Let $A_1A_2 . . . A_{12}$ be a regular dodecagon. Equilateral triangles $\vartriangle A_1A_2B_1$, $\vartriangle A_2A_3B_2$, $. . . $, and $\vartriangle A_{12}A_1B_{12}$ are drawn such that points $B_1$, $B_2$,$ . . . $, and B_{12} lie outside dodecagon $A_1A_2 . . . A_{12}$. Then, equilateral triangles $\vartriangle A_1A_2C_1$, $\vartriangle A_2A_3C_2$, $. . .$ , and $\vartriangle A_{12}A_1C_{12}$ are drawn such that points $C_1$, $C_2$, $. . .$ , and $C_{12}$ lie inside dodecagon $A_1A_2 . . . A_{12}$. Compute the ratio of the area of dodecagon $B_1B_2 . . . B_{12}$ to the area of dodecagon $C_1C_2 . . . C_{12}$.

On an infinite grid, a square with four vertices lie at $(m, n)$, $(m-1, n)$, $(m,n-1)$, $(m-1, n-1)$ is denoted as cell $(m,n)$ $(m, n \in Z)$. Some marbles are dropped on some cell. Each cell may have more than one marble or have no marble at all. Consider a "move" can be conducted in one of two following ways: i) Remove one marble from cell $(m,n)$ (if there is marble at that cell), then add one marble to each of cell $(m - 1, n- 2)$ and cell $(m -2, n - 1)$. ii) Remove two marbles from cell $(m,n)$ (if there is marble at that cell), then add one marble to each of cell $(m +1, n - 2)$ and cell $(m - 2, n +1)$. Assume that initially, there are $n$ marbles at the cell $(1,n), (2,n - 1),..., (n, 1)$ (each cell contains one marble). Can we conduct an finite amount of moves such that both cells $(n + 1, n)$ and $(n, n + 1)$ have marbles?

For real numbers $x,y, \in [1,2]$, prove the inequality $3(x + y)\ge 2xy + 4$

Are the following statements true? a) if the four vertices of a rectangle lie on the four sides of a rhombus, then the sides of the rectangle are parallel to the diagonals of the rhombus; b) if the four vertices of a square lie on the four sides of a rhombus that is not a square, then the sides of the square are parallel to the diagonals of the rhombus.

Let be three positive real numbers $ a,b,c. $ Show that the function $ f:\mathbb{R}\longrightarrow\mathbb{R} , $ $$ f(x)=\frac{a^x}{b^x+c^x} +\frac{b^x}{a^x+c^x} +\frac{c^x}{a^x+b^x} , $$ is nondecresing on the interval $ \left[ 0,\infty \right) $ and nonincreasing on the interval $ \left( -\infty ,0 \right] . $

Consider all right triangles with integer side lengths that form an arithmetic sequence. Compute the $2014$th smallest perimeter of all such right triangles.

Let $ ABC$ be an acute triangle, $ AD,BE,CZ$ its altitudes and $ H$ its orthocenter. Let $ AI,A \Theta$ be the internal and external bisectors of angle $ A$. Let $ M,N$ be the midpoints of $ BC,AH$, respectively. Prove that: (a) $MN$ is perpendicular to $EZ$ (b) if $ MN$ cuts the segments $ AI,A \Theta$ at the points $ K,L$, then $ KL\equal{}AH$

Let $ a,b,c$ be non-zero real numbers satisfying \[ \dfrac{1}{a}\plus{}\dfrac{1}{b}\plus{}\dfrac{1}{c}\equal{}\dfrac{1}{a\plus{}b\plus{}c}.\] Find all integers $ n$ such that \[ \dfrac{1}{a^n}\plus{}\dfrac{1}{b^n}\plus{}\dfrac{1}{c^n}\equal{}\dfrac{1}{a^n\plus{}b^n\plus{}c^n}.\]

