How many ways are there to completely fill a $3 \times 3$ grid of unit squares with the letters $B, M$, and $T$, assigning exactly one of the three letters to each of the squares, such that no $2$ adjacent unit squares contain the same letter? Two unit squares are adjacent if they share a side.

Prove that, for any integer $x$, $x^2 +5x+16$ is not divisible by $169$.

Let $\alpha$ and $\beta$ be the roots of $x^{2} - qx + 1$, where $q$ is a rational number larger than $2$. Let $s_1 = \alpha + \beta$, $t_1 = 1$, and for all integers $n \geq 2$: $s_n = \alpha^n + \beta^n$ $t_n = s_{n-1} + 2s_{n-2} + \cdot \cdot \cdot + (n - 1)s_{1} + n$ Prove that, for all odd integers $n$, $t_n$ is the square of a rational number.

A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure $x$. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}$. Find the difference between the largest and smallest possible values of $x$.

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove the inequality $$ \left ( \frac{1+a}{b}+2 \right ) \left ( \frac{1+b}{c}+2 \right ) \left ( \frac{1+c}{a}+2 \right )\geq 216.$$ When does equality hold? [i]Authored by Anastasija Trajanova[/i]

Find all polynomials with real coefficients such that one can find an integer valued series $a_0, a_1, \dots$ satisfying $\lfloor P(x) \rfloor = a_{ \lfloor x^2 \rfloor}$ for all $x$ real numbers.

$ a,b,c$ are sides of triangle $ ABC$. For any choosen triple from $ (a \plus{} 1,b,c),(a,b \plus{} 1,c),(a,b,c \plus{} 1)$ there exist a triangle which sides are choosen triple. Find all possible values of area which triangle $ ABC$ can take.

Let $S$ be a set of positive integers such that if $a$ and $b$ are elements of $S$ such that $a<b$, then $b-a$ divides the least common multiple of $a$ and $b$, and the quotient is an element of $S$. Prove that the cardinality of $S$ is less than or equal to $2$.

Let $d \ge 2$ be an integer. Prove that there exists a constant $C(d)$ such that the following holds: For any convex polytope $K\subset \mathbb{R}^d$, which is symmetric about the origin, and any $\varepsilon \in (0, 1)$, there exists a convex polytope $L \subset \mathbb{R}^d$ with at most $C(d) \varepsilon^{1-d}$ vertices such that \[(1-\varepsilon)K \subseteq L \subseteq K.\] Official definitions: For a real $\alpha,$ a set $T \in \mathbb{R}^d$ is a [i]convex polytope with at most $\alpha$ vertices[/i], if $T$ is a convex hull of a set $X \in \mathbb{R}^d$ of at most $\alpha$ points, i.e. $T = \{\sum\limits_{x\in X} t_x x | t_x \ge 0, \sum\limits_{x \in X} t_x = 1\}.$ Define $\alpha K = \{\alpha x | x \in K\}.$ A set $T \in \mathbb{R}^d$ is [i]symmetric about the origin[/i] if $(-1)T = T.$

Each of the positive integers $1,2,\dots,n$ is colored in one of the colors red, blue or yellow regarding the following rules: (1) A Number $x$ and the smallest number larger than $x$ colored in the same color as $x$ always have different parities. (2) If all colors are used in a coloring, then there is exactly one color, such that the smallest number in that color is even. Find the number of possible colorings.

For each $n$, define $L(n)$ to be the number of natural numbers $1\leq a\leq n$ such that $n\mid a^{n}-1$. If $p_{1},p_{2},\ldots,p_{k}$ are the prime divisors of $n$, define $T(n)$ as $(p_{1}-1)(p_{2}-1)\cdots(p_{k}-1)$. a) Prove that for each $n\in\mathbb N$ we have $n\mid L(n)T(n)$. b) Prove that if $\gcd(n,T(n))=1$ then $\varphi(n) | L(n)T(n)$.

For real numbers $B,M,$ and $T,$ we have $B^2+M^2+T^2 =2022$ and $B+M+T =72.$ Compute the sum of the minimum and maximum possible values of $T.$

Let $t$ be a real number, and let $$a_n=2\cos \left(\frac{t}{2^n}\right)-1,\quad n=1,2,3,\dots$$ Let $b_n$ be the product $a_1a_2a_3\cdots a_n$. Find a formula for $b_n$ that does not involve a product of $n$ terms, and deduce that $$\lim_{n\to \infty}b_n=\frac{2\cos t+1}{3}$$

Find all integers of the form $2^n$ (where $n$ is a natural number) such that after deleting the first digit of its decimal representation we again get a power of $2$.

