For what $n$ can the vertices of a regular $n$-gon be connected in a $n$-link closed polyline so that such a polyline does not have three equal links?

Let $AD$ be an internal angle bisector in triangle $\Delta ABC$. An arbitrary point $M$ is chosen on the closed segment $AD$. A parallel to $BC$ through $M$ cuts $AB$ at $N$. Let $AD, CM$ cut circumcircle of $\Delta ABC$ at $K, L$, respectively. Prove that $K,N,L$ are collinear.

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

Simplify \[\frac{1}{\log_a(abc)}+\frac{1}{\log_b(abc)}+\frac{1}{\log_c(abc)},\] where $a, b, c$ are positive real numbers.

If $a$ , $b$ , $c$ , $d$ are integers and $A=2(a-2b+c)^4+2(b-2c+a)^4+2(c-2a+b)^4$ , $B=d(d+1)(d+2)(d+3)+1$ , then prove that $\left (\sqrt{A}+1 \right )^2 +B$ cannot be a perfect square.

$I$ is incenter of $\triangle{ABC}$. The incircle touches $BC,CA,AB$ at $D,E,F$, respectively . Let $M,N,K=BI,CI,DI \cap EF$ respectively and $BN\cap CM=P,AK\cap BC=G$. Point $Q$ is intersection of the perpendicular line to $PG$ through $I$ and the perpendicular line to $PB$ through $P$. Prove that $BI$ bisect segment $PQ$.

Calculate the sum of digit of the number $$9 \times 99 \times 9999 \times ... \times \underbrace{ 99...99}_{2^n}$$ where the number of nines doubles in each factor.

Find all real values of $a$ for which the equation $x(x+1)^3=(2x+a)(x+a+1)$ has four distinct real roots.

Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths \[ a, f(b) \text{ and } f(b \plus{} f(a) \minus{} 1).\] (A triangle is non-degenerate if its vertices are not collinear.) [i]Proposed by Bruno Le Floch, France[/i]

In order to earn her vacation spending money, Alexis helped her mother remove weeds from the garden. When she was done, she came into the house to put away her gardening gloves and change into clean clothes. On her way to her room she notices Joshua with his face to the floor in the family room, looking pretty silly. "Josh, did you know you lose IQ points for sniffing the carpet?" "Shut up. I'm $\textit{not}$ sniffing the carpet. I'm $\textit{doing something}$." "Sure, if $\textit{sniffing the carpet}$ counts as $\textit{doing something}.$" At this point Alexis stands over her twin brother grinning, trying to see how silly she can make him feel. Joshua climbs to his feet and stands on his toes to make himself a half inch taller than his sister, who is ordinarily a half inch taller than Joshua. "I'm measuring something. I'm $\textit{designing}$ something." Alexis stands on her toes too, reminding her brother that she is still taller than he. "When you're done, can you design me a dress?" "Very funny." Joshua walks to the table and points to some drawings. "I'm designing the sand castle I want to build at the beach. Everything needs to be measured out so that I can build something awesome." "And this requires sniffing carpet?" inquires Alexis, who is just a little intrigued by her brother's project. "I was imagining where to put the base of a spiral staircase. Everything needs to be measured out correctly. See, the castle walls will be in the shape of a rectangle, like this room. The center of the staircase will be $9$ inches from one of the corners, $15$ inches from another, $16$ inches from another, and some whole number of inches from the furthest corner." Joshua shoots Alexis a wry smile. The twins liked to challenge each other, and Alexis knew she had to find the distance from the center of the staircase to the fourth corner of the castle on her own, or face Joshua's pestering, which might last for hours or days. Find the distance from the center of the staircase to the furthest corner of the rectangular castle, assuming all four of the distances to the corners are described as distances on the same plane (the ground).

Given a quadratic trinomial $f\left(x\right)=x^2+ax+b$. Assume that the equation $f\left(f\left(x\right)\right)=0$ has four different real solutions, and that the sum of two of these solutions is $-1$. Prove that $b\leq -\frac14$.

