paths from the corner cell to this one (you can only walk up and to the right and you cannot pass through the removed cells). Prove that for each removed cell the numbers to the left and below it differ by exactly $2$.

Alex and Brian take turns shooting free throws until they each shoot twice. Alex and Brian have 80% and 60% chances of making their free throws, respectively. What is the probability that after each free throw they take, Alex has made at least as many free throws as Brian if Brian shoots first?

Find all real functions $f$ defined on $ \mathbb R$, such that \[f(f(x)+y) = f(f(x)-y)+4f(x)y ,\] for all real numbers $x,y$.