For any natural number $n$, consider a $1\times n$ rectangular board made up of $n$ unit squares. This is covered by $3$ types of tiles : $1\times 1$ red tile, $1\times 1$ green tile and $1\times 2$ domino. (For example, we can have $5$ types of tiling when $n=2$ : red-red ; red-green ; green-red ; green-green ; and blue.) Let $t_n$ denote the number of ways of covering $1\times n$ rectangular board by these $3$ types of tiles. Prove that, $t_n$ divides $t_{2n+1}$.

$a,b,c\in\mathbb{R^+}$ and $a^2+b^2+c^2=48$. Prove that \[a^2\sqrt{2b^3+16}+b^2\sqrt{2c^3+16}+c^2\sqrt{2a^3+16}\le24^2\]

Let $a, b, c$ be integers such that \[\frac ab+\frac bc+\frac ca= 3\] Prove that $abc$ is a cube of an integer.

There are $9$ visually indistinguishable coins, and one of them is fake and thus lighter. We are given $3$ indistinguishable balance scales to find the fake coin; however, one of the scales is defective and shows a random result each time. Show that the fake coin can still be found with $4$ weighings.

Prove that if $ H $ is the point of intersection of the altitudes of a non-right triangle $ ABC $, then the circumcircles of the triangles $ AHB $, $ BHC $, $ CHA $ and $ ABC $ are equal.

Suppose $a_i, b_i, c_i, i=1,2,\cdots ,n$, are $3n$ real numbers in the interval $\left [ 0,1 \right ].$ Define $$S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}.$$ Now we know that $\left | S \right |\ge 2018,\, \left | T \right |\ge 2018.$ Try to find the minimal possible value of $n$.

Simon and Garfunkle play in a round-robin golf tournament. Each player is awarded one point for a victory, a half point for a tie, and no points for a loss. Simon beat Garfunkle in the first game by a record margin as Garfunkle sent a shot over the bridge and into troubled waters on the final hole. Garfunkle went on to score $8$ total victories, but no ties at all. Meanwhile, Simon wound up with exactly $8$ points, including the point for a victory over Garfunkle. Amazingly, every other player at the tournament scored exactly $n$. Find the sum of all possible values of $n$.

Determine all integers $a$ and $b$ such that \[(19a + b)^{18} + (a + b)^{18} + (a + 19b)^{18}\] is a perfect square.

Is it possible to arrange numbers from 1 to 2004 in some order so that the sum of any 10 consecutive numbers is divisble by 10?

Find all numbers $ n $ for which there exist three (not necessarily distinct) roots of unity of order $ n $ whose sum is $ 1. $

Let$f:\mathbb{R}\to \mathbb{R}$be a monotone function. a) Prove that$f$ have side limits in each point ${{x}_{0}}\in \mathbb{R}$. b) We define the function $g:\mathbb{R}\to \mathbb{R}$, $g\left( x \right)=\underset{t\nearrow x}{\mathop{\lim }}\,f\left( t \right)$( $g\left( x \right)$ with limit at at left in $x$). Prove that if the $g$ function is continuous, than the function $f$ is continuous.

Find all pairs $(p, n)$ with $n>p$, consisting of a positive integer $n$ and a prime $p$, such that $n^{n-p}$ is an $n$-th power of a positive integer.

Consider the set $F$ of all polynomials whose coefficients are in the set of $\{0,1\}$. Let $q(x) = x^3 + x +1$. The number of polynomials $p(x)$ in $F$ of degree $14$ such that the product $p(x)q(x)$ is also in $F$ is:

Let $\Gamma$ be a circle, and let $ABCD$ be a square lying inside the circle $\Gamma$. Let $\mathcal{C}_a$ be a circle tangent interiorly to $\Gamma$, and also tangent to the sides $AB$ and $AD$ of the square, and also lying inside the opposite angle of $\angle BAD$. Let $A'$ be the tangency point of the two circles. Define similarly the circles $\mathcal{C}_b$, $\mathcal{C}_c$, $\mathcal{C}_d$ and the points $B',C',D'$ respectively. Prove that the lines $AA'$, $BB'$, $CC'$ and $DD'$ are concurrent.

