Given positive integer $ n \ge 5 $ and a convex polygon $P$, namely $ A_1A_2...A_n $. No diagonals of $P$ are concurrent. Proof that it is possible to choose a point inside every quadrilateral $ A_iA_jA_kA_l (1\le i<j<k<l\le n) $ not on diagonals of $P$, such that the $ \tbinom{n}{4} $ points chosen are distinct, and any segment connecting these points intersect with some diagonal of P.

Let $p$, $q$, and $r$ be distinct positive prime numbers. Show that if \[pqr\mid (pq)^r+(qr)^p+(rp)^q-1,\] then \[(pqr)^3\mid 3((pq)^r+(qr)^p+(rp)^q-1).\]

If 100 dice are rolled, what is the probability that the sum of the numbers rolled is even? [i]Lightning 1.1[/i]

Let be two positive real numbers $ a,b $ whose product is $ 1$ and whose sum is irrational. Prove that for any natural number $ n\ge 2 $ the epression $ \sqrt[n]{a}+\sqrt[n]{b} $ is irrational. [i]Râmbu Gheorghe[/i]

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

Three segments $2$ cm, $5$ cm and $12$ cm long are constructed on the plane. Construct a trapezoid with bases of $2$ cm and $5$ cm, the sum of the sides of which is $12$ cm, and one of the angles is $60^o$. (Bogdan Rublev)

Find the positive integer $k$ such that the roots of $x^3 - 15x^2 + kx -1105$ are three distinct collinear points in the complex plane.

In triangle $ABC$ with incenter $I$ and circumcenter $O$, let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$-mixtilinear circles, respectively. Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$, and define $\omega_B, \omega_C$ similarly. Prove that $\omega_A,\omega_B,\omega_C$ have a common point $X$ other than $I$, and that $\angle AXO = \angle OXA'$. [i]Proposed by Sammy Luo[/i]

In the following diagram (not to scale), $A$, $B$, $C$, $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$. Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$. Find $\angle OPQ$ in degrees. [asy] pathpen = rgb(0,0,0.6)+linewidth(0.7); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6)+ linewidth(0.7) + linetype("4 4"); real n = 10, start = 360/n*6-15; pair O=(0,0), A=dir(start), B=dir(start+360/n), C=dir(start+2*360/n), D=dir(start+3*360/n), P=(A+C)/2, Q=(O+D)/2; D(D("O",O,NE)--D("A",A,W)--D("B",B,SW)--D("C",C,S)--D("D",D,SE)--O--D("P",P,1.6*dir(95))--D("Q",Q,NE)); D(A--C); D(A--(A+dir(start-360/n))/2, dd); D(D--(D+dir(start+4*360/n))/2, dd); [/asy]

The head of the Mint wants to release 12 coins denominations (each - a natural number rubles) so that any amount from 1 to 6543 rubles could be paid without having to pass, using no more than 8 coins. Can he do it? (If the payment amount you can use a few coins of the same denomination.)

A chess tournament had 10 participants. Each round, the participants split into pairs, and each pair played a game. In total, each participant played with every other participant exactly once, and in at least half of the games both the players were from the same town. Prove that during each round there was a game played by two participants from the same town. [i](Boris Frenkin)[/i]

On a unit circle with center $O$, $AB$ is an arc with the central angle $\alpha<90^\circ$. Point $H$ is the foot of the perpendicular from $A$ to $OB$, $T$ is a point on arc $AB$, and $l$ is the tangent to the circle at $T$. The line $l$ and the angle $AHB$ form a triangle $\Delta$. (a) Prove that the area of $\Delta$ is minimal when $T$ is the midpoint of arc $AB$. (b) Prove that if $S_\alpha$ is the minimal area of $\Delta$ then the function $\frac{S_\alpha}\alpha$ has a limit when $\alpha\to0$ and find this limit.

Let $ABC$ be a triangle with incenter $I$. Let $P$ and $Q$ be points such that $IP\perp AC$, $IQ\perp AB$, and $IA\perp PQ$. Assume that $BP$ and $CQ$ intersect at the point $R\neq A$ on the circumcircle of $ABC$ such that $AR\parallel BC$. Given that $\angle B-\angle C=36^\circ$, the value of $\cos A$ can be expressed in the form $\frac{m-\sqrt n}{p}$ for positive integers $m,n,p$ and where $n$ is not divisible by the square of any prime. Find the value of $100m+10n+p$. [i]Proposed by Michael Ren[/i]

Find the sum of all the real values of x satisfying $(x+\frac{1}{x}-17)^2$ $= x + \frac{1}{x} + 17.$

2. A unit square paper has a triangle-shaped hole (vertices of the hole are not on the border of the paper). Prove that a triangle with area of $1 / 6$ can be cut from the remaining paper. Alexandr Yuran

Triangle $ ABC$ has a right angle at $ B$. Point $ D$ is the foot of the altitude from $ B$, $ AD\equal{}3$, and $ DC\equal{}4$. What is the area of $ \triangle{ABC}$? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21)); pair D=foot(B,A,C); pair[] ps={B,C,A,D}; draw(A--B--C--cycle); draw(B--D); draw(rightanglemark(B,D,C)); dot(ps); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NE); label("$3$",midpoint(A--D),NE); label("$4$",midpoint(D--C),NE);[/asy]$ \textbf{(A)}\ 4\sqrt3 \qquad \textbf{(B)}\ 7\sqrt3 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 14\sqrt3 \qquad \textbf{(E)}\ 42$