We color the numbers $1, 2, 3,....,20$ with two colors white and black in such a way that both colors are used. Find the number of ways, we can perform this coloring if the product of white numbers and the product of black numbers have greatest common divisor equal to $1$.

Let $\Delta ABC$ be an acute triangle and $\Omega$ its circumscribed circle, with diameter $AP$. Points $E$ and $F$ are the orthogonal projections from $B$ on $AC$ and $AP$, points $M$ and $N$ are the midpoints of segments $EF$ and $CP$. Prove that $\angle BMN=90$.

Let $ \varphi(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$ be a polynomial with real coefficients. Given that $ \varphi(x)$ has three positive real roots and that $ \varphi(0) < 0$, prove that \[ 2b^3 \plus{} 9a^2d \minus{} 7abc \leq 0. \]

Suppose that $2018$ numbers $1$ and $-1$ are written around a circle. For every two adjacent numbers, their product is taken. Suppose that the sum of all $2018$ products is negative. Find all possible values of sum of $2018$ given numbers.

Consider a natural number $ n\equiv 9\pmod {25}. $ Prove that there exist three nonnegative integers $ a,b,c $ having the property that: $$ n=\frac{a(a+1)}{2} +\frac{b(b+1)}{2} +\frac{c(c+1)}{2} $$

Prove that the numbers from $1$ to $ 15$ cannot be divided into two groups: $A$ of $2$ numbers and $B$ of $13$ numbers such that the sum of the numbers in group $B$ is equal to product of numbers in group $A$.

Find all integer values can take $n$ such that $$\cos(2x)=\cos^nx - \sin^nx$$ for every real number $x$.

Let X be a $\kappa$ weighted compact $T_2$ space. Prove that for every $\omega\leq\lambda<\kappa$, X has a continuous image of a $T_2$ space of weight $\lambda$. (The weight of a space X is the smallest infinite cardinality of a base of X.)

An integer $m\ge 3$ and an infinite sequence of positive integers $(a_n)_{n\ge 1}$ satisfies the equation \[a_{n+2} = 2\sqrt[m]{a_{n+1}^{m-1} + a_n^{m-1}} - a_{n+1}. \] for all $n\ge 1$. Prove that $a_1 < 2^m$.

Determine the sum of all positive integers $n$ that satisfy the following condition: when $6n + 1$ is written in base $10$, all its digits are equal.

Consider the two mutually tangent parabolas $y=x^2$ and $y=-x^2$. The upper parabola rolls without slipping around the fixed lower parabola. Find the locus of the focus of the moving parabola.

In a forest there are $5$ trees $A, B, C, D, E$ that are in that order on a straight line. At the midpoint of $AB$ there is a daisy, at the midpoint of $BC$ there is a rose bush, at the midpoint of $CD$ there is a jasmine, and at the midpoint of $DE$ there is a carnation. The distance between $A$ and $E$ is $28$ m; the distance between the daisy and the carnation is $20$ m. Calculate the distance between the rose bush and the jasmine.

There is a row of $n$ chairs, numbered in order from left to right from $1$ to $n$. Additionally, the $n$ numbers from $1$ to $n$ are distributed on the backs of the chairs, one number per chair, such that the number on the back of a chair never matches the number of the chair itself. There is a child sitting on each chair. Every time the teacher claps, each child checks the number on the back of the chair they are sitting on and moves to the chair corresponding to that number. Prove that for any $m$ that is not a power of a prime, with $1 < m \leqslant n$, it is possible to distribute the numbers on the backrests such that, after the teacher claps $m$ times, for the first time, all the children are sitting in the chairs where they initially started. (During the process, it may happen that some children return to their original chairs, but they do not all do so simultaneously until the $m^{\text{th}}$ clap.)

