Given a convex pentagon $ABCDE$, prove that if an arbitrary point $M$ inside the pentagon is connected by lines with all the pentagon’s vertices, then either one or three or five of these lines cross the sides of the pentagon opposite the vertices they pass. Note: In reality, we need to exclude the points of the diagonals, because that in this case the drawn lines can pass not through the internal points of the sides, but through the vertices. But if the drawn diagonals are not considered or counted twice (because they are drawn from two vertices), then the statement remains true.

Prove that \[\arctan \frac 12 +\arctan \frac 13 = \frac{\pi}{4}.\]

Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker $ \$ \underline{1}$ $ \underline{A}$ $ \underline{2}$. What is the missing digit $A$ of this $3$-digit number? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4$

A convex polygon is divided into finitely many quadrilaterals. Prove that at least one of these quadrilaterals must also be convex.

In triangle $ ABC$ the bisector of angle $ BCA$ intersects the circumcircle again at $ R$, the perpendicular bisector of $ BC$ at $ P$, and the perpendicular bisector of $ AC$ at $ Q$. The midpoint of $ BC$ is $ K$ and the midpoint of $ AC$ is $ L$. Prove that the triangles $ RPK$ and $ RQL$ have the same area. [i]Author: Marek Pechal, Czech Republic[/i]

Real numbers $a,b,c$ satisfy $a+b \ne 0$, $b+c \ne 0$ and $c+a \ne 0$. Show that \[\left(\frac{a^2c}{a+b}+\frac{b^2a}{b+c}+\frac{c^2b}{c+a}\right) \cdot \left(\frac{b^2c}{a+b}+\frac{c^2a}{b+c}+\frac{a^2b}{c+a}\right) \ge 0.\]

A two-digit number is [i]nice [/i] if it is both a multiple of the product of its digits and a multiple of the sum of its digits. How many numbers satisfy this property? What is the ratio of the number to the sum of digits for each of the nice numbers?

Given a plane $P$ and two fixed points $A$ and $B$ that do not lie in this plane. Denote two points $A'$ and $B'$ on plane $P$ and $M ,N$ the midpoints of the segments $AA'$, $BB'$. a) Determine the locus of the midpoint of the segment MN if the points are $A'$ and $B'$ move arbitrarily in plane $P$. b) A circle $O$ is considered in the plane $P$. Determine the locus $L$ of the midpoint of the segment $MN$ if the points $A'$ and $B'$ lie on the circle $O$ or inside it . c) $A'$ is assumed to be fixed on the circle $O$ or inside it and $B'$ is assumed to be movable inside it , except for $O$. Determine the locus of the point $B'$ such the above certain locus $L$ remains the same . Note: For b) and c) the following cases should be considered: 1. $A'$ and $B'$ are different, 2. $A'$ and $B'$ coincide.

Points $P,Q,R$ lie on the sides $AB,BC,CA$ of triangle $ABC$ in such a way that $AP=PR, CQ=QR$. Let $H$ be the orthocenter of triangle $PQR$, and $O$ be the circumcenter of triangle $ABC$. Prove that $$OH||AC$$.

On the coordinate plane, is there a line family of infinitely many lines $l_1,l_2,\cdots,l_n,\cdots$, satisfying the following? (1) Point$(1,1)\in l_n$ for all $n\in \mathbb{Z}_{+}$. (2) For all $n\in \mathbb{Z}_{+}$,$k_{n+1}=a_n-b_n$, where $k_{n+1}$ is the slope of $l_{n+1}$, $a_n,b_n$ are intercepts of $l_n$ on $x$-axis, $y$-axis. (3) $k_nk_{n+1}\geq0$ for all $n\in \mathbb{Z}_{+}$.

A subset $M$ of $\{1, 2, . . . , 2006\}$ has the property that for any three elements $x, y, z$ of $M$ with $x < y < z$, $x+ y$ does not divide $z$. Determine the largest possible size of $M$.

