Let $h_k$ be an apothem of the regular $k$-gon inscribed into a circle with radius $R$. Prove that $$(n + 1)h_{n+1} - nh_n > R$$

A broken line $A_1A_2 \ldots A_n$ is drawn in a $50 \times 50$ square, so that the distance from any point of the square to the broken line is less than $1$. Prove that its total length is greater than $1248.$

Below is a diagram showing a $6 \times 8$ rectangle divided into four $6 \times 2$ rectangles and one diagonal line. Find the total perimeter of the four shaded trapezoids.

Suppose that \[ \prod_{n=1}^{\infty}\left(\frac{1+i\cot\left(\frac{n\pi}{2n+1}\right)}{1-i\cot\left(\frac{n\pi}{2n+1}\right)}\right)^{\frac{1}{n}} = \left(\frac{p}{q}\right)^{i \pi}, \] where $p$ and $q$ are relatively prime positive integers. Find $p+q$. [i]Note: for a complex number $z = re^{i \theta}$ for reals $r > 0, 0 \le \theta < 2\pi$, we define $z^{n} = r^{n} e^{i \theta n}$ for all positive reals $n$.[/i]

The velocities of a submerged and surfaced submarine are, respectively, $v$ and $kv$. It is situated at a point $P$ at $30$ miles from the center $O$ of a circle of $60$ mile radius. The surveillance of an enemy squadron forces him to navigate submerged while inside the circle. Discuss, according to the values of $k$, the fastest path to move to the opposite end of the diameter that passes through $P$ . (Consider the case particular $k =\sqrt5$.)

(A.Zaslavsky) Given three points $ C_0$, $ C_1$, $ C_2$ on the line $ l$. Find the locus of incenters of triangles $ ABC$ such that points $ A$, $ B$ lie on $ l$ and the feet of the median, the bisector and the altitude from $ C$ coincide with $ C_0$, $ C_1$, $ C_2$.

A crate with a base of $4 \times 2$ and a height of $2$ is open at the top. Tomas wants to completely fill the crate with some of his cubes. It has $16$ equal cubes of volume $1$ and two equal cubes of volume $8$. A cube of volume $1$ can only be placed on the top layer if the cube on the bottom layer has already been placed. In how many ways can Tom'as fill the box with cubes, placing them one by one?

Let $k$ be a fixed even positive integer, $N$ is the product of $k$ distinct primes $p_1,...,p_k$, $a,b$ are two positive integers, $a,b\leq N$. Denote $S_1=\{d|$ $d|N, a\leq d\leq b, d$ has even number of prime factors$\}$, $S_2=\{d|$ $d|N, a\leq d\leq b, d$ has odd number of prime factors$\}$, Prove: $|S_1|-|S_2|\leq C^{\frac{k}{2}}_k$

There are two non-decreasing sequences $\{a_i\}$ and $\{b_i\}$ of $n$ real numbers each, such that $a_i\le a_{i+1}$ for each $1\le i\le n-1$, and $b_i\le b_{i+1}$ for each $1\le i\le n-1$, and $\sum_{k=1}^{m}{a_k}\ge \sum_{k=1}^{m}{b_k}$ where $m\le n$ with equality for $m=n$. For a convex function $f$ defined on the real numbers, prove that $\sum_{k=1}^{n}{f(a_k)}\le \sum_{k=1}^{n}{f(b_k)}$.

Determine all integers $m$ for which the $m \times m$ square can be dissected into five rectangles, the side lengths of which are the integers $1,2,3,\ldots,10$ in some order.

Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$

Let n be a positve integer. prove there exists a positive integer n st $n^{2013}-n^{20}+n^{13}-2013$ has at least N distinct prime factors.

Bessie and her $2015$ bovine buddies work at the Organic Milk Organization, for a total of $2016$ workers. They have a hierarchy of bosses, where obviously no cow is its own boss. In other words, for some pairs of employees $(A, B)$, $B$ is the boss of $A$. This relationship satisfies an obvious condition: if $B$ is the boss of $A$ and $C$ is the boss of $B$, then $C$ is also a boss of $A$. Business has been slow, so Bessie hires an outside organizational company to partition the company into some number of groups. To promote growth, every group is one of two forms. Either no one in the group is the boss of another in the group, or for every pair of cows in the group, one is the boss of the other. Let $G$ be the minimum number of groups needed in such a partition. Find the maximum value of $G$ over all possible company structures. [i]Proposed by Yang Liu[/i]

A convex polygon $ A_1,A_2,\cdots ,A_n$ is inscribed in a circle with center $ O$ and radius $ R$ so that $ O$ lies inside the polygon. Let the inradii of the triangles $ A_1A_2A_3, A_1A_3A_4, \cdots , A_1A_{n \minus{} 1}A_n$ be denoted by $ r_1,r_2,\cdots ,r_{n \minus{} 2}$. Prove that $ r_1 \plus{} r_2 \plus{} ... \plus{} r_{n \minus{} 2}\leq R(n\cos \frac {\pi}{n} \minus{} n \plus{} 2)$.

