$S_n$ is a square side $\frac{1}{n}$. Find the smallest $k$ such that the squares $S_1, S_2,S_3, ...$ can be put into a square side $k$ without overlapping.

The line $l$ intersects the extension of $AB$ in $D$ ($D$ is nearer to $B$ than $A$) and the extension of $AC$ in $E$ ($E$ is nearer to $C$ than $A$) of triangle $ABC$. Suppose that reflection of line $l$ to perpendicular bisector of side $BC$ intersects the mentioned extensions in $D'$ and $E'$ respectively. Prove that if $BD+CE=DE$, then $BD'+CE'=D'E'$.

Let $ f$ be a finite real function of one variable. Let $ \overline{D}f$ and $ \underline{D}f$ be its upper and lower derivatives, respectively, that is, \[ \overline{D}f\equal{}\limsup_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}\] , \[ \underline{D}f\equal{}\liminf_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}.\] Show that $ \overline{D}f$ and $ \underline{D}f$ are Borel-measurable functions. [A. Csaszar]

A regular hexagon and an equilateral triangle have equal perimeter. If the area of the triangle is $4\sqrt3$ square units, the area of the hexagon is (A): $5\sqrt3$, (B): $6\sqrt3$, (C): $7\sqrt3$, (D): $8\sqrt3$, (E): None of the above.

Let $A_{1}$, $B_{1}$, $C_{1}$ be the feet of the altitudes of an acute-angled triangle $ABC$ issuing from the vertices $A$, $B$, $C$, respectively. Let $K$ and $M$ be points on the segments $A_{1}C_{1}$ and $B_{1}C_{1}$, respectively, such that $\measuredangle KAM = \measuredangle A_{1}AC$. Prove that the line $AK$ is the angle bisector of the angle $C_{1}KM$.

For each integer $m$, consider the polynomial \[ P_m(x)=x^4-(2m+4)x^2+(m-2)^2. \] For what values of $m$ is $P_m(x)$ the product of two non-consant polynomials with integer coefficients?

The persons $P_1, P_2, . . . , P_{n-1}, P_n$ sit around a table, in this order, and each one of them has a number of coins. In the start, $P_1$ has one coin more than $P_2, P_2$ has one coin more than $P_3$, etc., up to $P_{n-1}$ who has one coin more than $P_n$. Now $P_1$ gives one coin to $P_2$, who in turn gives two coins to $P_3 $ etc., up to $ Pn$ who gives n coins to $ P_1$. Now the process continues in the same way: $P_1$ gives $n+ 1$ coins to $P_2$, $P_2$ gives $n+2$ coins to $P_3$; in this way the transactions go on until someone has not enough coins, i.e. a person no more can give away one coin more than he just received. At the moment when the process comes to an end in this manner, it turns out that there are two neighbours at the table such that one of them has exactly five times as many coins as the other. Determine the number of persons and the number of coins circulating around the table.

Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality \[ 1 < a < b+2 < 10. \] [i]Proposed by Lewis Chen [/i]

Let $\odot O$ , $\odot I$ be the circumcircle and inscribed circles of triangle$ABC$ . Prove that : From every point $D$ on $\odot O$ ,we can construct a triangle $DEF$ such that $ABC$ and $DEF$ have the same circumcircle and inscribed circles

In the figure, polygons $ A$, $ E$, and $ F$ are isosceles right triangles; $ B$, $ C$, and $ D$ are squares with sides of length $ 1$; and $ G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is $ \textbf{(A)}\ 1/2\qquad \textbf{(B)}\ 2/3\qquad \textbf{(C)}\ 3/4\qquad \textbf{(D)}\ 5/6\qquad \textbf{(E)}\ 4/3$ [asy] size(180); defaultpen(linewidth(.7pt)+fontsize(10pt)); draw((-1,1)--(2,1)); draw((-1,0)--(1,0)); draw((-1,1)--(-1,0)); draw((0,-1)--(0,3)); draw((1,2)--(1,0)); draw((-1,1)--(1,1)); draw((0,2)--(1,2)); draw((0,3)--(1,2)); draw((0,-1)--(2,1)); draw((0,-1)--((0,-1) + sqrt(2)*dir(-15))); draw(((0,-1) + sqrt(2)*dir(-15))--(1,0)); label("$\textbf{A}$",foot((0,2),(0,3),(1,2)),SW); label("$\textbf{B}$",midpoint((0,1)--(1,2))); label("$\textbf{C}$",midpoint((-1,0)--(0,1))); label("$\textbf{D}$",midpoint((0,0)--(1,1))); label("$\textbf{E}$",midpoint((1,0)--(2,1)),NW); label("$\textbf{F}$",midpoint((0,-1)--(1,0)),NW); label("$\textbf{G}$",midpoint((0,-1)--(1,0)),2SE);[/asy]

Let $m$ and $n$ be positive integers such that $m - n = 189$ and such that the least common multiple of $m$ and $n$ is equal to $133866$. Find $m$ and $n$.

