Let $a,b,c$ be positive real numbers for which $ab+bc+ca=\frac{1}{3}$. Prove the inequality \[ \frac{a}{a^2-bc+1}+\frac{b}{b^2-ca+1}+\frac{c}{c^2-ab+1}\ge\frac{1}{a+b+c}\]

Let $m, n$ be natural numbers and let $m\cdot n$ be a multiple of $4$. A chessboard with $m \times n$ fields are covered with $1 \times 2$ large dominoes without gaps and without overlapping. Show that the number of dominoes that are parallel to a edge of the chess board is fixed . [hide=original wording] Seien m, n natu¨rliche Zahlen und sei m · n ein Vielfaches von 4. Ein Schachbrett mit m × n Feldern sei mit 1 × 2 großen Dominosteinen lu¨ckenlos und u¨berlappungsfrei u¨berdeckt. Zeige, dass die Anzahl der Dominosteine, die zu einer fest gew¨ahlten Kante des Schachbrettes parallel sind, gerade ist. [/hide]

Determine all integers $ n > 1$ such that \[ \frac {2^n \plus{} 1}{n^2} \] is an integer.

Given $A(-2,2)$, and $B$ is a moving point on ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$. $F$ is the left focal point of the ellipse, find the coordinate of $B$ when $|AB|+\frac{5}{3}|BF|$ takes its minumum value.

Show an infinite sequence $a_1, a_2, \ldots$ of integers with both of the following properties: • $a_i \neq 0$ for every positive integer $i$, that is, no term in the sequence is equal to zero; • for all positive integer $n$, $a_n + a_{2n} + \ldots + a_{2023n} = 0$.

Let $n$ be a constant positive integer. Show that for only non-negative integers $k$, the Diophantine equation $\sum_{i=1 }^{n}{ x_i ^3}=y^{3k+2}$ has infinitely many solutions in the positive integers $x_i, y$.

Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]

Let $D$ and $E$ be points on respectively $[AB]$ and $[AC]$ of $\triangle ABC$ where $|AB|=|AC|$, $m(\widehat{BAC})=40^\circ$. Let $F$ be a point on $BC$ such that $C$ is between $B$ and $F$. If $|BE|=|CF|$, $|AD|=|AE|$, and $m(\widehat{BEC})=60^\circ$, then what is $m(\widehat{DFB})$ ? $ \textbf{(A)}\ 45^\circ \qquad\textbf{(B)}\ 40^\circ \qquad\textbf{(C)}\ 35^\circ \qquad\textbf{(D)}\ 30^\circ \qquad\textbf{(E)}\ 25^\circ $

Let $\alpha,\beta,$ and $\gamma$ be three real numbers. Suppose that $$\cos\alpha+\cos\beta+\cos\gamma=1$$ $$\sin\alpha+\sin\beta+\sin\gamma=1.$$ Find the smallest possible value of $\cos \alpha.$

There are rhombus $ABCD$ and circle $\Gamma_B$, which is centred at $B$ and has radius $BC$, and circle $\Gamma_C$, which is centred at $C$ and has radius $BC$. Circles $\Gamma_B$ and $\Gamma_C$ intersect at point $E$. The line $ED$ intersects $\Gamma_B$ at point $F$. Find all possible values of $\angle AFB$.

In the figure below the triangles $BCD, CAE$ and $ABF$ are equilateral, and the triangle $ABC$ is right-angled with $\angle A = 90^o$. Prove that $|AD| = |EF|$. [img]https://1.bp.blogspot.com/-QMMhRdej1x8/XzP18QbsXOI/AAAAAAAAMUI/n53OsE8rwZcjB_zpKUXWXq6bg3o8GUfSwCLcBGAsYHQ/s0/2019%2Bmohr%2Bp5.png[/img]

Show that given any $4$ points in the plane we can find two whose distance apart is not an odd integer.

Sketch the evolvent of the cubic parabola $y=x^3$ (the evolvent is the locus of the points $\overrightarrow{r}(s)+(c-s)\dot{\overrightarrow{r}}(s)$, where $s$ is the arc-length of the curve $\overrightarrow{r}(s)$ and $c$ is a constant).

Each of the numbers $a_1,...,a_n$ is equal to $1$ or $-1$. If $a_1a_2 + a_2a_3 + ··· + a_{n-1}a_n + a_na_1 = 0$, proves that $n$ is divisible by $4$.