Consider the equation \[\frac{xy-C}{x+y}= k ,\] where all symbols used are positive integers. 1. Show that, for any (fixed) values $C, k$ this equation has at least a solution $x, y$; 2. Show that, for any (fixed) values $C, k$ this equation has at most a finite number of solutions $x, y$; 3. Show that, for any $C, n$ there exists $k = k(C,n)$ such that the equation has more than $n$ solutions $x, y$.

Juan writes the list of pairs $(n, 3^n)$, with $n=1, 2, 3,...$ on a chalkboard. As he writes the list, he underlines the pairs $(n, 3^n)$ when $n$ and $3^n$ have the same units digit. What is the $2013^{th}$ underlined pair?

Let $A_1, A_2, ... , A_{100}$ be subsets of a line, each a union of $100$ pairwise disjoint closed segments. Prove that the intersection of all hundred sets is a union of at most $9901$ disjoint closed segments.

A circle is inscribed in a quadrilateral $ABCD$, which is tangent to its sides $AB$, $BC$, $CD$, and $DC$ in points $M$, $N$, $P$, and $Q$ respectively. Prove that the lines $MP$, $NQ$, $AC$, and $BD$ intersect in one point.

Let $m$ be a convex polygon in a plane, $l$ its perimeter and $S$ its area. Let $M\left( R\right) $ be the locus of all points in the space whose distance to $m$ is $\leq R,$ and $V\left(R\right) $ is the volume of the solid $M\left( R\right) .$ [i]a.)[/i] Prove that \[V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.\] Hereby, we say that the distance of a point $C$ to a figure $m$ is $\leq R$ if there exists a point $D$ of the figure $m$ such that the distance $CD$ is $\leq R.$ (This point $D$ may lie on the boundary of the figure $m$ and inside the figure.) additional question: [i]b.)[/i] Find the area of the planar $R$-neighborhood of a convex or non-convex polygon $m.$ [i]c.)[/i] Find the volume of the $R$-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron. [b]Note by Darij:[/b] I guess that the ''$R$-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is $\leq R.$

$a_1,a_2,a_3,\dots$ and $b_1,b_2,b_3,\dots$ are sequences of positive integers. Show that we can find $m < n$ such that $a_m \leq a_n$ and $b_m \leq b_n$.

Natural numbers $m$ and $n$ are given. Prove that the number $2^n-1$ is divisible by the number $(2^m -1)^2$ if and only if the number $n$ is divisible by the number $m(2^m-1)$.

Decide whether for every polynomial $P$ of degree at least $1$, there exist infinitely many primes that divide $P(n)$ for at least one positive integer $n$. [i](Walther Janous)[/i]

The sequence $ (x_n)$, $ n\in\mathbb{N}^*$ is defined by $ |x_1|<1$, and for all $ n \ge 1$, \[ x_{n\plus{}1} \equal{}\frac{\minus{}x_n \plus{}\sqrt{3\minus{}3x_n^2}}{2}\] (a) Find the necessary and sufficient condition for $ x_1$ so that each $ x_n > 0$. (b) Is this sequence periodic? And why?

Find all triples of positive real numbers $(a, b, c)$ so that the expression $M = \frac{(a + b)(b + c)(a + b + c)}{abc}$ gets its least value.

Find all function $f:\mathbb{R}^+$ $\rightarrow \mathbb{R}^+$ such that: $f(f(x) + y)f(x) = f(xy + 1) \forall x, y \in \mathbb{R}^+$

Let $p$ be a prime and let $M$ be an $n\times m$ matrix with integer entries such that $Mv\not\equiv 0\pmod{p}$ for any column vector $v\neq 0$ whose entries are $0$ are $1$. Show that there exists a row vector $x$ with integer entries such that no entry of $xM$ is $0\pmod{p}$. (translated by L. Erdős)

Given integers $a_1, \ldots, a_{10},$ prove that there exist a non-zero sequence $\{x_1, \ldots, x_{10}\}$ such that all $x_i$ belong to $\{-1,0,1\}$ and the number $\sum^{10}_{i=1} x_i \cdot a_i$ is divisible by 1001.

Find all pairs $ (m, n)$ of positive integers such that $ m^n \minus{} n^m \equal{} 3$.