Two parallel chords $c, d$ in a circle have lengths $10$ and $14$ respectively, and the distance between them is $6$. If the length of the chord that is equidistant from $c$ and $d$ and parallel to $c$ and $d$ is $x$, find $x^2$.

Find all real solutions to the equation $4x^2 - 40 \lfloor x \rfloor + 51 = 0$.

The sum of the real values of $x$ satisfying the equality $|x+2|=2|x-2|$ is: $\textbf{(A)}\ \dfrac{1}{3} \qquad \textbf{(B)}\ \dfrac{2}{3} \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 6\dfrac{1}{3} \qquad \textbf{(E)}\ 6\dfrac{2}{3} $

The inscribed circle of $\Delta ABC$ $(AC<BC)$ is tangent to $AC$ and $BC$ in points $X$ and $Y$ respectively. A line is constructed through the middle point $M$ of $AB$, parallel to $XY$, which intersects $BC$ in $N$. Let $L\in BC$ be such that $NL=AC$ and $L$ is between $C$ and $N$. The lines $ML$ and $AC$ intersect in point $K$. Prove that $BN=CK$.

An infinite set $B$ consisting of positive integers has the following property. For each $a,b \in B$ with $a>b$ the number $\frac{a-b}{(a,b)}$ belongs to $B$. Prove that $B$ contains all positive integers. Here, $(a,b)$ is the greatest common divisor of numbers $a$ and $b$.

In $ \triangle ABC$, $ AB \equal{} BC$, and $ BD$ is an altitude. Point $ E$ is on the extension of $ \overline{AC}$ such that $ BE \equal{} 10$. The values of $ \tan CBE$, $ \tan DBE$, and $ \tan ABE$ form a geometric progression, and the values of $ \cot DBE$, $ \cot CBE$, $ \cot DBC$ form an arithmetic progression. What is the area of $ \triangle ABC$? [asy]unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair D=(0,0), C=(3,0), A=(-3,0), B=(0, 8), Ep=(6,0); draw(A--B--Ep--cycle); draw(D--B--C); label("$A$",A,S); label("$D$",D,S); label("$C$",C,S); label("$E$",Ep,S); label("$B$",B,N);[/asy]$ \textbf{(A)}\ 16 \qquad \textbf{(B)}\ \frac {50}{3} \qquad \textbf{(C)}\ 10\sqrt3 \qquad \textbf{(D)}\ 8\sqrt5 \qquad \textbf{(E)}\ 18$

Let $n\geq 3$ be an integer. Find the largest real number $M$ such that for any positive real numbers $x_1,x_2,\cdots,x_n$, there exists an arrangement $y_1,y_2,\cdots,y_n$ of real numbers satisfying \[\sum_{i=1}^n \frac{y_i^2}{y_{i+1}^2-y_{i+1}y_{i+2}+y_{i+2}^2}\geq M,\] where $y_{n+1}=y_1,y_{n+2}=y_2$.

In a country, mathematicians chose an $\alpha> 2$ and issued coins in denominations of 1 ruble, as well as $\alpha ^k$ rubles for each positive integer k. $\alpha$ was chosen so that the value of each coins, except the smallest, was irrational. Is it possible that any natural number of rubles can be formed with at most 6 of each denomination of coins?

Let $ABC$ be an acute triangle and $A_1, B_1$ and $C_1$, points on the sides $BC, CA$ and $AB$, respectively, such that $CB_1 = A_1B_1$ and $BC_1 = A_1C_1$. Let $D$ be the symmetric of $A_1$ with respect to $B_1C_1, O$ and $O_1$ are the circumcenters of triangles $ABC$ and $A_1B_1C_1$, respectively. If $A \ne D, O \ne O_1$ and $AD$ is perpendicular to $OO_1$, prove that $AB = AC$.

Find the product $ \cos a \cdot \cos 2a\cdot \cos 3a \cdots \cos 1006a$ where $a=\frac{2\pi}{2013}$.

An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. [quote]For example, 4 can be partitioned in five distinct ways: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1[/quote] The number of partitions of n is given by the partition function $p\left ( n \right )$. So $p\left ( 4 \right ) = 5$ . Determine all the positive integers so that $p\left ( n \right )+p\left ( n+4 \right )=p\left ( n+2 \right )+p\left ( n+3 \right )$.