The sequence $a_n$ is given as $$a_1=1, a_2=2 \;\;\; \text{and} \;\;\;\; a_{n+2}=a_n(a_{n+1}+1) \quad \forall n\geq 1$$ Prove that $a_{a_n}$ is divisible by $(a_n)^n$ for $n\geq 100$.

Point $O$ is chosen in a triangle $ABC$ such that ${d_a},{d_b},{d_c}$ are distance from point $O$ to sides $BC,CA,AB$, respectively. Find position of point $O$ so that product ${d_a} \cdot {d_b} \cdot {d_c}$ becomes maximum.

The base of a pyramid is an isosceles triangle with the vertex angle $\alpha$. The pyramid’s lateral edges are at angle $\phi$ to the base. Find the dihedral angle $\theta$ at the edge connecting the pyramid’s vertex to that of angle $\alpha$.

Let $n,k$ be positive integers, satisfying $n$ is even, $k\geq 2$ and $n>4k.$ There are $n$ points on the circumference of a circle. If the endpoints of $\frac{n}{2}$ chords in a circle that do not intersect with each other are exactly the $n$ points, we call these chords a matching.Determine the maximum of integer $m,$ such that for any matching, there exists $k$ consecutive points, satisfying all the endpoints of at least $m$ chords are in the $k$ points.

Connecting the vertices of a regular $ n$-gon we obtain a closed (not necessarily convex) $ n$-gon. Show that if $ n$ is even, then there are two parallel segments among the connecting segments and if $ n$ is odd then there cannot be exactly two parallel segments.

Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which \[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\] is a rational number.

Reconstruct an acute-angled triangle given the orthocenter and midpoints of two sides. (A. Zaslavsky)

If $x \in \left(0,\frac{\pi}{2}\right)$ then$$\left(\frac{\sin \left(\frac{\pi}{2}\sin x\right)}{\sin x}\right)^2+\left(\frac{\sin \left(\frac{\pi}{2}\cos x\right)}{\cos x}\right)^2\geq 3$$

Let $f_{n}(x)=\sum_{k=1}^{n}\frac{\sin kx}{\sqrt{k(k+1)}}.$ Find $\lim_{n\to\infty}\int_{0}^{2\pi}\{f_{n}(x)\}^{2}dx.$

Anna has a magical compass which can point only in four directions: North, East, South, West. Initially, the compass points North. After each minute, the compass can either turn left, turn right, or stay at its current orientation, with each action occurring with equal probability. What is the probability that the compass points South after $6$ minutes?

A fenced, rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partitioned using all or some of the 1994 meters of fence?

Two trapezoid angles and diagonals are respectively equal. Is it true that such are the trapezoid equal?

a) Given two squares $ABCD$ and $DEFG$, with point $E$ lying on the segment $CD$, and points$ F,G$ outside the square $ABCD$. Find the angle between lines $AE$ and $BF$. b) Two regular pentagons $OKLMN$ and $OPRST$ are given, and the point $P$ lies on the segment $ON$, and the points $R, S, T$ are outside the pentagon $OKLMN$. Find the angle between straight lines $KP$ and $MS$.

Find the minimum value of $\sqrt{x^2+y^2}$ if $5x+12y=60$. ${{ \textbf{(A)}\ \frac{60}{13} \qquad\textbf{(B)}\ \frac{13}{5} \qquad\textbf{(C)}\ \frac{13}{12} \qquad\textbf{(D)}\ 1}\qquad\textbf{(E)}\ 0 } $

a)Does there exist for any rational number $\frac{a}{b}$ some rational numbers $x_1,x_2,....x_n$ such that $x_1*x_2*....*x_n=1$ and $x_1+x_2+....+x_n=\frac{a}{b}$ a)Does there exist for any rational number $\frac{a}{b}$ some rational numbers $x_1,x_2,....x_n$ such that $x_1*x_2*....*x_n=\frac{a}{b}$ and $x_1+x_2+....+x_n=1$

Statement 1: In a cuboid, we can always find a point, its distances to eight vertexes are equal. Statement 2: In a cuboid, we can always find a point, its distances to twelve edges are equal. Statement 3: In a cuboid, we can always find a point, its distances to six surfaces are equal. In the statements above, how many of them are true? $\text{(A)}0\qquad\text{(B)}1\qquad\text{(C)}2\qquad\text{(D)}3$

The colonizers of a spherical planet have decided to build $N$ towns, each having area $1/1000$ of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of $N$?

For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots? $\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }14 \qquad \textbf{(E) }16$

The vertices of two regular octagons are numbered from $1$ to $8$, in some order, which may vary between both octagons (each octagon must have all numbers from $1$ to $8$). After this, one octagon is placed on top of the other so that every vertex from one octagon touches a vertex from the other. Then, the numbers of the vertices which are in contact are multiplied (i.e., if vertex $A$ has a number $x$ and is on top of vertex $A'$ that has a number $y$, then $x$ and $y$ are multiplied), and the $8$ products are then added. Prove that, for any order in which the vertices may have been numbered, it is always possible to place one octagon on top of the other so that the final sum is at least $162$. Note: the octagons can be rotated.

Prove that the area of a unit cube's projection on any plane equals the length of its projection on the perpendicular to this plane.

Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that \[ \frac {\left( f(w) \right)^2 \plus{} \left( f(x) \right)^2}{f(y^2) \plus{} f(z^2) } \equal{} \frac {w^2 \plus{} x^2}{y^2 \plus{} z^2} \] for all positive real numbers $ w,x,y,z,$ satisfying $ wx \equal{} yz.$ [i]Author: Hojoo Lee, South Korea[/i]

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous and bounded function such that \[x\int_{x}^{x+1}f(t)\, \text{d}t=\int_{0}^{x}f(t)\, \text{d}t,\quad\text{for any}\ x\in\mathbb{R}.\] Prove that $f$ is a constant function.

A triangle with perimeter $2p$ is inscribed in a circle of radius $R$ and also circumscribed on a circle of radius $r$. Prove that $p < 2(R+r)$.

A trapezoid with bases $AB$ and $CD$ is inscribed into a circle centered at $O$. Let $AP$ and $AQ$ be the tangents from $A$ to the circumcircle of triangle $CDO$. Prove that the circumcircle of triangle $APQ$ passes through the midpoint of $AB$.

Given a real number $a> 0$. How many positive real solutions of the equation is $ a^{x}=x^{a} $