Let $x, y, z$ be positive real numbers such that $\sqrt {x}, \sqrt {y}, \sqrt {z}$ are sides of a triangle and $\frac {x}{y}+\frac {y}{z}+\frac {z}{x}=5$. Prove that $\frac {x(y^2-2z^2)}{z}+\frac {y(z^2-2x^2)}{x}+\frac {z(x^2-2y^2)}{y}\geqslant0$

Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}.$

Let $a_1, . . . , a_{2020}$ be a sequence of real numbers such that $a_1 = 2^{-2019}$, and $a^2_{n-1}a_n = a_n-a_{n-1}$. Prove that $a_{2020} <\frac{1}{2^{2019} -1}$

Find all polynomials $p(x)$ with real coeffcients such that \[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\] for all $a, b, c\in\mathbb{R}$. [i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i]

How many positive integer solutions does the following system of equations have? $$\begin{cases}\sqrt{2020}(\sqrt{a}+\sqrt{b})=\sqrt{(c+2020)(d+2020)}\\\sqrt{2020}(\sqrt{b}+\sqrt{c})=\sqrt{(d+2020)(a+2020)}\\\sqrt{2020}(\sqrt{c}+\sqrt{d})=\sqrt{(a+2020)(b+2020)}\\\sqrt{2020}(\sqrt{d}+\sqrt{a})=\sqrt{(b+2020)(c+2020)}\\ \end{cases}$$

Say an integer polynomial is $\textit{primitive}$ if the greatest common divisor of its coefficients is $1$. For example, $2x^2+3x+6$ is primitive because $\gcd(2,3,6)=1$. Let $f(x)=a_2x^2+a_1x+a_0$ and $g(x) = b_2x^2+b_1x+b_0$, with $a_i,b_i\in\{1,2,3,4,5\}$ for $i=0,1,2$. If $N$ is the number of pairs of polynomials $(f(x),g(x))$ such that $h(x) = f(x)g(x)$ is primitive, find the last three digits of $N$.

We know that the function $ f: \left[ 0,\frac{\pi }{2}\right]\longrightarrow [a,b], f(x)=\sqrt[n]{\cos x } +\sqrt[n]{\sin x} , $ is surjective for a given natural number $ n\ge 2. $ Determine the numbers $ a,b, $ and the monotony of $ f. $

[u]Round 1[/u] [b]p1. [/b]The Queen of Bees invented a new language for her hive. The alphabet has only $6$ letters: A, C, E, N, R, T; however, the alphabetic order is different than in English. A word is any sequence of $6$ different letters. In the dictionary for this language, the word TRANCE immediately follows NECTAR. What is the last word in the dictionary? [b]p2.[/b] Is it possible to solve the equation $\frac{1}{x}= \frac{1}{y} +\frac{1}{z}$ with $x,y,z$ integers (positive or negative) such that one of the numbers $x,y,z$ has one digit, another has two digits, and the remaining one has three digits? [b]p3.[/b] The $10,000$ dots in a $100\times 100$ square grid are all colored blue. Rekha can paint some of them red, but there must always be a blue dot on the line segment between any two red dots. What is the largest number of dots she can color red? The picture shows a possible coloring for a $5\times 7$ grid. [img]https://cdn.artofproblemsolving.com/attachments/0/6/795f5ab879938ed2a4c8844092b873fb8589f8.jpg[/img] [b]p4.[/b] Six flies rest on a table. You have a swatter with a checkerboard pattern, much larger than the table. Show that there is always a way to position and orient the swatter to kill at least five of the flies. Each fly is much smaller than a swatter square and is killed if any portion of a black square hits any part of the fly. [b]p5.[/b] Maryam writes all the numbers $1-81$ in the cells of a $9\times 9$ table. Tian calculates the product of the numbers in each of the nine rows, and Olga calculates the product of the numbers in every column. Could Tian's and Olga's lists of nine products be identical? [u]Round 2[/u] [b]p6.[/b] A set of points in the plane is epic if, for every way of coloring the points red or blue, it is possible to draw two lines such that each blue point is on a line, but none of the red points are. The figure shows a particular set of $4$ points and demonstrates that it is epic. What is the maximum possible size of an epic set? [img]https://cdn.artofproblemsolving.com/attachments/e/f/44fd1679c520bdc55c78603190409222d0b721.jpg[/img] [b]p7.[/b] Froggy Chess is a game played on a pond with lily pads. First Judit places a frog on a pad of her choice, then Magnus places a frog on a different pad of his choice. After that, they alternate turns, with Judit moving first. Each player, on his or her turn, selects either of the two frogs and another lily pad where that frog must jump. The jump must reduce the distance between the frogs (all distances between the lily pads are different), but both frogs cannot end up on the same lily pad. Whoever cannot make a move loses. The picture below shows the jumps permitted in a particular situation. Who wins the game if there are $2017$ lily pads? [img]https://cdn.artofproblemsolving.com/attachments/a/9/1a26e046a2a614a663f9d317363aac61654684.jpg[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that \[\displaystyle \sum_{\{i,j,k\}=\{1,2,3\}} \csc ^{13} \frac{2^i \pi}{7}\csc ^{14} \frac{2^j \pi}{7}\csc ^{2005} \frac{2^k\pi}{7}\] is rational. Here, $(i,j,k)$ is summed over all possible permutations of $(1,2,3)$.

