Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .

Around the outside of a $4$ by $4$ square, construct four semicircles (as shown in the figure) with the four sides of the square as their diameters. Another square, $ABCD$, has its sides parallel to the corresponding sides of the original square, and each side of $ABCD$ is tangent to one of the semicircles. The area of the square $ABCD$ is [asy] pair A,B,C,D; A = origin; B = (4,0); C = (4,4); D = (0,4); draw(A--B--C--D--cycle); draw(arc((2,1),(1,1),(3,1),CCW)--arc((3,2),(3,1),(3,3),CCW)--arc((2,3),(3,3),(1,3),CCW)--arc((1,2),(1,3),(1,1),CCW)); draw((1,1)--(3,1)--(3,3)--(1,3)--cycle); dot(A); dot(B); dot(C); dot(D); dot((1,1)); dot((3,1)); dot((1,3)); dot((3,3)); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); [/asy] $\text{(A)}\ 16 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 48 \qquad \text{(E)}\ 64$

Show that for all integers $n>1$, \[ \dfrac {1}{2ne} < \dfrac {1}{e} - \left( 1 - \dfrac {1}{n} \right)^n < \dfrac {1}{ne}. \]

Two circles $\Gamma_1$ and $\Gamma_2$ meet at two distinct points $A$ and $B$. A line passing through $A$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ respectively, such that $A$ lies between $C$ and $D$. The tangent at $A$ to $\Gamma_2$ meets $\Gamma_1$ again at $E$. Let $F$ be a point on $\Gamma_2$ such that $F$ and $A$ lie on different sides of $BD$, and $2\angle AFC=\angle ABC$. Prove that the tangent at $F$ to $\Gamma_2$, and lines $BD$ and $CE$ are concurrent.

Let $K, K_1$ be two circles with the same center and their radii equal to $R$ and $R_1 (R_1>R)$ respectively. Quadrilateral $ABCD$ is inscribed in circle $K$. Quadrilateral $A_1B_1C_1D_1$ is inscribed in circle $K_1$ where $A_1,B_1,C_1,D_1$ lie on rays $CD,DA,AB,BC$ respectively. Show that $\dfrac{S_{A_1B_1C_1D_1}}{S_{ABCD}}\ge \dfrac{R^2_1}{R^2}$.

A club with $x$ members is organized into four committees in accordance with these two rules: $ \text{(1)}\ \text{Each member belongs to two and only two committees}\qquad$ $\text{(2)}\ \text{Each pair of committees has one and only one member in common}$ Then $x$: $\textbf{(A)} \ \text{cannont be determined} \qquad$ $\textbf{(B)} \ \text{has a single value between 8 and 16} \qquad$ $\textbf{(C)} \ \text{has two values between 8 and 16} \qquad$ $\textbf{(D)} \ \text{has a single value between 4 and 8} \qquad$ $\textbf{(E)} \ \text{has two values between 4 and 8} \qquad$

Let $AD, BE$ be the altitudes and $CF$ be the angle bissector of acute non-isosceles triangle $ABC$ and $AE+BD=AB.$ Denote by $I_A, I_B, I_C$ the incentres of triangles $AEF,$ $BDF,$ $CDE$ respectively. Prove that points $D, E, F, I_A, I_B$ and $I_C$ lie on the same circle.

An unfair coin, which has the probability of $a\ \left(0<a<\frac 12\right)$ for showing $Heads$ and $1-a$ for showing $Tails$, is flipped $n\geq 2$ times. After $n$-th trial, denote by $A_n$ the event that heads are showing on at least two times and by$B_n$ the event that are not showing in the order of $tails\rightarrow heads$, until the trials $T_1,\ T_2,\ \cdots ,\ T_n$ will be finished . Answer the following questions: (1) Find the probabilities $P(A_n),\ P(B_n)$. (2) Find the probability $P(A_n\cap B_n )$. (3) Find the limit $\lim_{n\to\infty} \frac{P(A_n) P(B_n)}{P(A_n\cap B_n )}.$ You may use $\lim_{n\to\infty} nr^n=0\ (0<r<1).$

Find the greatest integer less than the number $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{1000000}}$

Let $\alpha$ and $\beta$ be real numbers with $\beta \ne 0$. Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[f(\alpha f(x)+f(y))=\beta x+f(y)\] holds for all real $x$ and $y$. [i](Walther Janous)[/i]

We call a diagonal of a polygon [i]nice[/i], if it is entirely inside the polygon or entirely outside the polygon. Let $P$ be an $n$–gon with no three of its vertices being on the same line. Prove that $P$ has at least $3(n-3)/2$ nice diagonals. [i]Proposed by Bálint Hujter, Budapest and Gábor Szűcs, Szikszó[/i]

Are there $10$ consecutive positive integers, such that if we consider the digits that appear in the decimal representations of those numbers as a multiset, every digit appears the same number of times in this multiset?