$(MON 3)$ Let $A_k (1 \le k \le h)$ be $n-$element sets such that each two of them have a nonempty intersection. Let $A$ be the union of all the sets $A_k,$ and let $B$ be a subset of $A$ such that for each $k (1\le k \le h)$ the intersection of $A_k$ and $B$ consists of exactly two different elements $a_k$ and $b_k$. Find all subsets $X$ of the set $A$ with $r$ elements satisfying the condition that for at least one index $k,$ both elements $a_k$ and $b_k$ belong to $X$.

Let $ABC$ be a triangle with $AB > AC$ with incenter $I{}$. The internal bisector of the angle $BAC$ intersects the $BC$ at the point $D{}$. Let $M{}$ the midpoint of the segment $AD{}$, and let $F{}$ be the second intersection point of $MB$ with the circumcircle of the triangle $BIC$. Prove that $AF$ is perpendicular to $FC$.

A student steps onto a stationary elevator and stands on a bathroom scale. The elevator then travels from the top of the building to the bottom. The student records the reading on the scale as a function of time. At what time(s) does the student have maximum downward velocity? $\textbf{(A)}$ At all times between $2 s$ and $4 s$ $\textbf{(B)}$ At $4 s$ only $\textbf{(C)}$ At all times between $4 s$ and $22 s$ $\textbf{(D)}$ At $22 s$ only $\textbf{(E)}$ At all times between $22 s$ and $24 s$

Determine all possible values of integer $k$ for which there exist positive integers $a$ and $b$ such that $\dfrac{b+1}{a} + \dfrac{a+1}{b} = k$.

a) We have a geometric sequence of $3$ terms. If the sum of these terms is $26$ , and their sum of squares is $364$ , find the terms of the sequence. b) Suppose that $a,b,c,u,v,w$ are positive real numbers , and each of $a,b,c$ and $u,v,w$ are geometric sequences. Suppose also that $a+u,b+v,c+w$ are an arithmetic sequence. Prove that $a=b=c$ and $u=v=w$ c) Let $a,b,c,d$ be real numbers (not all zero), and let $f(x,y,z)$ be the polynomial in three variables defined by$$f(x,y,z) = axyz + b(xy + yz + zx) + c(x+y+z) + d$$.Prove that $f(x,y,z)$ is reducible if and only if $a,b,c,d$ is a geometric sequence.

A league with $12$ teams holds a round-robin tournament, with each team playing every other team once. Games either end with one team victorious or else end in a draw. A team scores $2$ points for every game it wins and $1$ point for every game it draws. Which of the following is $\textbf{not}$ a true statement about the list of $12$ scores? $\textbf{(A) }\text{There must be an even number of odd scores.}$ $\textbf{(B) }\text{There must be an even number of even scores.}$ $\textbf{(C) }\text{There cannot be two scores of 0.}$ $\textbf{(D) }\text{The sum of the scores must be at least 100.}$ $\textbf{(E) }\text{The highest score must be at least 12.}$

For positive numbers $a, b, c$ define $A = \frac{(a + b + c)}{3}$, $G = \sqrt[3]{abc}$, $H = \frac{3}{(a^{-1} + b^{-1} + c^{-1})}.$ Prove that \[ \left( \frac AG \right)^3 \geq \frac 14 + \frac 34 \cdot \frac AH.\]

Let $ABCD$ be a convex quadrilateral inscibed in circle $(O)$ such that $DB = DA + DC$. The point $P$ lies on the ray $AC$ such that $AP = BC$. The point $E$ is on $(O)$ such that $BE \perp AD$. Prove that $DP$ is parallel to the angle bisector of $\angle BEC$.

Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ \[ a \cos^2{x}+b \cos{x}+c=0. \] Using the numbers $a,b,c$ form a quadratic equation in $\cos{2x}$ whose roots are the same as those of the original equation. Compare the equation in $\cos{x}$ and $\cos{2x}$ for $a=4$, $b=2$, $c=-1$.

You start with a number. Every second, you can add or subtract any number of the form $n!$ to your current number to get a new number. In how many ways can you get from $0$ to $100$ in $4$ seconds? ($n!$ is dened as $n\times (n -1)\times(n - 2) ... 2\times1$, so $1! = 1$, $2! = 2$, $3! = 6$, $4! = 24$, etc.)

On the coordinate plane in the first quarter there are $100$ non-intersecting single unit segments parallel to the coordinate axes. These segments aremirrors (on both sides), they reflect the light according to the rule. "The angle of incidence is equal to the angle of reflection." (If you hit the edge of the mirror, the beam of light does not change its direction.) From the point lying in the unit circle with the center at the origin, a ray of light in the direction of the bisector of the first coordinate angle. Prove that, that this initial point can be chosen so that the ray is reflected from the mirrors not more than $150$ times.