Let $k$ be a natural number. Determine the number of non-congruent triangles with the vertices at vertices of a given regular $6k$-gon.

Ana and Luca play the following game. Ana writes a list of $n$ different integer numbers. Luca wins if he can choose four different numbers, $a, b, c$ and $d$, so that the number $a+b-(c+d)$ is multiple of $20$. Determine the minimum value of $n$ for which, whatever Ana's list, Luca can win.

Two circles touch internally at point $P$. A line tangent to one of the circles at point $A$ intersects the other circle at points $B$ and $C$. Prove that the line $ PA $ is the bisector of the angle $ BPC $.

The solutions of the equation $ z^4 \plus{} 4z^3i \minus{} 6z^2 \minus{} 4zi \minus{} i \equal{} 0$ are the vertices of a convex polygon in the complex plane. What is the area of the polygon? $ \textbf{(A)}\ 2^{5/8} \qquad \textbf{(B)}\ 2^{3/4} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2^{5/4} \qquad \textbf{(E)}\ 2^{3/2}$

Find $f(2)$ given that $f$ is a real-valued function that satisfies the equation $$4f(x)+\left(\frac23\right)(x^2+2)f\left(x-\frac2x\right)=x^3+1.$$

All the squares of a board of $(n+1)\times(n-1)$ squares are painted with [b]three colors[/b] such that, for any two different columns and any two different rows, the 4 squares in their intersections they don't have all the same color. Find the greatest possible value of $n$.

The integer 72 is the first of three consecutive integers 72, 73, and 74, that can each be expressed as the sum of the squares of two positive integers. The integers 72, 288, and 800 are the first three members of an infinite increasing sequence of integers with the above property. Find a function that generates the sequence and give the next three members.

In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lenghts, and the area of either of them can be expressed uniquley in the form $m\pi-n\sqrt{d},$ where $m, n,$ and $d$ are positive integers and $d$ is not divisible by the square of any prime number. Find $m+n+d.$

Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals. Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded? [i]Proposed by Mihai Bălună, Romania[/i]

Show that for $0 <\epsilon <1$ the expression $(x+y)^n(x^2-(2-\epsilon)xy+y^2)$ is a polynomial with positive coefficients for $n$ sufficiently large and integral. For $\epsilon =.002$ find the smallest admissible value of $n$.

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

Let $\{ z_n \}_{n \ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$, $|z_k z_{k+1}|=2^k$. Denote $f_n=|z_1+z_2+\cdots+z_n|,$ for $n=1,2,\cdots$ (1) Find the minimum of $f_{2020}$. (2) Find the minimum of $f_{2020} \cdot f_{2021}$.

Let $a,b,c$ be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: $ax^2+bx+c, bx^2+cx+a,$ and $cx^2+ax+b $.

On an infinite checkered plane $100$ chips in form of a $10\times 10$ square are given. These chips are rearranged such that any two adjacent (by side) chips are again adjacent, moreover no two chips are in the same cell. Prove that the chips are again in form of a square.

Let $V_n=\sqrt{F_n^2+F_{n+2}^2}$, where $F_n$ is the Fibonacci sequence ($F_1=F_2=1,F_{n+2}=F_{n+1}+F_{n}$) Show that $V_n,V_{n+1},V_{n+2}$ are the sides of a triangle with area $1/2$

$ ABCD$ is convex $ AD\parallel BC$, $ AC\perp BD$. $ M$ is interior point of $ ABCD$ which is not a intersection of diagonals $ AC$ and $ BD$ such that $ \angle AMB \equal{}\angle CMD \equal{}\frac{\pi}{2}$ .$ P$ is intersection of angel bisectors of $ \angle A$ and $ \angle C$. $ Q$ is intersection of angel bisectors of $ \angle B$ and $ \angle D$. Prove that $ \angle PMB \equal{}\angle QMC$.

Let $ABC$ be a triangle without obtuse angles, $M$ the midpoint of $BC$, $K$ the midpoint of $BM$. What is the largest value of the angle $\angle KAM$?

Prove that in a triangle one of the sides is twice as large as the other if and only if a median and an angle bisector of this triangle are perpendicular

Solve the system of equations \[\begin{cases}x^5=y+y^5\\ y^5=z+z^5\\ z^5=t+t^5\\ t^5=x+x^5.\end{cases}\]

The sequence of real numbers $\{a_n\}$, $n \in \mathbb{N}$ satisfies the following condition: $a_{n+1}=a_n(a_n+2)$ for any $n \in \mathbb{N}$. Find all possible values for $a_{2004}$.

Let $ f:\mathbb{R}\longrightarrow (0,\infty ) $ be a continuous and periodic function having a period of $ 2, $ and such that the integral $ \int_0^2 \frac{f(1+x)}{f(x)} dx $ exists. Show that $$ \int_0^2 \frac{f(1+x)}{f(x)} dx\ge 2, $$ with equality if and only if $ 1 $ is also a period of $ f. $

Let $ABCD$ be a quadrilateral inscribed in the unit circle such that $\angle BAD$ is $30$ degrees. Let $m$ denote the minimum value of $CP + PQ + CQ$, where $P$ and $Q$ may be any points lying along rays $AB$ and $AD$, respectively. Determine the maximum value of $m$.

For a positive integer $n$, an [i]$n$-branch[/i] $B$ is an ordered tuple $(S_1, S_2, \dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$. An integer $x$ is said to [i]appear[/i] in $B$ if it is an element of the last set $S_m$. Define an [i]$n$-plant[/i] to be an (unordered) set of $n$-branches $\{ B_1, B_2, \dots, B_k\}$, and call it [i]perfect[/i] if each of $1$, $2$, \dots, $n$ appears in exactly one of its branches. Let $T_n$ be the number of distinct perfect $n$-plants (where $T_0=1$), and suppose that for some positive real number $x$ we have the convergence \[ \ln \left( \sum_{n \ge 0} T_n \cdot \frac{\left( \ln x \right)^n}{n!} \right) = \frac{6}{29}. \] If $x = \tfrac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Yang Liu[/i]

Points $K$ and $L$ are chosen on the sides $AB$ and $BC$ of the isosceles $\triangle ABC$ ($AB = BC$) so that $AK +LC = KL$. A line parallel to $BC$ is drawn through midpoint $M$ of the segment $KL$, intersecting side $AC$ at point $N$. Find the value of $\angle KNL$.

Triangle $ABC$ is inscribed in a circle of radius 2 with $\angle ABC \geq 90^\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC$, $b=CA$, $c=AB$. Find all possible values of $x$.