Solve the system \begin{align*} 36x^2y-27y^3&~=~8,\\ 4x^3-27xy^2&~=~4.\end{align*}

Prove that for each $\Delta ABC$ with an acute $\angle C$ the following inequality is true: $(a^2+b^2) cos(\alpha -\beta )\leq 2ab$.

Determine all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q^+}$ that satisfy: \[f\left(x+\frac{y}{x}\right) = f(x)+f\left(\frac{y}{x}\right)+2y \:\text{for all}\: x, y \in \mathbb{Q^+}\]

Let $a$ be a real number and $A = \{(x, y) \in R \times R | \, x + y = a\}$, $B = \{(x,y) \in R \times R | \, x^3 + y^3 < a\}$ . Find all values of $a$ such that $A \cap B = \emptyset$ .

As a gift, Dilhan was given the number $n=1^1\cdot2^2\cdots2021^{2021}$, and each day, he has been dividing $n$ by $2021!$ exactly once. One day, when he did this, he discovered that, for the first time, $n$ was no longer an integer, but instead a reduced fraction of the form $\frac{a}b$. What is the sum of all distinct prime factors of $b$? [i]Proposed by Adam Bertelli[/i]

Does there exist a function $f : \mathbb N \to \mathbb N$, such that $f(f(n)) =n + 1987$ for every natural number $n$? [i](IMO Problem 4)[/i] [i]Proposed by Vietnam.[/i]

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

A biologist found a pond with frogs. When classifying them by their mass, he noticed the following: [i]The $50$ lightest frogs represented $30\%$ of the total mass of all the frogs in the pond, while the $44$ heaviest frogs represented $27\%$ of the total mass. [/i]As fate would have it, the frogs escaped and the biologist only has the above information. How many frogs were in the pond?

[u]Round 1 [/u] [b]p1.[/b] Pirate Jim had $8$ boxes with gun powder weighing $1, 2, 3, 4, 5, 6, 7$, and $8$ pounds (the weight is printed on top of every box). Pirate Bob hid a $1$-pound gold bar in one of these boxes. Pirate Jim has a balance scale that he can use, but he cannot open any of the boxes. Help him find the box with the gold bar using two weighings on the balance scale. [b]p2.[/b] James Bond will spend one day at Dr. Evil's mansion to try to determine the answers to two questions: a) Is Dr. Evil at home? b) Does Dr. Evil have an army of ninjas? The parlor in Dr. Evil's mansion has three windows. At noon, Mr. Bond will sneak into the parlor and use open or closed windows to signal his answers. When he enters the parlor, some windows may already be opened, and Mr. Bond will only have time to open or close one window (or leave them all as they are). Help Mr. Bond and Moneypenny design a code that will tell Moneypenny the answers to both questions when she drives by later that night and looks at the windows. Note that Moneypenny will not have any way to know which window Mr. Bond opened or closed. [b]p3.[/b] Suppose that you have a triangle in which all three side lengths and all three heights are integers. Prove that if these six lengths are all different, there cannot be four prime numbers among them. p4. Fred and George have designed the Amazing Maze, a $5\times 5$ grid of rooms, with Adorable Doors in each wall between rooms. If you pass through a door in one direction, you gain a gold coin. If you pass through the same door in the opposite direction, you lose a gold coin. The brothers designed the maze so that if you ever come back to the room in which you started, you will find that your money has not changed. Ron entered the northwest corner of the maze with no money. After walking through the maze for a while, he had $8$ shiny gold coins in his pocket, at which point he magically teleported himself out of the maze. Knowing this, can you determine whether you will gain or lose a coin when you leave the central room through the north door? [b]p5.[/b] Bill and Charlie are playing a game on an infinite strip of graph paper. On Bill’s turn, he marks two empty squares of his choice (not necessarily adjacent) with crosses. Charlie, on his turn, can erase any number of crosses, as long as they are all adjacent to each other. Bill wants to create a line of $2013$ crosses in a row. Can Charlie stop him? [u]Round 2 [/u] [b]p6.[/b] $1000$ non-zero numbers are written around a circle and every other number is underlined. It happens that each underlined number is equal to the sum of its two neighbors and that each non-underlined number is equal to the product of its two neighbors. What could the sum of all the numbers written on the circle be? [b]p7.[/b] A grasshopper is sitting at the edge of a circle of radius $3$ inches. He can hop exactly $4$ inches in any direction, as long as he stays within the circle. Which points inside the circle can the grasshopper reach if he can make as many jumps as he likes? [img]https://cdn.artofproblemsolving.com/attachments/1/d/39b34b2b4afe607c1232f4ce9dec040a34b0c8.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $f(-f(x)-f(y)) = 1-x-y$ $\quad \forall x,y \in \mathbb{Z}$

Prove the theorem: the lengths $ a$, $ b $, $ c $ of the sides of a triangle and the arc measures $ \alpha $, $ \beta $, $ \gamma $of its opposite angles satisfy the inequalities $$\frac{\pi}{3}\leq \frac{a \alpha + b \beta +c \gamma}{a+b+c}<\frac{\pi }{ 2}.$$

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function, and let $f^m$ be $f$ applied $m$ times. Suppose that for every $n \in \mathbb{N}$ there exists a $k \in \mathbb{N}$ such that $f^{2k}(n)=n+k$, and let $k_n$ be the smallest such $k$. Prove that the sequence $k_1,k_2,\ldots $ is unbounded. [i]Proposed by Palmer Mebane, United States[/i]

Alexa wrote the first $16$ numbers of a sequence: \[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\] Then she continued following the same pattern, until she had $2015$ numbers in total. What was the last number she wrote?