A single segment contains several non-intersecting red segments, the total length of which is greater than $0.5$. Are there necessarily two red dots at the distance: a) $1/99$ b) $1/100$ ?

Let $\mathbb{Q}$ denote the set of rational numbers. A nonempty subset $S$ of $\mathbb{Q}$ has the following properties: (a) $0$ is not in $S$; (b) for each $s_1,s_2$ in $S$, the rational number $s_1/s_2$ is in $S$; (c) there exists a nonzero number $q\in \mathbb{Q} \backslash S$ that has the property that every nonzero number in $\mathbb{Q} \backslash S$ is of the form $qs$ for some $s$ in $S$. Prove that if $x$ belongs to $S$, then there exists elements $y,z$ in $S$ such that $x=y+z$.

Prove that the triangle ABC is right-angled if it holds: \[ \sin^2 A+\sin^2 B+\sin^2 C = 2 \]

Let $E$ be the midpoint of the side $[BC]$ of $\triangle ABC$ with $|AB|=7$, $|BC|=6$, and $|AC|=5$. The line, which passes through $E$ and is perpendicular to the angle bisector of $\angle A$, intersects $AB$ at $D$. What is $|AD|$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ \frac 92 \qquad\textbf{(D)}\ 3\sqrt 2 \qquad\textbf{(E)}\ \text{None of above} $

In a convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ are equal. We construct four equilateral triangles with centers $O_1,O_2,O_3,O_4$ on the sides sides $AB, BC, CD, DA$ outside of this quadrilateral, respectively. Show that $O_1O_3 \perp O_2O_4$.

Let $\gamma_1$ and $\gamma_2$ be two circles tangent at point $T$, and let $\ell_1$ and $\ell_2$ be two lines through $T$. The lines $\ell_1$ and $\ell_2$ meet again $\gamma_1$ at points $A$ and $B$, respectively, and $\gamma_2$ at points $A_1$ and $B_1$, respectively. Let further $X$ be a point in the complement of $\gamma_1 \cup \gamma_2 \cup \ell_1 \cup \ell_2$. The circles $ATX$ and $BTX$ meet again $\gamma_2$ at points $A_2$ and $B_2$, respectively. Prove that the lines $TX$, $A_1B_2$ and $A_2B_1$ are concurrent. [i]***[/i]

The diagram below contains eight line segments, all the same length. Each of the angles formed by the intersections of two segments is either a right angle or a $45$ degree angle. If the outside square has area $1000$, find the largest integer less than or equal to the area of the inside square. [asy] size(130); real r = sqrt(2)/2; defaultpen(linewidth(0.8)); draw(unitsquare^^(r,0)--(0,r)^^(1-r,0)--(1,r)^^(r,1)--(0,1-r)^^(1-r,1)--(1,1-r)); [/asy]

The sides of triangle are $x$, $2x+1$ and $x+2$ for some positive rational $x$. Angle of triangle is $60$ degree. Find perimeter

How many ordered sequences of $36$ digits have the property that summing the digits to get a number and taking the last digit of the sum results in a digit which is not in our original sequence? (Digits range from $0$ to $9$.)

Let $ANC$, $CLB$ and $BKA$ be triangles erected on the outside of the triangle $ABC$ such that $\angle NAC = \angle KBA = \angle LCB$ and $\angle NCA = \angle KAB = \angle LBC$. Let $D$, $E$, $G$ and $H$ be the midpoints of $AB$, $LK$, $CA$ and $NA$ respectively. Prove that $DEGH$ is a parallelogram.

(a) In this drawing, there are three squares on each side of the square. Place a natural number in each of the boxes so that the sum of the numbers of two adjacent boxes is always odd. [img]https://cdn.artofproblemsolving.com/attachments/e/6/75517b7d49857abd3f8f0430a70ae5b0eb1554.gif[/img] (b) In this drawing, there are now four squares on each side of the triangle. Justify why a natural number cannot be placed in each box so that the sum of the numbers in two adjacent boxes is always odd. [img] https://cdn.artofproblemsolving.com/attachments/c/8/061895b9c1cdcb132f7d37087873b7de3fb5f3.gif[/img] (c) If you now draw a polygon with$ 51$ sides and on each side you place $50$ boxes, taking care that there is a box at each vertex. Can you place a natural number in each box so that the sum of the numbers in two adjacent boxes is always odd? Why?

In the figure, a semicircle is folded along the $ AN $ string and intersects the $ MN $ diameter in $ B $. $ MB: BN = 2: 3 $ and $ MN = 10 $ are known to be. If $ AN = x $, what is the value of $ x ^ 2 $?

Given a board consists of $n \times n$ unit squares ($n \ge 3$). Each unit square is colored black and white, resembling a chessboard. In each step, TOMI can choose any $2 \times 2$ square and change the color of every unit square chosen with the other color (white becomes black and black becomes white). Find every $n$ such that after a finite number of moves, every unit square on the board has a same color.