Let $ABC$ be an acute triangle with orthocenter $H$, incenter $I$, and excenters $I_A, I_B, I_C$. Show that $II_A * II_B * II_C \ge 8 AH * BH * CH$.

Determine whether the inequality $$ \left|\sqrt{x^2+2x+5}-\sqrt{x^2-4x+8}\right|<3$$ is valid for all real numbers $x$.

What is the smallest integer multiple of 9997, other than 9997 itself, which contains only odd digits?

A rectangle is divided into $200\times 3$ unit squares. Prove that the number of ways of splitting this rectangle into rectangles of size $1\times 2$ is divisible by $3$.

Working alone, the expert can paint a car in one day, the amateur can paint a car in two days, and the beginner can paint a car in three days. If the three painters work together at these speeds to paint three cars, it will take them $\frac{m}{n}$ days where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Triangle $ABC$ in the plane $\Pi$ is called [i]good [/i] if it has the following property: For any point $D$ in space outside the plane $\Pi$, it is possible to construct a triangle with sides of lengths $CD,BD,AD$. Find all good triangles

We have big multivolume encyclopaedia about dogs on the shelf in alphabetical order, each volume in its specially selected place. Near each place there is an instruction that prescribes one of four actions: to rearrange this volume is one or two places left or right. If you simultaneously run all instructions, volumes will be placed in the same places in another order. The cynologist Dima performs all the instructions every morning. Once he discovered, that the volume of "Bichons" stands still, which was initially occupied by the volume of "Terriers". Prove , that after some time the volume of "Mudies" will stand on the original place of the volume "Poodles".

Determine all polynomials with integral coefficients $P(x)$ such that if $a,b,c$ are the sides of a right-angled triangle, then $P(a), P(b), P(c)$ are also the sides of a right-angled triangle. (Sides of a triangle are necessarily positive. Note that it's not necessary for the order of sides to be preserved; if $c$ is the hypotenuse of the first triangle, it's not necessary that $P(c)$ is the hypotenuse of the second triangle, and similar with the others.)

A semicircle has diameter $AB$ and center $S$,with a point $M$ on the circumference.$U,V$ are the incircles of sectors $ASM$ and $BSM$.Prove that circles $U,V$ can be seperated by a line perpendicular to $AB$.

Given convex quadrilateral $ABCD$. We construct the similar triangles $APB, BQC, CRD, DSA$ outside $ABCD$ so that $\angle PAB = \angle QBC = \angle RCD = \angle SDA, \angle PBA = \angle QCB = \angle RDC = \angle SAD$. Prove that if $PQRS$ is a parallelogram, so is $ABCD$.

Two circles $ \Omega_{1}$ and $ \Omega_{2}$ are externally tangent to each other at a point $ I$, and both of these circles are tangent to a third circle $ \Omega$ which encloses the two circles $ \Omega_{1}$ and $ \Omega_{2}$. The common tangent to the two circles $ \Omega_{1}$ and $ \Omega_{2}$ at the point $ I$ meets the circle $ \Omega$ at a point $ A$. One common tangent to the circles $ \Omega_{1}$ and $ \Omega_{2}$ which doesn't pass through $ I$ meets the circle $ \Omega$ at the points $ B$ and $ C$ such that the points $ A$ and $ I$ lie on the same side of the line $ BC$. Prove that the point $ I$ is the incenter of triangle $ ABC$. [i]Alternative formulation.[/i] Two circles touch externally at a point $ I$. The two circles lie inside a large circle and both touch it. The chord $ BC$ of the large circle touches both smaller circles (not at $ I$). The common tangent to the two smaller circles at the point $ I$ meets the large circle at a point $ A$, where the points $ A$ and $ I$ are on the same side of the chord $ BC$. Show that the point $ I$ is the incenter of triangle $ ABC$.

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

Consider an acute-angled triangle $ABC$ and its circumcircle. Let $D$ be a point on the arc $AB$ which does not include point $C$ and let $A_1$ and $B_1$ be points on the lines $DA$ and $DB$, respectively, such that $CA_1 \perp DA$ and $CB_1 \perp DB$. Prove that $|AB| \ge |A_1B_1|$.

Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$. The common external tangent of the two circles closer to $X$ intersects $\omega_1$ and $\omega_2$ at $A$ and $B,$ respectively. Given that $AB=6,$ the radius of $\omega_1$ is $3,$ and $AY$ is tangent to $\omega_2,$ find $XY^2$.

Let $U$ be the center of the circle inscribed in the triangle $ABC$, $O_1$, $O_2$ and $O_3$ the centers of the circles circumscribed by the triangles $BCU$, $CAU$ and $ABU$ respectively. Prove that the circles circumscribed around the triangles $ABC$ and $O_1O_2O_3$ have the same center.

There are $315$ marbles divided into three piles of $81, 115$ and $119$. In each move Ah Meng can either merge several piles into a single pile or divide a pile with an even number of marbles into $2$ equal piles. Can Ah Meng divide the marbles into $315$ piles, each with a single marble?

For natural numbers $m,n$, set $a=(n+1)^m-n$ and $b=(n+1)^{m+3}-n$. (a) Prove that $a$ and $b$ are coprime if $m$ is not divisible by $3$. (b) Find all numbers $m,n$ for which $a$ and $b$ are not coprime.

Find all functions $f:R\to R$ such that for any real $x, y$ , $$f(x+2^y)=f(2^x)+f(y)$$

Prove that for any three real numbers $x,y,z$ the following inequality holds: $$|x|+|y|+|z|-|x+y|-|y+z|-|z+x|+|x+y+z|\ge0.$$

Let $ABC$ be a triangle and let $D\in (BC)$ be the foot of the $A$- altitude. The circle $w$ with the diameter $[AD]$ meet again the lines $AB$ , $AC$ in the points $K\in (AB)$ , $L\in (AC)$ respectively. Denote the meetpoint $M$ of the tangents to the circle $w$ in the points $K$ , $L$ . Prove that the ray $[AM$ is the $A$-median in $\triangle ABC$ ([b][u]Serbia[/u][/b]).

Solve the following equation in $\mathbb{Z}$: \[3^{2a + 1}b^2 + 1 = 2^c\]

Positive real numbers $x, y, z$ satisfy $xyz+xy+yz+zx = x+y+z+1$. Prove that \[ \frac{1}{3} \left( \sqrt{\frac{1+x^2}{1+x}} + \sqrt{\frac{1+y^2}{1+y}} + \sqrt{\frac{1+z^2}{1+z}} \right) \le \left( \frac{x+y+z}{3} \right)^{5/8} . \]

a) Prove that for all positive reals $u,v,x,y$ the following inequality takes place: \[ \frac ux + \frac vy \geq \frac {4(uy+vx)}{(x+y)^2} . \] b) Let $a,b,c,d>0$. Prove that \[ \frac a{b+2c+d} + \frac b{c+2d+a} + \frac c{d+2a+b} + \frac d{a+2b+c} \geq 1.\] [i]Traian Tămâian[/i]

Find all pairs of non-negative integers $(m,n)$ satisfying $\frac{n(n+2)}{4}=m^4+m^2-m+1$

The figure below shows a barn in the shape of two congruent pentagonal prisms that intersect at right angles and have a common center. The ends of the prisms are made of a 12 foot by 7 foot rectangle surmounted by an isosceles triangle with sides 10 feet, 10 feet, and 12 feet. Each prism is 30 feet long. Find the volume of the barn in cubic feet. [center][img]https://snag.gy/Ox9CUp.jpg[/img][/center]