Prove that in Euclidean space the surfaces \[((A-\lambda E)^{-1}x,x)=1\] passing through the point $x$ and corresponding to different values of $\lambda$ are pairwise orthogonal ($A$ is a symmetric operator without multiple eigenvalues).

Let $n$ be a positive integer, and let $p$ be a prime number. Prove that if $p^p | n!$, then $p^{p+1} | n!$.

Call a polynomial $ P(x_{1}, \ldots, x_{k})$ [i]good[/i] if there exist $ 2\times 2$ real matrices $ A_{1}, \ldots, A_{k}$ such that $ P(x_{1}, \ldots, x_{k}) = \det \left(\sum_{i=1}^{k}x_{i}A_{i}\right).$ Find all values of $ k$ for which all homogeneous polynomials with $ k$ variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)

Let $n\geq2$ be an integer. $n$ is a prime if it is only divisible by $1$ and $n$. Prove that there are infinitely many prime numbers.

Each of the points $G$ and $H$ lying from different sides of the plane of hexagon $ABCDEF$ is connected with all vertices of the hexagon. Is it possible to mark 18 segments thus formed by the numbers $1, 2, 3, \ldots, 18$ and arrange some real numbers at points $A, B, C, D, E, F, G, H$ so that each segment is marked with the difference of the numbers at its ends? [i]Proposed by A. Golovanov[/i]

Let $S$ be a subset of $\{0,1,2,\ldots,98 \}$ with exactly $m\geq 3$ (distinct) elements, such that for any $x,y\in S$ there exists $z\in S$ satisfying $x+y \equiv 2z \pmod{99}$. Determine all possible values of $m$.

For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

Let $S$ be a set of 6 integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$? $\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7$

Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be a point on the line $AB$, distinct from $B$, such that $|CG| = |CB|$. Let $H$ be a point on the line $BC$, distinct from $B$, such that $|AB| =|AH|$. Prove that triangle $DGH$ is isosceles. [asy] unitsize(1.5 cm); pair A, B, C, D, G, H; A = (0,0); B = (2,0); D = (0.5,1.5); C = B + D - A; G = reflect(A,B)*(C) + C - B; H = reflect(B,C)*(H) + A - B; draw(H--A--D--C--G); draw(interp(A,G,-0.1)--interp(A,G,1.1)); draw(interp(C,H,-0.1)--interp(C,H,1.1)); draw(D--G--H--cycle, dashed); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, E); dot("$D$", D, NW); dot("$G$", G, NE); dot("$H$", H, SE); [/asy]

Show that there exists an integer divisible by $1996$ such that the sum of the its decimal digits is $1996$.

All letters in the word $VUQAR$ are different and chosen from the set $\{1,2,3,4,5\}$. Find all solutions to the equation \[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\]

If $a_1 ,a_2 ,\ldots, a_n$ are complex numbers such that $$ |a_1| =|a_2 | =\cdots = |a_n| =r \ne 0,$$ and if $T_s$ denotes the sum of all products of these $n$ numbers taken $s$ at a time, prove that $$ \left| \frac{T_s }{T_{n-s}}\right| =r^{2s-n}$$ whenever the denominator of the left-hand side is different from $0$.

The sequence $a_1, a_2, a_3,...$ is defined by $$a_1 = 1\,\,\,, \,\,\,a_{n+1} =\frac{1}{16}(1 + 4a_n +\sqrt{1 + 24a_n}) \,\,\,(n \in N^* ).$$ Determine and prove a formula with which for every natural number $n$ the term $a_n$ can be computed directly without having to determine preceding terms of the sequence.

Show that there exists a positive number t such that for all positive numbers $a,b,c,d$ with $abcd = 1$, $$\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+d}> t.$$ and find the largest $t$ with this property.