Let $ m$ and $ n$ be two distinct positive integers. Prove that there exists a real number $ x$ such that $ \frac {1}{3}\le\{xn\}\le\frac {2}{3}$ and $ \frac {1}{3}\le\{xm\}\le\frac {2}{3}$. Here, for any real number $ y$, $ \{y\}$ denotes the fractional part of $ y$. For example $ \{3.1415\} \equal{} 0.1415$.

A cube with edge length $3$ is divided into $27$ unit cubes. The numbers $1, 2,\ldots ,27$ are distributed arbitrarily over the unit cubes, with one number in each cube. We form the $27$ possible row sums (there are nine such sums of three integers for each of the three directions parallel with the edges of the cube). At most how many of the $27$ row sums can be odd?

Zeus’s lightning is made of a copper rod of length $60$ by bending it $4$ times in alternating directions so that the angle between two adjacent parts is always $60^o$. What is the minimum value of the square of the distance between the two endpoints of the lightning? All five segments of the lightning lie in the same plane. [img]https://cdn.artofproblemsolving.com/attachments/5/1/a18206df4fde561421022c0f2b4332f5ac44a2.png[/img]

The square $ABCD$ is divided into $8$ equal right triangles and the square $KLMN$, as shown in the figure. Find the area of the square $ABCD$ if $KL = 5, PS = 8$. [img]https://1.bp.blogspot.com/-B2QIHvPcIx0/X4BhUTMDhSI/AAAAAAAAMj4/4h0_q1P6drskc5zSvtfTZUskarJjRp5LgCLcBGAsYHQ/s0/Yasinsky%2B2020%2Bp1.png[/img]

In the Cartesian plane, let $G_1$ and $G_2$ be the graphs of the quadratic functions $f_1(x) = p_1x^2 + q_1x + r_1$ and $f_2(x) = p_2x^2 + q_2x + r_2$, where $p_1 > 0 > p_2$. The graphs $G_1$ and $G_2$ cross at distinct points $A$ and $B$. The four tangents to $G_1$ and $G_2$ at $A$ and $B$ form a convex quadrilateral which has an inscribed circle. Prove that the graphs $G_1$ and $G_2$ have the same axis of symmetry.

Alex the Kat and Kelvin the Frog play a game on a complete graph with $n$ vertices. Kelvin goes first, and the players take turns selecting either a single edge to remove from the graph, or a single vertex to remove from the graph. Removing a vertex also removes all edges incident to that vertex. The player who removes the final vertex wins the game. Assuming both players play perfectly, for which positive integers $n$ does Kelvin have a winning strategy?

Let $ABC$ be a triangle and $L$, $M$, $N$ be the midpoints of $BC$, $CA$ and $AB$, respectively. The tangent to the circumcircle of $ABC$ at $A$ intersects $LM$ and $LN$ at $P$ and $Q$, respectively. Show that $CP$ is parallel to $BQ$.

The complex number w has positive imaginary part and satisfies $|w| = 5$. The triangle in the complex plane with vertices at $w, w^2,$ and $w^3$ has a right angle at $w$. Find the real part of $w^3$.

An equilateral triangle $ABC$ has side a. A square is constructed on the outside of each side of the triangle. A right regular pyramid with sloping side $a$ is placed on each square. These pyramids are rotated about the sides of the triangle so that the apex of each pyramid comes to a common point above the triangle. Show that when this has been done, the other vertices of the bases of the pyramids (apart from the vertices of the triangle) form a regular hexagon.

Bahman wants to build an area next to his garden's wall for keeping his poultry. He has three fences each of length $10$ meters. Using the garden's wall, which is straight and long, as well as the three pieces of fence, what is the largest area Bahman can enclose in meters squared? $\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 50+25 \sqrt 3\qquad\textbf{(C)}\ 50 + 50\sqrt 2\qquad\textbf{(D)}\ 75 \sqrt 3 \qquad\textbf{(E)}\ 300$

Prove that there exist infinitely many positive integers $n$, for which at least one of the numbers $2^{2^n}+1$ and $2018^{2^n}+1$ is composite.