We call number as funny if it divisible by sum its digits $+1$.(for example $ 1+2+1|12$ ,so $12$ is funny) What is maximum number of consecutive funny numbers ? [i] O. Podlipski [/i]

$P$ is a point on the base of an isosceles triangle. Lines parallel to the sides through $P$ meet the sides at $Q$ and $R$. Show that the reflection of $P$ in the line $QR$ lies on the circumcircle of the triangle.

A natural number $n$ is called "good" if there exists natural numbers $a$ and $b$ such that $a+b=n$ and $ab \mid n^2+n+1$. Show that there are infinitely many "good" numbers

Let $ y_0$ be chosen randomly from $ \{0, 50\}$, let $ y_1$ be chosen randomly from $ \{40, 60, 80\}$, let $ y_2$ be chosen randomly from $ \{10, 40, 70, 80\}$, and let $ y_3$ be chosen randomly from $ \{10, 30, 40, 70, 90\}$. (In each choice, the possible outcomes are equally likely to occur.) Let $ P$ be the unique polynomial of degree less than or equal to $ 3$ such that $ P(0) \equal{} y_0$, $ P(1) \equal{} y_1$, $ P(2) \equal{} y_2$, and $ P(3) \equal{} y_3$. What is the expected value of $ P(4)$?

$\triangle{ABC}$ is equailateral. $E$ is any point on $\overline{AC}$ produced and the equilateral $\triangle{ECD}$ is drawn. If $M$ and $N$ are the midpoints of $\overline{AD}$ and $\overline{EB}$ respectively then show that $\triangle{CMN}$ is equilateral.

Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca\leq 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\leq \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)\] [i]Proposed by Dzianis Pirshtuk, Belarus[/i]

Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 \\ (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 \\ (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 \\ (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 \\ (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \\ \end{eqnarray*}

If for numbers $a$, $b$ and $c$ holds $a : b=4:3$ and $b : c=2:5$, find the value $$(3a-2b):(b+2c)$$

A $m\times n$ matrix $(a_{ij})$ of real numbers satisfies $|a_{ij}| <1$ and $\sum_{i=1}^m a_{ij}= 0$ for all$ j$. Show that one can permute the entries in each column in such a way that the obtained matrix $(b_{ij})$ satisfies $\sum_{j=1}^n b_{ij} < 2$ for all $i$.

Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a,b,c$ by \[a=\root3\of n\kern1.5pt,\qquad b={1\over a-[a]}\kern1pt,\qquad c={1\over b-[b]}\kern1.5pt,\] where $[x]$ denotes the integer part of $x$. Prove that there are infinitely many such integers $n$ with the property that there exist integers $r,s,t$, not all zero, such that $ra+sb+tc=0$.

Let $ABC$ be an acute-angled triangle with incircle $\omega$ and incenter $I$. Let $\omega$ touch $AB, BC$ and $CA $ at points $D, E, F$ respectively. The circles $\omega_1$ and $\omega_2$ centered at $J_1$ and $J_2$ respectively are inscribed into A$DIF$ and $BDIE$. Let $J_1J_2$ intersect $AB$ at point $M$. Prove that $CD$ is perpendicular to $IM$.

$S$ is a family of balls in $\mathbb{R}^{n}$ ($n > 1$) such that the intersection of any two contains at most one point. Show that the set of points belonging to at least two members of $S$ is countable.

Let $F=.48181\cdots$ be an infinite repeating decimal with the digits $8$ and $1$ repeating. When $F$ is written as a fraction in lowest terms, the denominator exceeds the numerator by $\textbf{(A) }13\qquad\textbf{(B) }14\qquad\textbf{(C) }29\qquad\textbf{(D) }57\qquad \textbf{(E) }126$

In a pyramid, the base is a right-angled triangle with integer sides. The height of the pyramid is also integer. Show that the volume of the pyramid is even.

Call a $9$-digit number a [i]cassowary[/i] if it uses each of the digits $1$ through $9$ exactly once. Compute the number of cassowaries that are prime.

Let $ABC$ be a scalene triangle and let its incircle touch sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$ respectively. Let line $AD$ intersect this incircle at point $X$. Point $M$ is chosen on the line $FX$ so that the quadrilateral $AFEM$ is cyclic. Let lines $AM$ and $DE$ intersect at point $L$ and let $Q$ be the midpoint of segment $AE$. Point $T$ is given on the line $LQ$ such that the quadrilateral $ALDT$ is cyclic. Let $S$ be a point such that the quadrilateral $TFSA$ is a parallelogram, and let $N$ be the second point of intersection of the circumcircle of triangle $ASX$ and the line $TS$. Prove that the circumcircles of triangles $TAN$ and $LSA$ are tangent to each other.