For any positive real numbers $a,b$ and $c$, prove: \[ \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} \geq \dfrac{2}{a+b} + \dfrac{2}{b+c} + \dfrac{2}{c+a} \geq \dfrac{9}{a+b+c} \]

Prove that if $g \ge 2$ is an integer, then two series \[\sum_{n=0}^{\infty}\frac{1}{g^{n^{2}}}\;\; \text{and}\;\; \sum_{n=0}^{\infty}\frac{1}{g^{n!}}\] both converge to irrational numbers.

A triangle $ABC$ is given. Let $C'$ and $C'_{a}$ be the touching points of sideline $AB$ with the incircle and with the excircle touching the side $BC$. Points $C'_{b}$, $C'_{c}$, $A'$, $A'_{a}$, $A'_{b}$, $A'_{c}$, $B'$, $B'_{a}$, $B'_{b}$, $B'_{c}$ are defined similarly. Consider the lengths of $12$ altitudes of triangles $A'B'C'$, $A'_{a}B'_{a}C'_{a}$, $A'_{b}B'_{b}C'_{b}$, $A'_{c}B'_{c}C'_{c}$. (a) (8-9) Find the maximal number of different lengths. (b) (10-11) Find all possible numbers of different lengths.

A circle inscribed in a square, Has two chords as shown in a pair. It has radius $2$, And $P$ bisects $TU$. The chords' intersection is where? Answer the question by giving the distance of the point of intersection from the center of the circle. [asy] size(100); defaultpen(linewidth(0.8)); draw(unitcircle); draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle); label("$A$",(-1,1),SE); label("$B$",(1,1),SE); label("$C$",(1,-1),SE); label("$D$",(-1,-1),SE); pair M=(1,0),N=(0,-1),T=(-1,0),U=(0,1),P=dir(135); draw(P--M^^(-1,-1)--(1,1)); label("$M$",M,SE); label("$N$",N,SE); label("$T$",T,SE); label("$U$",U,SE); label("$P$",P,dir(270)); dot(origin^^(-1,1)^^(-1,-1)^^(1,-1)^^(1,1)^^M^^N^^T^^U^^P); [/asy]

Two concentric spheres have radii $r$ and $R,r < R$. We try to select points $A, B$ and $C$ on the surface of the larger sphere such that all sides of the triangle $ABC$ would be tangent to the surface of the smaller sphere. Show that the points can be selected if and only if $R \le 2r$.

Compute the minimum value of $cos(a-b) + cos(b-c) + cos(c-a)$ as $a,b,c$ ranges over the real numbers.

In a group of five people any two are either friends or enemies , no three of them are friends of each other and no three of them are enemies of each other . Prove that every person in this group has exactly two friends .

Find every real values that $a$ can assume such that $$\begin{cases} x^3 + y^2 + z^2 = a\\ x^2 + y^3 + z^2 = a\\ x^2 + y^2 + z^3 = a \end{cases}$$ has a solution with $x, y, z$ distinct real numbers.

Let $ABC$ be a triangle, $O$ its circumcenter and $\Gamma$ its circumcircle. Let $E$ and $F$ be points on $AB$ and $AC$, respectively, such that $O$ is the midpoint of $EF$. Let $A'=AO\cap \Gamma$, with $A'\ne A$. Finally, let $P$ be the point on line $EF$ such that $A'P\perp EF$. Prove that the lines $EF,BC$ and the tangent to $\Gamma$ at $A'$ are concurrent and that $\angle BPA' = \angle CPA'$.

Let $\mathbb{Z}_{\ge 0}$ denote the set of nonnegative integers. Define a function $f:\mathbb{Z}_{\ge 0} \to\mathbb{Z}$ with $f\left(0\right)=1$ and \[ f\left(n\right)=512^{\left\lfloor n/10 \right\rfloor}f\left(\left\lfloor n/10 \right\rfloor\right)\] for all $n \ge 1$. Determine the number of nonnegative integers $n$ such that the hexadecimal (base $16$) representation of $f\left(n\right)$ contains no more than $2500$ digits. [i]Proposed by Tristan Shin[/i]

Let $n$ be a positive integer and $a_1,...,a_n$ be positive real numbers. Let $g(x)$ denote the product $(x + a_1)\cdot ... \cdot (x + a_n)$ . Let $a_0$ be a real number and let $f(x) = (x - a_0)g(x)= x^{n+1} + b_1x^n + b_2x^{n-1}+...+ b_nx + b_{n+1}$ . Prove that all the coeffcients $b_1,b_2,..., b_{n+1}$ of the polynomial $f(x)$ are negative if and only if $a_0 > a_1 + a_2 +...+ a_n$.