[b]8.1[/b] $x$ and $y$ are the roots of the equation $t^2-ct-c=0$. Prove that holds the inequality $x^3 + y^3 + (xy)^3 \ge 0.$ [b]8.2.[/b] Two circles touch internally at point $A$ . Through a point $B$ of the inner circle, different from $A$, a tangent to this circle intersecting the outer circle at points C and $D$. Prove that $AB$ is a bisector of angle $CAD$. [img]https://cdn.artofproblemsolving.com/attachments/2/8/3bab4b5c57639f24a6fd737f2386a5e05e6bc7.png[/img] [b]8.3[/b] Prove that $2^{3^{100}} + 1$ is divisible by $3^{101}$. [b]8.4 / 7.5[/b] An entire arc of circle is drawn through the vertices $A$ and $C$ of the rectangle $ABCD$ lying inside the rectangle. Draw a line parallel to $AB$ intersecting $BC$ at point $P$, $AD$ at point $Q$, and the arc $AC$ at point $R$ so that the sum of the areas of the figures $AQR$ and $CPR$ is the smallest. [img]https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png [/img] [b]8.5[/b] In a certain group of people, everyone has one enemy and one Friend. Prove that these people can be divided into two companies so that in every company there will be neither enemies nor friends. [b]8.6[/b] Numbers $a_1, a_2, . . . , a_{100}$ are such that $$a_1 - 2a_2 + a_3 \le 0$$ $$a_2-2a_3 + a_ 4 \le 0$$ $$...$$ $$a_{98}-2a_{99 }+ a_{100} \le 0$$ and at the same time $a_1 = a_{100}\ge 0$. Prove that all these numbers are non-negative. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

$f(x)=x^3-6x^2+17x$. If $f(a)=16, f(b)=20$, find $a+b$.

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that if $a,b,c$ are the length sides of a triangle, and $r$ is the radius of its incircle, then $f(a),f(b),f(c)$ also form a triangle where its radius of the incircle is $f(r)$.

Alice thinks of four positive integers $a\leq b\leq c\leq d$ satisfying $\{ab+cd,ac+bd,ad+bc\}=\{40,70,100\}$. What are all the possible tuples $(a,b,c,d)$ that Alice could be thinking of?

Let $n \ne 0$ be a natural number. A sequence of numbers is briefly called a sequence “$F_n$” if $n$ different numbers $z_1$, $z_2$, $...$, $z_n$ exist so that the following conditions are fulfilled: (1) Each term of the sequence is one of the numbers $z_1$, $z_2$, $...$, $z_n$. (2) Each of the numbers $z_1$, $z_2$, $...$, $z_n$ occurs at least once in the sequence. (3) Any two immediately consecutive members of the sequence are different numbers. (4) No subsequence of the sequence has the form $\{a, b, a, b\}$ with $a \ne b$. Note: A subsequence of a given sequence $\{x_1, x_2, x_3, ...\}$ or $\{x_1, x_2, x_3, ..., x_s\}$ is called any sequence of the form $\{x_{m1}, x_{m2}, x_{m3}, ...\}$ or $\{x_{m1}, x_{m2}, x_{m3}, ..., x_{mt}\}$ with natural numbers $m_1 < m_2 < m_3 < ...$ Answer the following questions: a) Given $n$, are there sequences $F_n$ of arbitrarily long length? b) If question (a) is answered in the negative for an $n$: What is the largest possible number of terms that a sequence $F_n$ can have (given $n$)?

Let $\{a_k\}^{2011}_{k=1}$ be the sequence of real numbers defined by $$a_1=0.201, \quad a_2=(0.2011)^{a_1},\quad a_3=(0.20101)^{a_2},\quad a_4=(0.201011)^{a_3},$$ and more generally \[ a_k = \begin{cases}(0.\underbrace{20101\cdots0101}_{k+2 \ \text{digits}})^{a_{k-1}}, &\text {if } k \text { is odd,} \\ (0.\underbrace{20101\cdots01011}_{k+2 \ \text{digits}})^{a_{k-1}}, &\text {if } k \text { is even.}\end{cases} \] Rearranging the numbers in the sequence $\{a_k\}^{2011}_{k=1}$ in decreasing order produces a new sequence $\{b_k\}^{2011}_{k=1}$. What is the sum of all the integers $k$, $1\le k \le 2011$, such that $a_k = b_k$? $ \textbf{(A)}\ 671\qquad\textbf{(B)}\ 1006\qquad\textbf{(C)}\ 1341\qquad\textbf{(D)}\ 2011\qquad\textbf{(E)}\ 2012 $

Prove that the equation $x^2 + y^2 + z^2 = x + y + z + 1$ has no rational solutions.

Let $ x,y,z $ be numbers distinct from $ -1 $ that verify the equation $$ \frac{1}{1+a} +\frac{1}{1+b} +\frac{1}{1+c} =\frac{3}{2} . $$ Prove that if $ abc=1, $ then $ a $ or $ b $ or $ c $ is equal to $ 1. $

Prove that for each $n\geq 3$ the equation: $x^n+y^n+z^n+u^n=v^{n-1}$ has infinitely many solutions in natural numbers.

Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

There are $ n $ natural numbers ($ n\geq 2 $) whose sum is equal to their product. Prove that this common value does not exceed $2n$.

In an infinite sequence $a_1, a_2, a_3, \cdots$, the number $a_1$ equals $1$, and each $a_n, n > 1$, is obtained from $a_{n-1}$ as follows: [list]- if the greatest odd divisor of $n$ has residue $1$ modulo $4$, then $a_n = a_{n-1} + 1,$ - and if this residue equals $3$, then $a_n = a_{n-1} - 1.$[/list] Prove that in this sequence [b](a) [/b] the number $1$ occurs infinitely many times; [b](b)[/b] each positive integer occurs infinitely many times. (The initial terms of this sequence are $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, \cdots$ )