Evaluate $\sum_{n=0}^{\infty}\frac{1}{(2n+1)2^{2n+1}}.$

Assume a triangle $ABC$ satisfies $|AB| = 1, |AC| = 2$ and $\angle ABC = \angle ACB + 90^\circ.$ What is the area of $ABC?$ \[ \mathrm a. ~ 6/7\qquad \mathrm b.~5/7\qquad \mathrm c. ~1/2 \qquad \mathrm d. ~4/5 \qquad \mathrm e. ~3/5 \]

Let $H$ denote the set of the matrices from $\mathcal{M}_n(\mathbb{N})$ and let $P$ the set of matrices from $H$ for which the sum of the entries from any row or any column is equal to $1$. a)If $A\in P$, prove that $\det A=\pm 1$. b)If $A_1,A_2,\ldots,A_p\in H$ and $A_1A_2\cdot \ldots\cdot A_p\in P$, prove that $A_1,A_2,\ldots,A_p\in P$.

Let a function $g:\mathbb{N}_0\to\mathbb{N}_0$ satisfy $g(0)=0$ and $g(n)=n-g(g(n-1))$ for all $n\ge 1$. Prove that: a) $g(k)\ge g(k-1)$ for any positive integer $k$. b) There is no $k$ such that $g(k-1)=g(k)=g(k+1)$.

A computer program generated $175$ positive integers at random, none of which had a prime divisor grater than $10.$ Prove that there are three numbers among them whose product is the cube of an integer.

A piece of graph paper is folded once so that $ (0,2)$ is matched with $ (4,0)$ and $ (7,3)$ is matched with $ (m,n)$. Find $ m \plus{} n$. $ \textbf{(A)}\ 6.7\qquad \textbf{(B)}\ 6.8\qquad \textbf{(C)}\ 6.9\qquad \textbf{(D)}\ 7.0\qquad \textbf{(E)}\ 8.0$

A segment $AB$ is marked on a line $t$. The segment is moved on the plane so that it remains parallel to $t$ and that the traces of points $A$ and $B$ do not intersect. The segment finally returns onto $t$. How far can point $A$ now be from its initial position?

$n$ people numbered from $1$ to $n$ are arranged in a row. An [i]acceptable movement[/i] consists of each person changing at most once its place with another or remains in its place. For example $\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline initial position & 1 & 2 & 3 & 4 & 5 & 6 & ... & n-2 & n-1 & n \\ \hline final position & 2 & 1 & 3 & 6 & 5 & 4 & ... & n & n-1 & n-2 \\ \hline \end{tabular}$ is an a[i]cceptable movement[/i]. Is it possible that starting from the position $\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 2 & 3 & 4 & 5 & 6 & ... & n-2 & n-1 & n \\ \hline \end{tabular}$ to reach to $\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & ... & n-2 & n-1 \\ \hline \end{tabular}$ through two [i]acceptable movements[/i]?

Suppose two positive real numbers are given. Prove that if their sum is less than their product then their sum is greater than four. (N Vasiliev, Moscow)

Two players take turns to write natural numbers on a board. The rules forbid writing numbers greater than $p$ and also divisors of previously written numbers. The player who has no move loses. Determine which player has a winning strategy for $p = 10$ and describe this strategy.

Source: 1976 Euclid Part A Problem 9 ----- A circle has an inscribed triangle whose sides are $5\sqrt{3}$, $10\sqrt{3}$, and $15$. The measure of the angle subtended at the centre of the circle by the shortest side is $\textbf{(A) } 30 \qquad \textbf{(B) } 45 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } \text{none of these}$

You are at one vertex of a equilateral triangle with side length $ 1 $. All of the edges of the equilateral triangle will reflect the laser beam perfectly (angle of incidence is equal to angle of reflection). Given that the laser beam bounces off exactly $ 137 $ edges and returns to the original vertex without touching any other vertices, let $ M $ be the maximum possible distance the beam could have traveled, and $ m $ be the minimum possible distance the beam could have traveled. Find $ M^2 - m^2 $.

Let $ABCD$ be a parallelogram and $P$ a point in the plane. The line $BP$ intersects the circumcircle of $ABC$ again at $X$ and the line $DP$ intersects the circumcircle of $DAC$ again at $Y$. Let $M$ be the midpoint of the side $AC$. The point $N$ lies on the circumcircle of $PXY$ so that $MN$ is a tangent to this circle. Prove that the segments $MN$ and $AM$ have the same length. [i]Proposed by David Anghel[/i]

Let $I$ be the incenter of a triangle $ABC$, $D$ be an arbitrary point of segment $AC$, and $A_{1}, A_{2}$ be the common points of the perpendicular from $D$ to the bisector $CI$ with $BC$ and $AI$ respectively. Define similarly the points $C_{1}$, $C_{2}$. Prove that $B$, $A_{1}$, $A_{2}$, $I$, $C_{1},$ $C_{2}$ are concyclic. Proposed by:D.Shvetsov