Let $ABC$ be an acute-angled triangle and $M$ be the midpoint of the side $BC$. Let $N$ be a point in the interior of the triangle $ABC$ such that $\angle NBA=\angle BAM$ and $\angle NCA=\angle CAM$. Prove that $\angle NAB=\angle MAC$. [i]Gabriel Nagy[/i]

Let $ S $ be the first quadrant and $ T:S\longrightarrow S $ be a transformation that takes the reciprocal of the coordinates of the points that belong to its domain. Define an [i]S-line[/i] to be the intersection of a line with $ S. $ [b]a)[/b] Show that the fixed points of $ T $ lie on any fixed S-line of $ T. $ [b]b)[/b] Find all fixed S-lines of $ T. $ [i]Gabriel Popa[/i]

Several boys and girls are dancing a waltz at a ball. Is it possible that each girl can always get to dance the next dance with a boy who is either more handsome or more clever than for the previous dance, and that each time one of the girls gets to dance the next dance with a boy who is more handsome and more clever? (The numbers of boys and girls are equal and all are dancing.) (AY Belov)

Andrew chooses three (not necessarily distinct) integers $a$, $b$, and $c$ independently and uniformly at random from $\{1,2,3,4,5,6,7\}$. Let $p$ be the probability that $abc(a+b+c)$ is divisible by $4$. If $p$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$. [i]Proposed by Andrew Wen[/i]

Determine the set of real numbers $\alpha$ that can be expressed in the form \[\alpha=\sum_{n=0}^{\infty}\frac{x_{n+1}}{x_n^3}\] where $x_0,x_1,x_2,\dots$ is an increasing sequence of real numbers with $x_0=1$.

Is there a $3 \times 3$ magic square consisting of distinct Fibonacci numbers (both $f_{1}$ and $f_{2}$ may be used; thus two $1$s are allowed)?

Find the least natural number $n$, which has at least 6 different divisors $1=d_1<d_2<d_3<d_4<d_5<d_6<...$, for which $d_3+d_4=d_5+6$ and $d_4+d_5=d_6+7$.

In the triangle $ABC$, $\angle{C}=90^{\circ},\angle {A}=30^{\circ}$ and $BC=1$. Find the minimum value of the longest side of all inscribed triangles (i.e. triangles with vertices on each of three sides) of the triangle $ABC$.

For a positive integer $n$ let $n\#$ denote the product of all primes less than or equal to $n$ (or $1$ if there are no such primes), and let $F(n)$ denote the largest integer $k$ for which $k\#$ divides $n$. Find the remainder when $F(1) + F(2) +F(3) + \dots + F(2015\#-1) + F(2015\#)$ is divided by $3999991$. [i]Proposed by Evan Chen[/i]

If $n>4$ is a composite number, then $2n$ divides $(n-1)!$.

Let $D$ be the midpoint of $\overline{BC}$ in $\Delta ABC$. Let $P$ be any point on $\overline{AD}$. If the internal angle bisector of $\angle ABP$ and $\angle ACP$ intersect at $Q$. Prove that, if $BQ \perp QC$, then $Q$ lies on $AD$

[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 6 Determine all pairs $(x,y)$ of positive integers such that $\frac{x^{2}y+x+y}{xy^{2}+y+11}$ is an integer.

Prove that $\binom{n+k}{n}$ can be written as product of $n$ pairwise coprime numbers $a_1,a_2,\dots,a_n$ such that $k+i$ is divisible by $a_i$ for all indices $i$.

Lines in the xy-plane are drawn through the point $(3,4)$ and the trisection points of the line segment joining the points $(-4,5)$ and $(5,-1).$ One of these lines has the equation $\textbf{(A) }3x-2y-1=0\qquad\textbf{(B) }4x-5y+8=0\qquad\textbf{(C) }5x+2y-23=0\qquad$ $\textbf{(D) }x+7y-31=0\qquad \textbf{(E) }x-4y+13=0$

Let $(R,+,\cdot)$ be a ring with center $Z=\{a\in\mathbb{R}:ar=ra,\forall r\in\mathbb{R}\}$ with the property that the group $U=U(R)$ of its invertible elements is finite. Given that $G$ is the group of automorphisms of the additive group $(R,+),$ prove that \[|G|\geq\frac{|U|^2}{|Z\cap U|}.\][i]Dragoș Crișan[/i]

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

Evaluate $$\int_{0}^{ \pi \slash 2} \frac{ dx}{1+( \tan x)^{\sqrt{2}} }\;.$$

Call a natural number $n$ faithful if there exist natural numbers $a<b<c$ such that $a|b,$ and $b|c$ and $n=a+b+c.$ $(i)$ Show that all but a finite number of natural numbers are faithful. $(ii)$ Find the sum of all natural numbers which are not faithful.