[b]p1.[/b] Suppose that $20\%$ of a number is $17$. Find $20\%$ of $17\%$ of the number. [b]p2.[/b] Let $A, B, C, D$ represent the numbers $1$ through $4$ in some order, with $A \ne 1$. Find the maximum possible value of $\frac{\log_A B}{C +D}$. Here, $\log_A B$ is the unique real number $X$ such that $A^X = B$. [b]p3. [/b]There are six points in a plane, no four of which are collinear. A line is formed connecting every pair of points. Find the smallest possible number of distinct lines formed. [b]p4.[/b] Let $a,b,c$ be real numbers which satisfy $$\frac{2017}{a}= a(b +c), \frac{2017}{b}= b(a +c), \frac{2017}{c}= c(a +b).$$ Find the sum of all possible values of $abc$. [b]p5.[/b] Let $a$ and $b$ be complex numbers such that $ab + a +b = (a +b +1)(a +b +3)$. Find all possible values of $\frac{a+1}{b+1}$. [b]p6.[/b] Let $\vartriangle ABC$ be a triangle. Let $X,Y,Z$ be points on lines $BC$, $CA$, and $AB$, respectively, such that $X$ lies on segment $BC$, $B$ lies on segment $AY$ , and $C$ lies on segment $AZ$. Suppose that the circumcircle of $\vartriangle XYZ$ is tangent to lines $AB$, $BC$, and $CA$ with center $I_A$. If $AB = 20$ and $I_AC = AC = 17$ then compute the length of segment $BC$. [b]p7. [/b]An ant makes $4034$ moves on a coordinate plane, beginning at the point $(0, 0)$ and ending at $(2017, 2017)$. Each move consists of moving one unit in a direction parallel to one of the axes. Suppose that the ant stays within the region $|x - y| \le 2$. Let N be the number of paths the ant can take. Find the remainder when $N$ is divided by $1000$. [b]p8.[/b] A $10$ digit positive integer $\overline{a_9a_8a_7...a_1a_0}$ with $a_9$ nonzero is called [i]deceptive [/i] if there exist distinct indices $i > j$ such that $\overline{a_i a_j} = 37$. Find the number of deceptive positive integers. [b]p9.[/b] A circle passing through the points $(2, 0)$ and $(1, 7)$ is tangent to the $y$-axis at $(0, r )$. Find all possible values of $ r$. [b]p10.[/b] An ellipse with major and minor axes $20$ and $17$, respectively, is inscribed in a square whose diagonals coincide with the axes of the ellipse. Find the area of the square. PS. You had better use hide for answers.

Inscribe an equilateral triangle of minimum side in a given acute-angled triangle $ABC$ (one vertex on each side).

Let $p_{1}$, $p_{2}$, ..., $p_{k}$ be different prime numbers. Determine the number of positive integers of the form $p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}...p_{k}^{\alpha_{k}}$, $\alpha_{i}$ $\in$ $\mathbb{N}$ for which $\alpha_{1} \alpha_{2}...\alpha_{k}=p_{1}p_{2}...p_{k}$.

Compute the number of ordered pairs of integers $(x,y)$ such that $x^2 + y^2 < 2019$ and $$x^2 + min(x,y) = y^2 + max(x, y) .$$

For a class of $20$ students several field trips were arranged. In each trip at least one student participated. Prove that there was a field trip such that each student who participated in it took part in at least $1/20$-th of all field trips.

Let $C$ be the curve $y^2 = x^3$ (where $x$ takes all non-negative real values). Let $O$ be the origin, and $A$ be the point where the gradient is $1.$ Find the length of the curve from $O$ to $A.$

In a non-isosceles acute triangle $ABC$, $D$ is the midpoint of the edge $[BC]$. The points $E$ and $F$ lie on $[AC]$ and $[AB]$, respectively, and the circumcircles of $CDE$ and $AEF$ intersect in $P$ on $[AD]$. The angle bisector from $P$ in triangle $EFP$ intersects $EF$ in $Q$. Prove that the tangent line to the circumcirle of $AQP$ at $A$ is perpendicular to $BC$.

The sequence $ a_n$ satisfies $ a_{m\plus{}n}\plus{} a_{m\minus{}n}\equal{}\frac12(a_{2m}\plus{}a_{2n})$ for all $ m\geq n\geq 0$. If $ a_1\equal{}1$, find $ a_{1995}$.

Prove that if, in parallel projection of one plane onto another plane, the image of a certain square is a square, then the image of every figure is the figure congruent to it.

Let $x_1, x_2,... x_n$ be positive numbers such that $S = x_1+x_2+...+x_n =\frac{1}{x_1}+...+\frac{1}{x_n}$ Prove that $$\sum_{i=1}^{n} \frac{1}{n - 1 + x_i} \ge \sum_{i=1}^{n} \frac{1}{1+S - x_i}$$

Ava and Tiffany participate in a knockout tournament consisting of a total of $32$ players. In each of $5$ rounds, the remaining players are paired uniformly at random. In each pair, both players are equally likely to win, and the loser is knocked out of the tournament. The probability that Ava and Tiffany play each other during the tournament is $\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Compute $100a + b.$

The side lengths of an equiangular octagon alternate between $20$ and $22$. Find its area.

Find all the functions $f : \mathbb R^+ \to \mathbb R$ satisfying the identity \[f(x)f(y)=y^{\alpha}f\left(\frac x2 \right) + x^{\beta} f\left(\frac y2 \right) \qquad \forall x,y \in \mathbb R^+\] Where $\alpha,\beta$ are given real numbers.

In the Republic of Unistan there are $n$ national roads, each of them links two cities exactly. You can travel from one city to another of your choice using a sequence of roads. The President of Unistan ordered to label the national roads with the integers from $1$ to $n$ by an old law: if a city is adjacent to more than one road, the greatest common divisor of the numbers of that roads must be one. Show that you can label the national roads without breaking the law.

The sides of a rhombus have length $a$ and the area is $S$. What is the length of the shorter diagonal?

Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.

In the set of real numbers solve the system of equations $x^4+y^2+4=5yz$ $y^4+z^2+4=5zx$ $z^4+x^2+4=5xy$