$ 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.

The following diagram shows four adjacent $2\times 2$ squares labeled $1, 2, 3$, and $4$. A line passing through the lower left vertex of square $1$ divides the combined areas of squares $1, 3$, and $4$ in half so that the shaded region has area $6$. The difference between the areas of the shaded region within square $4$ and the shaded region within square $1$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [img]https://cdn.artofproblemsolving.com/attachments/7/4/b9554ccd782af15c680824a1fbef278a4f736b.png[/img]

Find the number of ways a series of $+$ and $-$ signs can be inserted between the numbers $0,1,2,\cdots, 12$ such that the value of the resulting expression is divisible by 5. [i]Proposed by Matthew Lerner-Brecher[/i]

Let $a_0, b_0, c_0$ be complex numbers, and define \begin{align*}a_{n+1} &= a_n^2 + 2b_nc_n \\ b_{n+1} &= b_n^2 + 2c_na_n \\ c_{n+1} &= c_n^2 + 2a_nb_n\end{align*}for all nonnegative integers $n.$ Suppose that $\max{\{|a_n|, |b_n|, |c_n|\}} \leq 2022$ for all $n.$ Prove that $$|a_0|^2 + |b_0|^2 + |c_0|^2 \leq 1.$$

Given the trapezoid $ABCD$ with the $BC\parallel AD$, let $C_1$ and $C_2$ be circles with diameters $AB$ and $CD$ respectively. Let $M$ and $N$ be the intersection points of $C_1$ with $AC$ and $BD$ respectively. Let $K$ and $L$ be the intersection points of $C_2$ with $AC$ and $BD$ respectively. Given $M\ne A$, $N\ne B$, $K\ne C$, $L\ne D$. Prove that $NK \parallel ML$.

Consider two quadrilaterals $ABCD$ and $A'B'C'D'$ in an affine Euclidian plane such that $AB = A'B', BC = B'C', CD = C'D'$, and $DA = D'A'$. Prove that the following two statements are true: [b](a)[/b] If the diagonals $BD$ and $AC$ are mutually perpendicular, then the diagonals $B'D'$ and $A'C'$ are also mutually perpendicular. [b](b)[/b] If the perpendicular bisector of $BD$ intersects $AC$ at $M$, and that of $B'D'$ intersects $A'C'$ at $M'$, then $\frac{\overline{MA}}{\overline{MC}}=\frac{\overline{M'A'}}{\overline{M'C'}}$ (if $MC = 0$ then $M'C' = 0$).

A sequence $\{z_n\}$ satisfies that for any positive integer $i,$ $z_i\in\{0,1,\cdots,9\}$ and $z_i\equiv i-1 \pmod {10}.$ Suppose there is $2021$ non-negative reals $x_1,x_2,\cdots,x_{2021}$ such that for $k=1,2,\cdots,2021,$ $$\sum_{i=1}^kx_i\geq\sum_{i=1}^kz_i,\sum_{i=1}^kx_i\leq\sum_{i=1}^kz_i+\sum_{j=1}^{10}\dfrac{10-j}{50}z_{k+j}.$$ Determine the least possible value of $\sum_{i=1}^{2021}x_i^2.$