Let $n \ge 3$ be an integer. Prove that there exist positive integers $\ge 2$, $a_1,a_2,..,a_n$, such that $a_1 a_2 ... \widehat{a_i}... a_n \equiv 1$ (mod $a_i$), for $i = 1,..., n$. Here $\widehat{a_i}$ means the term $a_i$ is omitted.

In a convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ are equal, and they intersect at $E$. Perpendicular bisectors of $AB$ and $CD$ intersect at point $P$ lying inside triangle $AED$, and perpendicular bisectors of $BC$ and $DA$ intersect at point $Q$ lying inside triangle $CED$. Prove that $\angle PEQ = 90^\circ$.

Let $a_1, a_2, \dots, a_6$ be a permutation of $1, 2, 2, 3, 4, 4$. Let $b_i = 5 - a_i$. Find the minimum value of \[\sum_{i=1}^7{\left(\prod_{j=0}^{i-1}{b_j}\right)\left(\prod_{j=i}^{6}{a_j}\right)}.\] [i]Lightning 4.1[/i]

Prove that if the function $ ax^2 + bx + c $ takes an integer value for every integer value of the variable $ x $, then $ 2a $, $ a + b $, $ c $ are integers and vice versa.

The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle? [asy] size(5cm); filldraw((4,0)--(8,3)--(8-3/4,4)--(1,4)--cycle,mediumgray); draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.1)); draw((1,0)--(1,4)--(4,0)--(8,3)--(5,7)--(1,4),linewidth(1.1)); label("$4$", (8,2), E); label("$8$", (4,0), S); label("$5$", (3,11/2), NW); draw((1,.35)--(1.35,.35)--(1.35,0),linewidth(.4)); draw((5,7)--(5+21/100,7-28/100)--(5-7/100,7-49/100)--(5-28/100,7-21/100)--cycle,linewidth(.4)); [/asy] $\textbf{(A) } 15\dfrac{1}{8} \qquad \textbf{(B) } 15\dfrac{3}{8} \qquad \textbf{(C) } 15\dfrac{1}{2} \qquad \textbf{(D) } 15\dfrac{5}{8} \qquad \textbf{(E) } 15\dfrac{7}{8}$

Let $M,N$ and $P$ be the midpoints of sides $BC,CA$ and $AB$ respectively, of the acute triangle $ABC.$ Let $A',B'$ and $C'$ be the antipodes of $A,B$ and $C$ in the circumcircle of triangle $ABC.$ On the open segments $MA',NB'$ and $PC'$ we consider points $X,Y$ and $Z$ respectively such that \[\frac{MX}{XA'}=\frac{NY}{YB'}=\frac{PZ}{ZC'}.\][list=a] [*]Prove that the lines $AX,BY,$ and $CZ$ are concurrent at some point $S.$ [*]Prove that $OS<OG$ where $O$ is the circumcenter and $G$ is the centroid of triangle $ABC.$ [/list]

(a) Prove that $\{n\sqrt3\} >\frac{1}{n\sqrt3}$ for any positive integer $n$. (b) Is there a constant $c > 1$ such that $\{n\sqrt3\} >\frac{c}{n\sqrt3}$ for all $n \in N$?

For a positive integer $n$, let $\sigma (n)$ be the sum of the divisors of $n$ (for example $\sigma (10) = 1 + 2 + 5 + 10 = 18$). For how many $n \in \{1, 2,. .., 100\}$, do we have $\sigma (n) < n+ \sqrt{n}$?

Let $a$, $b$, $c$ be positive real numbers such that $a + b + c = 1$. Prove that $$\sqrt[3]{\left (\frac{1+a}{b+c}\right )^{\frac{1-a}{bc}}\left (\frac{1+b}{c+a}\right )^{\frac{1-b}{ca}}\left (\frac{1+c}{a+b}\right )^{\frac{1-c}{ab}}} \geq 64 $$

The roots of $ x(x^2\plus{}8x\plus{}16)(4\minus{}x)\equal{}0$ are: $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 0,4 \qquad\textbf{(C)}\ 0,4,\minus{}4 \qquad\textbf{(D)}\ 0,4,\minus{}4,\minus{}4 \qquad\textbf{(E)}\ \text{none of these}$

An arbitary point $D$ is selected on arc $BC$ not containing $A$ on $(ABC)$. $P$ and $Q$ are the reflections of point $B$ and $C$ with respect to $AD$, respectively. Circumcircles of $ABQ$ and $ACP$ intersect at $E\neq A$. Prove that $A;D;E$ is colinear

$ OA, OB, OC, OD$ are 4 rays in space such that the angle between any two is the same. Show that for a variable ray $ OX,$ the sum of the cosines of the angles $ XOA, XOB, XOC, XOD$ is constant and the sum of the squares of the cosines is also constant.