Is it true that for each natural number $n$ there exist a circle, which contains exactly $n$ points with integer coordinates?

Prove that there exist infinitely many positive integers $n$ such that $\frac{4^n+2^n+1}{n^2+n+1}$ is a positive integer.

$$Problem1:$$ A two-digit number which is not a multiple of $10$ is given. Assuming it is divisible by the sum of its digits, prove that it is also divisible by $3$. Does the statement hold for three-digit numbers as well?

In acute triangle $ABC$, $\angle{A}<\angle{B}$ and $\angle{A}<\angle{C}$. Let $P$ be a variable point on side $BC$. Points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $BP=PD$ and $CP=PE$. Prove that as $P$ moves along side $BC$, the circumcircle of triangle $ADE$ passes through a fixed point other than $A$.

In a circumference $\Gamma$ a chord $PQ$ is considered such that the segment that joins the midpoint of the smallest arc $PQ$ and the midpoint of the segment $PQ$ measures $1$. Let $\Gamma_1, \Gamma_2$ and $\Gamma_3$ be three tangent circumferences to the chord $PQ$ that are in the same half plane than the center of $\Gamma$ with respect to the line $PQ$. Furthermore, $\Gamma_1$ and $\Gamma_3$ are internally tangent to $\Gamma$ and externally tangent to$ \Gamma_2$, and the centers of $\Gamma_1$ and $\Gamma_3$ are on different halfplanes with respect to the line determined by the centers of $\Gamma$ and $\Gamma_2$. If the sum of the radii of $\Gamma_1, \Gamma_2$ and $\Gamma_3$ is equal to the radius of $\Gamma$, calculate the radius of $\Gamma_2$.

Let $a,b,$ and $c$ be real numbers such that $a^2(b+1)=1, b^2(c+a)=2,$ and $c^2(a+b)=5.$ Given that there are three possible values for $abc,$ compute the minimum possible value of $abc.$

Joe is given a permutation $p = (a_1, a_2, a_3, a_4, a_{5})$ of $(1, 2, 3, 4, 5)$. A [i]swap[/i] is an ordered pair $(i, j)$ with $1 \le i < j \le 5$, and this allows Joe to swap the positions $i$ and $j$ in the permutation. For example, if Joe starts with the permutation $(1, 2, 3, 4, 5)$, and uses the swaps $(1, 2)$ and $(1, 3)$, the permutation becomes \[(1, 2, 3, 4, 5) \rightarrow (2, 1, 3, 4, 5) \rightarrow (3, 1, 2, 4, 5). \]Out of all $\tbinom{5}{2} = 10$ swaps, Joe chooses $4$ of them to be in a set of swaps $S$. Joe notices that from $p$ he could reach any permutation of $(1, 2, 3, 4, 5)$ using only the swaps in $S$. How many different sets are possible? [i]Proposed by Yang Liu[/i]

Prove that for $ x,y\in\mathbb{R^ \plus{} }$, $ \frac {1}{(1 \plus{} \sqrt {x})^{2}} \plus{} \frac {1}{(1 \plus{} \sqrt {y})^{2}} \ge \frac {2}{x \plus{} y \plus{} 2}$

For a given positive integer $m$, the series $$\sum_{k=1,k\neq m}^{\infty}\frac{1}{(k+m)(k-m)}$$ evaluates to $\frac{a}{bm^2}$, where $a$ and $b$ are positive integers. Compute $a+b$.

Eight points are equally spaced around a circle of radius $r$. If we draw a circle of radius $1$ centered at each of the eight points, then each of these circles will be tangent to two of the other eight circles that are next to it. IF $r^2=a+b\sqrt{2}$, where $a$ and $b$ are integers, then what is $a+b$? $\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) }6\qquad\text{(E) }7$

Inside the parallelogram $ABCD$, point $E$ is chosen, such that $AE = DE$ and $\angle ABE = 90^o$. Point $F$ is the midpoint of the side $BC$ . Find the measure of the angle $\angle DFE$.

On the $ AB $ side of the triangle $ ABC $, points $ X $ and $ Y $ are chosen, on the side of $ AC $ is a point of $ Z $, and on the side of $ BC $ is a point of $ T $. Wherein $ XZ \parallel BC $, $ YT \parallel AC $. Line $ TZ $ intersects the circumscribed circle of triangle $ ABC $ at points $ D $ and $ E $. Prove that points $ X $, $ Y $, $ D $ and $ E $ lie on the same circle.

On a past Mathematical Olympiad the maximum possible score on a problem was 5. The average score of boys was 4, the average score of girls was 3.25, and the overall average score was 3.60. Find the total number of participants, knowing that it was in the range from 31 to 50.

Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.