[b]p1.[/b] Determine the number of ways to place $4$ rooks on a $4 \times 4$ chessboard such that: (a) no two rooks attack one another, and (b) the main diagonal (the set of squares marked $X$ below) does not contain any rooks. [img]https://cdn.artofproblemsolving.com/attachments/e/e/e3aa96de6c8ed468c6ef3837e66a0bce360d36.png[/img] The rooks are indistinguishable and the board cannot be rotated. (Two rooks attack each other if they are in the same row or column.) [b]p2.[/b] Seven students, numbered $1$ to $7$ in counter-clockwise order, are seated in a circle. Fresh Mann has 100 erasers, and he wants to distribute them to the students, albeit unfairly. Starting with person $ 1$ and proceeding counter-clockwise, Fresh Mann gives $i$ erasers to student $i$; for example, he gives $ 1$ eraser to student $ 1$, then $2$ erasers to student $2$, et cetera. He continues around the circle until he does not have enough erasers to give to the next person. At this point, determine the number of erasers that Fresh Mann has. [b]p3.[/b] Let $ABC$ be a triangle with $AB = AC = 17$ and $BC = 24$. Approximate $\angle ABC$ to the nearest multiple of $10$ degrees. [b]p4.[/b] Define a sequence of rational numbers $\{x_n\}$ by $x_1 =\frac35$ and for $n \ge 1$, $x_{n+1} = 2 - \frac{1}{x_n}$ . Compute the product $x_1x_2x_3... x_{2013}$. [b]p5.[/b] In equilateral triangle $ABC$, points $P$ and $R$ lie on segment $AB$, points $I$ and $M$ lie on segment $BC$, and points $E$ and $S$ lie on segment $CA$ such that $PRIMES$ is a equiangular hexagon. Given that $AB = 11$, $PR = 2$, $IM = 3$, and $ES = 5$, compute the area of hexagon $PRIMES$. [b]p6.[/b] Let $f(a, b) = \frac{a^2}{a+b}$ . Let $A$ denote the sum of $f(i, j)$ over all pairs of integers $(i, j)$ with $1 \le i < j \le 10$; that is, $$A = (f(1, 2) + f(1, 3) + ...+ f(1, 10)) + (f(2, 3) + f(2, 4) +... + f(2, 10)) +... + f(9, 10).$$ Similarly, let $B$ denote the sum of $f(i, j)$ over all pairs of integers $(i, j)$ with $1 \le j < i \le 10$, that is, $$B = (f(2, 1) + f(3, 1) + ... + f(10, 1)) + (f(3, 2) + f(4, 2) +... + f(10, 2)) +... + f(10, 9).$$ Compute $B - A$. [b]p7.[/b] Fresh Mann has a pile of seven rocks with weights $1, 1, 2, 4, 8, 16$, and $32$ pounds and some integer X between $1$ and $64$, inclusive. He would like to choose a set of the rocks whose total weight is exactly $X$ pounds. Given that he can do so in more than one way, determine the sum of all possible values of $X$. (The two $1$-pound rocks are indistinguishable.) [b]p8.[/b] Let $ABCD$ be a convex quadrilateral with $AB = BC = CA$. Suppose that point $P$ lies inside the quadrilateral with $AP = PD = DA$ and $\angle PCD = 30^o$. Given that $CP = 2$ and $CD = 3$, compute $CA$. [b]p9.[/b] Define a sequence of rational numbers $\{x_n\}$ by $x_1 = 2$, $x_2 = \frac{13}{2}$ , and for $n \ge 1$, $x_{n+2} = 3 -\frac{3}{x_{n+1}}+\frac{1}{x_nx_{n+1}}$. Compute $x_{100}$. [b]p10.[/b] Ten prisoners are standing in a line. A prison guard wants to place a hat on each prisoner. He has two colors of hats, red and blue, and he has $10$ hats of each color. Determine the number of ways in which the prison guard can place hats such that among any set of consecutive prisoners, the number of prisoners with red hats and the number of prisoners with blue hats differ by at most $2$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$ABCDE$ is a convex pentagon with $AB = CD = EA = 1$, $\angle ABC = \angle DEA = 90^o$, and $BC + DE = 1$. What is the area of the pentagon?

The numbers $1, 2, 3, . . . , 101$ are written in a row in some order. Prove that it is always possible to erase $90 $ of the numbers so that the remaining $11$ numbers remain arranged in either increasing or decreasing order.

In a polygon, every external angle is one sixth of its corresponding internal angle. How many sides does the polygon have?