A cylindrical container of revolution is partially filled with a liquid whose density we ignore. Placing it with the axis inclined $30^o$ with respect to the vertical, we observe that when removing liquid so that the level falls $1$ cm, the weight of the contents decreases $40$ g. How much will the weight of that content decrease for each centimeter that lower the level if the axis makes an angle of $45^o$ with the vertical? It is supposed that the horizontal surface of the liquid does not touch any of the bases of the container.

A running track is the ring formed by two concentric circles. It is $ 10$ feet wide. The circumference of the two circles differ by about: $ \textbf{(A)}\ 10\text{ feet} \qquad\textbf{(B)}\ 30\text{ feet} \qquad\textbf{(C)}\ 60\text{ feet} \qquad\textbf{(D)}\ 100\text{ feet} \\ \textbf{(E)}\ \text{none of these}$

Given a positive integer $ n\geq 2$, consider a set of $ n$ islands arranged in a circle. Between every two neigboring islands two bridges are built as shown in the figure. Starting at the island $ X_1$, in how many ways one can one can cross the $ 2n$ bridges so that no bridge is used more than once?

The symbol $ 25_b$ represents a two-digit number in the base $ b$. If the number $ 52_b$ is double the number $ 25_b$, then $ b$ is: $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 12$

Let $ a,b,c>0$. Prove that $ \frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}\geq 3\sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}}$

For which natural number $n$ is it possible to place natural number from 1 to $3n$ on the edges of a right $n$-angled prism (on each edge there is exactly one number placed and each one is used exactly 1 time) in such a way, that the sum of all the numbers, that surround each face is the same?

Let $ABC$ be a triangle for which $AB+BC = 3AC$. Let $D$ and $E$ be the points of tangency of the incircle with the sides $AB$ and $BC$ respectively, and let $K$ and $L$ be the other endpoints of the diameters originating from $D$ and $E.$ Prove that $C , A, L$, and $K$ lie on a circle.

Each vertex of a regular polygon is colored in one of three colors so that an odd number of vertices are colored in each of the three colors. Prove that the number of isosceles triangles whose vertices are colored in three different colors is odd. [i]From foreign Olympiads[/i]

Let $A',B',C'$ be the midpoints of the sides $BC, CA, AB$, respectively, of an acute non-isosceles triangle $ABC$, and let $D,E,F$ be the feet of the altitudes through the vertices $A,B,C$ on these sides respectively. Consider the arc $DA'$ of the nine point circle of triangle $ABC$ lying outside the triangle. Let the point of trisection of this arc closer to $A'$ be $A''$. Define analogously the points $B''$ (on arc $EB'$) and $C''$(on arc $FC'$). Show that triangle $A''B''C''$ is equilateral.

Let $ n$ be a positive integer. Show that the numbers \[ \binom{2^n \minus{} 1}{0},\; \binom{2^n \minus{} 1}{1},\; \binom{2^n \minus{} 1}{2},\; \ldots,\; \binom{2^n \minus{} 1}{2^{n \minus{} 1} \minus{} 1}\] are congruent modulo $ 2^n$ to $ 1$, $ 3$, $ 5$, $ \ldots$, $ 2^n \minus{} 1$ in some order. [i]Proposed by Duskan Dukic, Serbia[/i]

Let $ABC$ an acute triangle and $D,E$ and $F$ be the feet of altitudes from $A,B$ and $C$, respectively. The line $EF$ and the circumcircle of $ABC$ intersect at $P$, such that $F$ it´s between $E$ and $P$. Lines $BP$ and $DF$ intersect at $Q$. Prove that if $ED=EP$, then $CQ$ and $DP$ are parallel.

Let be a $ 5\times 5 $ complex matrix $ A $ whose trace is $ 0, $ and such that $ I_5-A $ is invertible. Prove that $ A^5\neq I_5. $

In a $3$ by $3$ table, by a $k$-worm, we mean a path of different cells $(S_1,S_2,...,S_k)$ such that each two consecutive cells have one side in common. The $k$-worm at each steep can go one cell forward and turn to the $(S,S_1,...,S_{k-1})$ if $S$ is an unfilled cell which is adjacent (has one side in common) with $S_1$. Find the maximum number of $k$ such that there is a $k$-worm $(S_1,...,S_k)$ such that after finitly many steps can be turned to $(S_k,...,S_1)$.