Find all pairs of real numbers $(x, y)$ that satisfy the equation $$\frac{(x+y)(2-\sin(x+y))}{4\sin^2(x+y)}=\frac{xy}{x+y}$$

Find all the functions $f: \mathbb{R} \to\mathbb{R}$ such that \[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\] for all $x,y \in \mathbb{R} $.

Given are the sequences \[ (..., a_{-2}, a_{-1}, a_0, a_1, a_2, ...); (..., b_{-2}, b_{-1}, b_0, b_1, b_2, ...); (..., c_{-2}, c_{-1}, c_0, c_1, c_2, ...)\] of positive real numbers. For each integer $n$ the following inequalities hold: \[a_n \geq \frac{1}{2} (b_{n+1} + c_{n-1})\] \[b_n \geq \frac{1}{2} (c_{n+1} + a_{n-1})\] \[c_n \geq \frac{1}{2} (a_{n+1} + b_{n-1})\] Determine $a_{2005}$, $b_{2005}$, $c_{2005}$, if $a_0 = 26, b_0 = 6, c_0 = 2004$.

$\mathbb{Q}$ is set of all rational numbers. Find all functions $f:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{Q}$ such that for all $x$, $y$, $z$ $\in\mathbb{Q}$ satisfy $f(x,y)+f(y,z)+f(z,x)=f(0,x+y+z)$

[b]p1.[/b] Find all integer solutions of the equation $a^2 - b^2 = 2002$. [b]p2.[/b] Prove that the disks drawn on the sides of a convex quadrilateral as on diameters cover this quadrilateral. [b]p3.[/b] $30$ students from one school came to Mathematical Olympiad. In how many different ways is it possible to place them in four rooms? [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve the system of equations $$ \qquad<br /> \begin{array}{c}<br /> x_1x_2 = 1\\<br /> x_2x_3 = 2\\<br /> x_3x_4 = 3\\<br /> \ldots\\<br /> x_nx_1 = n<br /> \end{array}$$

We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point). (a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$. (b) Find the largest possible size of a very set not contained in any line. (Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.) [i]Proposed by Sammy Luo[/i]

Let $n \ge 2$ be an integer. Let $a$ be the greatest positive integer such that $2^a | 5^n - 3^n$. Let $b$ be the greatest positive integer such that $2^b \le n$. Prove that $a \le b + 3$.

Find all polynomials $P(x, y)$ with real coefficients which for all real numbers $x$ and $y$ satisfy $P(x + y, x - y) = 2P(x, y)$.

Juan found an easy (but wrong) way to simplify fractions. He proposes to simplify a fraction $\frac{M}{N}$ , where $M <N$ are two natural numbers, erase simultaneously the equal digits in the numerator and denominator. For instance, $\frac{12356}{5789}$ transforms after simplification of Juan in $\frac{126}{789}$. Find out if there is at least one fraction $\frac{M}{N}$, with $10 <M <N <100$ for which this method gives a correct result.

Esmeralda writes $2n$ real numbers $x_1, x_2, \dots , x_{2n}$, all belonging to the interval $[0, 1]$, around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products $p_1 = x_1x_2$, $p_2 = x_2x_3$, $\dots$ , $p_{2n} = x_{2n}x_1$. She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?

Let $a, b, c > 0$ be real numbers. Show the following inequality: $$a^2 \cdot \frac{a - b}{a + b}+ b^2\cdot \frac{b - c}{b + c}+ c^2\cdot \frac{c - a}{c + a} \ge 0 .$$ When does equality holds?

Compute $$\prod^{2019}_{i=1} (2^{2^i}- 2^{2^{i-1}} + 1).$$

Find the least $k$ for which the number $2010$ can be expressed as the sum of the squares of $k$ integers.

Suppse that $n\geq 2$ and $0<x_1<x_2<...<x_n$ are integer numbers. We denote that :\[ S_k=\sum_{A\subset \{x_1,x_2,...,x_n\}} \frac{1}{\prod_{a\in A}a} , k=1,2,...,n. \] (where $A$ is a non-empty subset). Show that if $S_n ,S_{n-1}$ were positive integer numbers , then $\forall k : S_k$ is a positive integer.