We are given a row of $n\geq7$ tiles. In the leftmost 3 tiles, there is a white piece each, and in the rightmost 3 tiles, there is a black piece each. The white and black players play in turns (the white starts). In each move, a player may take a piece of their color, and move it to an adjacent tile, so long as it's not occupied by a piece of the [u]same color[/u]. If the new tile is empty, nothing happens. If the tile is occupied by a piece of the [u]opposite color[/u], both pieces are destroyed (both white and black). The player who destroys the last two pieces wins the game. Which player has a winning strategy, and what is it? (The answer may depend on $n$)

Let $P$ be a regular polygon with $ 4k + 1 $ sides (where $ k $ is a natural) whose vertices are $ A_1, A_2, ..., A_ {4k + 1} $ (in that order ). Each vertex $ A_j $ of $P$ is assigned a natural of the set $ \{1,2, ..., 4k + 1 \} $ such that no two vertices are assigned the same number. On $P$ the following operation is performed: Let $ B_j $ be the midpoint of the side $ A_jA_ {j + 1} $ for $ j = 1,2, ..., 4k + 1 $ (where is consider $ A_ {4k + 2} = A_1 $). If $ a $, $ b $ are the numbers assigned to $ A_ {j} $ and $ A_ {j + 1} $, respectively, the midpoint $ B_j $ is written the number $ 7a-3b $. By doing this with each of the $ 4k + 1 $ sides, the $ 4k + 1 $ vertices initially arranged are erased. We will say that a natural $ m $ is [i]fatal [/i] if for all natural $ k $ , no matter how the vertices of $P$ are initially arranged, it is impossible to obtain $ 4k + 1 $ equal numbers through a finite amount of operations from $ m $. a) Determine if the $ 2010 $ is fatal or not. Justify. b) Prove that there are infinite fatal numbers. [color=#f00]PS. A help in translation of the 2nd paragraph is welcome[/color]. [hide=Original wording]Diremos que un natural $m$ es fatal si no importa cómo se disponen inicialmente los vértices de ${P}$, es imposible obtener mediante una cantidad finita de operaciones $4k+1$ números iguales a $m$.[/hide]

Let $S$ be the sum of all positive integers $n$ such that $\frac{3}{5}$ of the positive divisors of $n$ are multiples of $6$ and $n$ has no prime divisors greater than $3$. Compute $\frac{S}{36}$.

Let $ M \equal{} \{ x_1..., x_{30}\}$ a set which consists of 30 distinct positive numbers, let $ A_n,$ $ 1 \leq n \leq 30,$ the sum of all possible products with $ n$ elements each of the set $ M.$ Prove if $ A_{15} > A_{10},$ then $ A_1 > 1.$

Let $FELDSPAR$ be a regular octagon, and let $I$ be a point in its interior such that $\angle FIL = \angle LID =$ $\angle DIS$ $=\angle SIA.$ Compute $\angle IAR$ in degrees.

Let $ K $ be the midpoint of the side $ AB $ of a given triangle $ ABC $. Let $ L $ and $ M$ be points on the sides $ AC $ and $ BC$, respectively, such that $ \angle CLK = \angle KMC $. Prove that the perpendiculars to the sides $ AB, AC, $ and $ BC $ passing through $ K,L, $ and $M$, respectively, are concurrent.

Let $D$ be the midpoint of the arc $B-A-C$ of the circumcircle of $\triangle ABC (AB<AC)$. Let $E$ be the foot of perpendicular from $D$ to $AC$. Prove that $|CE|=\frac{|BA|+|AC|}{2}$.