[b] Problem 5. [/b]Denote with $d(a,b)$ the numbers of the divisors of natural $a$, which are greater or equal to $b$. Find all natural $n$, for which $d(3n+1,1)+d(3n+2,2)+\ldots+d(4n,n)=2006.$ [i]Ivan Landgev[/i]

Suppose that $PQ$ and $RS$ are two chords of a circle intersecting at a point $O$. It is given that $PO=3 \text{cm}$ and $SO=4 \text{cm}$. Moreover, the area of the triangle $POR$ is $7 \text{cm}^2$. Find the area of the triangle $QOS$.

Let two ants stand on the perimeter of a regular $2019$-gon of unit side length. One of them stands on a vertex and the other one is on the midpoint of the opposite side. They start walking along the perimeter at the same speed counterclockwise. The locus of their midpoints traces out a figure $P$ in the plane with $N$ corners. Let the area enclosed by the convex hull of $P$ be $\tfrac{A}{B}\tfrac{\sin^m\left(\tfrac{\pi}{4038}\right)}{\tan\left(\tfrac{\pi}{2019}\right)}$, where $A$ and $B$ are coprime positive integers, and $m$ is the smallest possible positive integer such that this formula holds. Find $A+B+m+N$. [i]Note:[/i] The [i]convex hull[/i] of a figure $P$ is the convex polygon of smallest area which contains $P$.

Real numbers $x$ and $y$ satisfy $$xy+\frac{x}{y} = 3$$ $$\frac{1}{x^2y^2}+\frac{y^2}{x^2} = 4$$ If $x^2$ can be expressed in the form $\frac{a+\sqrt{b}}{c}$ for integers $a$, $b$, and $c$. Find $a+b+c$. [i]Proposed by Andy Xu[/i]

In acute-angled triangle $ABC,$ $AB>AC.$ Let $O,H$ be the circumcenter and orthocenter of $\triangle ABC,$ respectively. The line passing through $H$ and parallel to $AB$ intersects line $AC$ at $M,$ and the line passing through $H$ and parallel to $AC$ intersects line $AB$ at $N.$ $L$ is the reflection of the point $H$ in $MN.$ Line $OL$ and $AH$ intersect at $K.$ Prove that $K,M,L,N$ are concyclic.

Three different points $A,B$ and $C$ lie on a circle with center $M$ so that $| AB | = | BC |$. Point $D$ is inside the circle in such a way that $\vartriangle BCD$ is equilateral. Let $F$ be the second intersection of $AD$ with the circle . Prove that $| F D | = | FM |$.

Let $x,y,z$ be positive integers such that $x\neq y\neq z \neq x$ .Prove that $$(x+y+z)(xy+yz+zx-2)\geq 9xyz.$$ When does the equality hold? [i]Proposed by Dorlir Ahmeti, Albania[/i]

Six numbers are placed on a circle. For every number $A$ we have: $A$ equals the absolute value of $(B- C)$, where $B$ and $C$ follow $A$ clockwise. The total sum of the numbers equals $1$. Find all the numbers. (Folklore)

Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.

There are $169$ lamps, each equipped with an on/off switch. You have a remote control that allows you to change exactly $19$ switches at once. (Every time you use this remote control, you can choose which $19$ switches are to be changed.) (a) Given that at the beginning some lamps are on, can you turn all the lamps off, using the remote control? (b) Given that at the beginning all lamps are on, how many times do you need to use the remote control to turn all lamps off?

Let $a,b,c,d$ be positive real numbers such that $abcd=1$.Find with proof that $x=3 $ is the minimal value for which the following inequality holds: \[a^x+b^x+c^x+d^x\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}\]

Given a triangle $ABC$, let $P$ be a point inside the triangle such that $| BP | > | AP |, | BP | > | CP |$. Show that $\angle ABC <90^o$

What is the coefficient for $\text{O}_2$ when the following reaction $\_\text{As}_2\text{S}_3+\_\text{O}_2 \rightarrow \_\text{As}_2\text{O}_3+\_\text{SO}_2$ is correctly balanced with the smallest integer coefficients? $ \textbf{(A)} 5 \qquad\textbf{(B)} 6 \qquad\textbf{(C)} 8 \qquad\textbf{(D)} 9 \qquad $

Prove that for each positive integer n,the equation $x^{2}+15y^{2}=4^{n}$ has at least $n$ integer solution $(x,y)$

Let $S=\{1,2,3,4,5,6\}.$ Find the number of bijective functions $f:S\rightarrow S$ for which there exist exactly $6$ bijective functions $g:S\rightarrow S$ such that $f(g(x))=g(f(x))$ for all $x\in S$. [i]Proposed by Nathan Ramesh

Show that $x^4 > x - \frac12$ for all real $x$.

In triangle $ABC$ let $AD,BE$ and $CF$ be the altitudes. Let the points $P,Q,R$ and $S$ fulfil the following requirements: i) $P$ is the circumcentre of triangle $ABC$. ii) All the segments $PQ,QR$ and $RS$ are equal to the circumradius of triangle $ABC$. iii) The oriented segment $PQ$ has the same direction as the oriented segment $AD$. Similarly, $QR$ has the same direction as $BE$, and $Rs$ has the same direction as $CF$. Prove that $S$ is the incentre of triangle $ABC$.

Find all integers $x$ and $y$ such that \[x^3-117y^3=5.\]

Is there a six-link broken line in space that passes through all the vertices of a given cube?

Find all positive integer solutions of the equation $n^5+n^4=7^{m}-1$

Given are positive integers $a, b$ satisfying $a \geq 2b$. Does there exist a polynomial $P(x)$ of degree at least $1$ with coefficients from the set $\{0, 1, 2, \ldots, b-1 \}$ such that $P(b) \mid P(a)$?

$F_1,F_2$ are two focal points of ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, $P$ is a point on the ellipse, and $|PF_1|:|PF_2|=2:1$, then the area of $\triangle PF_1F_2$ is________.

Find the largest constant $K>0$ such that for any $0\le k\le K$ and non-negative reals $a,b,c$ satisfying $a^2+b^2+c^2+kabc=k+3$ we have $a+b+c\le 3$. (Dan Schwarz)

In triangle $\triangle ABC$, $D$ lies between $A$ and $C$ and $AC=3AD$, $E$ lies between $B$ and $C$ and $BC=4EC$. $B,G,F,D$ in that order, are on a straight line and $BD=5GF=5FD$. Suppose the area of $\triangle ABC$ is $900$, find the area of the triangle $\triangle EFG$.

The diagonals $ AC$ and $ BD$ of a convex quadrilateral $ ABCD$ intersect at point $ M$. The bisector of $ \angle ACD$ meets the ray $ BA$ at $ K$. Given that $ MA \cdot MC \plus{}MA \cdot CD \equal{} MB \cdot MD$, prove that $ \angle BKC \equal{} \angle CDB$.