[b]p1.[/b] Len's Spanish class has four tests in the first term. Len scores $72$, $81$, and $78$ on the first three tests. If Len wants to have an 80 average for the term, what is the minimum score he needs on the last test? [b]p2.[/b] In $1824$, the Electoral College had $261$ members. Andrew Jackson won $99$ Electoral College votes and John Quincy Adams won $84$ votes. A plurality occurs when no candidate has more than $50\%$ of the votes. Should a plurality occur, the vote goes to the House of Representatives to break the tie. How many more votes would Jackson have needed so that a plurality would not have occurred? [b]p3.[/b] $\frac12 + \frac16 + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}= 1 - \frac{1}{n}$. Find $n$. [b]p4.[/b] How many ways are there to sit Samuel, Esun, Johnny, and Prat in a row of $4$ chairs if Prat and Johnny refuse to sit on an end? [b]p5.[/b] Find an ordered quadruple $(w, x, y, z)$ that satisfies the following: $$3^w + 3^x + 3^y = 3^z$$ where $w + x + y + z = 2017$. [b]p6.[/b] In rectangle $ABCD$, $E$ is the midpoint of $CD$. If $AB = 6$ inches and $AE = 6$ inches, what is the length of $AC$? [b]p7.[/b] Call an integer interesting if the integer is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$, so $27$ is interesting. How many $2$-digit interesting integers are there? [b]p8.[/b] Let $a\#b = \frac{a^3-b^3}{a-b}$ . If $a, b, c$ are the roots of the polynomial $x^3 + 2x^2 + 3x + 4$, what is the value of $a\#b + b\#c + c\#a$? [b]p9.[/b] Akshay and Gowri are examining a strange chessboard. Suppose $3$ distinct rooks are placed into the following chessboard. Find the number of ways that one can place these rooks so that they don't attack each other. Note that two rooks are considered attacking each other if they are in the same row or the same column. [img]https://cdn.artofproblemsolving.com/attachments/f/1/70f7d68c44a7a69eb13ce12291c0600d11027c.png[/img] [b]p10.[/b] The Earth is a very large sphere. Richard and Allen have a large spherical model of Earth, and they would like to (for some strange reason) cut the sphere up with planar cuts. If each cut intersects the sphere, and Allen holds the sphere together so it does not fall apart after each cut, what is the maximum number of pieces the sphere can be cut into after $6$ cuts? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a, b, c$ be positive real numbers such that $a + b + c + ab + bc + ca + abc = 7$. Prove that $\sqrt{a^2 + b^2 + 2 }+\sqrt{b^2 + c^2 + 2 }+\sqrt{c^2 + a^2 + 2 } \ge 6$ .

A sequence $x_0, x_1, x_2, . . .$ is given by $x_0 = 8$ and $x_{n+1} =\frac{1 + x_n}{1- x_n}$ for $n = 0, 1, 2, . . . .$ Determine the number $x_{2013}$.

Prove that if $a,b,c>0,$ then \[ \frac{ab}{3a+4b+5c}+\frac{bc}{3b+4c+5a}+\frac{ca}{3c+4a+5b}\le \frac{a+b+c}{12}. \] [i] Nikolai Nikolov[/i]

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

The time-varying temperature of a certain body is given by a polynomial in the time of degree at most three. Show that the average temperature of the body between $9$ am and $3$ pm can always be found by taking the average of the temperatures at two fixed times, which are independent of the polynomial. Also, show that these two times are $10\colon \! 16$ am and $1\colon \!44$ pm to the nearest minute.

Let $u_1$, $u_2$, $u_3$, $\dots$ be a sequence of integers satisfying the recurrence relation $u_{n + 2} = u_{n + 1}^2 - u_n$. Suppose $u_1 = 39$ and $u_2 = 45$. Prove that 1986 divides infinitely many terms of the sequence.

Given a sequence of real numbers $a_1, a_2, ... , a_n$. Prove that it is possible to choose some of the numbers providing $3$ conditions: a) not a triple of successive members is chosen, b) at least one of every triple of successive members is chosen, c) the absolute value of chosen numbers sum is not less that one sixth part of the initial numbers' absolute values sum.

The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.

[b]p1.[/b] If Suzie has $6$ coins worth $23$ cents, how many nickels does she have? [b]p2.[/b] Let $a * b = (a - b)/(a + b)$. If $8 * (2 * x) = 4/3$, what is $x$? [b]p3.[/b] How many digits does $x = 100000025^2 - 99999975^2$ have when written in decimal form? [b]p4.[/b] A paperboy’s route covers $8$ consecutive houses along a road. He does not necessarily deliver to all the houses every day, but he always traverses the road in the same direction, and he takes care never to skip over $2$ consecutive houses. How many possible routes can he take? [b]p5.[/b] A regular $12$-gon is inscribed in a circle of radius $5$. What is the sum of the squares of the distances from any one fixed vertex to all the others? [b]p6.[/b] In triangle $ABC$, let $D, E$ be points on $AB$, $AC$, respectively, and let $BE$ and $CD$ meet at point $P$. If the areas of triangles $ADE$, $BPD$, and $CEP$ are $5$, $8$, and $3$, respectively, find the area of triangle ABC. [b]p7.[/b] Bob has $11$ socks in his drawer: $3$ different matched pairs, and $5$ socks that don’t match with any others. Suppose he pulls socks from the drawer one at a time until he manages to get a matched pair. What is the probability he will need to draw exactly $9$ socks? [b]p8.[/b] Consider the unit cube $ABCDEFGH$. The triangle bound to $A$ is the triangle formed by the $3$ vertices of the cube adjacent to $A$ (and similarly for the other vertices of the cube). Suppose we slice a knife through each of the $8$ triangles bound to vertices of the cube. What is the volume of the remaining solid that contains the former center of the cube? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].