Let $n$ and $k$ be positive integers. There are $nk$ objects (of the same size) and $k$ boxes, each of which can hold $n$ objects. Each object is coloured in one of $k$ different colours. Show that the objects can be packed in the boxes so that each box holds objects of at most two colours.

Let $a_0$ and $a_n$ be different divisors of a natural number $m$, and $a_0, a_1, \ldots, a_n$ be a sequence of natural numbers such that it satisfies \[a_{i+1} = |a_i\pm a_{i-1}|\text{ for }0 < i < n\] If $gcd(a_0,a_1,\ldots, a_n) = 1$, show that there exists a term of the sequence that is smaller than $\sqrt{m}$ . [i]Proposed by Dusan Djukic[/i]

Let $ABCA'B'C'$ a right triangular prism with the bases equilateral triangles. A plane $\alpha$ containing point $A$ intersects the rays $BB'$ and $CC'$ at points E and $F$, so that $S_ {ABE} + S_{ACF} = S_{AEF}$. Determine the measure of the angle formed by the plane $(AEF)$ with the plane $(BCC')$.

Alice and Bob are playing in the forest. They have six sticks of length 1, 2, 3, 4, 5, 6 inches. Somehow, they have managed to arrange these sticks, such that they form the sides of an equiangular hexagon. Compute the sum of all possible values of the area of the hexagon.

Find the smallest natural number which is a multiple of $2009$ and whose sum of (decimal) digits equals $2009$ [i]Proposed by Milos Milosavljevic[/i]

Let $\Gamma_1$ be a circle and $P$ a point outside of $\Gamma_1$. The tangents from $P$ to $\Gamma_1$ touch the circle at $A$ and $B$. Let $M$ be the midpoint of $PA$ and $\Gamma_2$ the circle through $P$, $A$ and $B$. Line $BM$ cuts $\Gamma_2$ at $C$, line $CA$ cuts $\Gamma_1$ at $D$, segment $DB$ cuts $\Gamma_2$ at $E$ and line $PE$ cuts $\Gamma_1$ at $F$, with $E$ in segment $PF$. Prove lines $AF$, $BP$, and $CE$ are concurrent.

Cards numbered from 1 to $2^n$ are distributed among $k$ children, $1\leq k\leq 2^n$, so that each child gets at least one card. Prove that the number of ways to do that is divisible by $2^{k-1}$ but not by $2^k$. [i] M. Ivanov [/i]

Let $f:\mathbb R\to\mathbb R$ be a continuous function. Do there exist continuous functions $g:\mathbb R\to\mathbb R$ and $h:\mathbb R\to\mathbb R$ such that $f(x)=g(x)\sin x+h(x)\cos x$ holds for every $x\in\mathbb R$?

$ABCD$ is a convex quadrangle, $G$ is its center of gravity as a homogeneous plate (i.e., the intersection point of two lines, each of which connects the centroids of triangles having a common diagonal). a) Suppose that around $ABCD$ we can circumscribe a circle centered on $O$. We define $H$ similarly to $G$, taking orthocenters instead of centroids. Then the points of $H, G, O$ lie on the same line and $HG: GO = 2: 1$. b) Suppose that in $ABCD$ we can inscribe a circle centered on $I$. The Nagel point N of the circumscribed quadrangle is the intersection point of two lines, each of which passes through points on opposite sides of the quadrangle that are symmetric to the tangent points of the inscribed circle relative to the midpoints of the sides. (These lines divide the perimeter of the quadrangle in half). Then $N, G, I$ lie on one straight line, with $NG: GI = 2: 1$.

The incenter and $A$-excenter of $\triangle{ABC}$ is $I$ and $O$. The foot from $A,I$ to $BC$ is $D$ and $E$. The intersection of $AD$ and $EO$ is $X$. The circumcenter of $\triangle{BXC}$ is $P$. Show that the circumcircle of $\triangle{BPC}$ is tangent to the $A$-excircle if $X$ is on the incircle of $\triangle{ABC}$.

There are $2001$ marked points in the plane. It is known that the area of any triangle with vertices at the given points is less than or equal than $1 \ cm^2$. Prove that there exists a triangle with area no more than $4 \ cm^2$, which contains all $2001$ points.