[b]p1.[/b] $4$ balls are distributed uniformly at random among $6$ bins. What is the expected number of empty bins? [b]p2.[/b] Compute ${150 \choose 20 }$ (mod $221$). [b]p3.[/b] On the right triangle $ABC$, with right angle at$ B$, the altitude $BD$ is drawn. $E$ is drawn on $BC$ such that AE bisects angle $BAC$ and F is drawn on $AC$ such that $BF$ bisects angle $CBD$. Let the intersection of $AE$ and $BF$ be $G$. Given that $AB = 15$,$ BC = 20$, $AC = 25$, find $\frac{BG}{GF}$ . [b]p4.[/b] What is the largest integer $n$ so that $\frac{n^2-2012}{n+7}$ is also an integer? [b]p5.[/b] What is the side length of the largest equilateral triangle that can be inscribed in a regular pentagon with side length $1$? [b]p6.[/b] Inside a LilacBall, you can find one of $7$ different notes, each equally likely. Delcatty must collect all $7$ notes in order to restore harmony and save Kanto from eternal darkness. What is the expected number of LilacBalls she must open in order to do so? PS. You had better use hide for answers.

Suppose $a_1, a_2, ... , a_{100}$ are positive real numbers such that $$a_k =\frac{ka_{k-1}}{a_{k-1} - (k - 1)}$$ for $k = 2, 3, ... , 100$. Given that $a_{20} = a_{23}$, compute $a_{100}$.

Do there exist two integer-valued functions $f$ and $g$ such that for every integer $x$ we have (a) $f(f(x)) = x, g(g(x)) = x, f(g(x)) > x, g(f(x)) > x$ ? (b) $f(f(x)) < x, g(g(x)) < x, f(g(x)) > x, g(f(x)) > x$ ?

Do either (1) or (2): (1) Show that any solution $f(t)$ of the functional equation $$f(x+y)f(x-y)=f(x)^{2} +f(y)^{2} -1$$ for $x,y\in \mathbb{R}$ satisfies $$f''(t)= \pm c^{2} f(t)$$ for a constant $c$, assuming the existence and continuity of the second derivative. Deduce that $f(t)$ is one of the functions $$ \pm \cos ct, \;\;\; \pm \cosh ct.$$ (2) Let $(a_{i})_{i=1,...,n}$ and $(b_{i})_{i=1,...,n}$ be real numbers. Define an $(n+1)\times (n+1)$-matrix $A=(c_{ij})$ by $$ c_{i1}=1, \; \; c_{1j}= x^{j-1} \; \text{for} \; j\leq n,\; \; c_{1n+1}=p(x), \;\; c_{ij}=a_{i-1}^{j-1} \; \text{for}\; i>1, j\leq n,\;\; c_{in+1}=b_{i-1}\; \text{for}\; i>1.$$ The polynomial $p(x)$ is defined by the equation $\det A=0$. Let $f$ be a polynomial and replace $(b_{i})$ with $(f(b_{i}))$. Then $\det A=0$ defines another polynomial $q(x)$. Prove that $f(p(x))-q(x)$ is a multiple of $$\prod_{i=1}^{n} (x-a_{i}).$$

Let the $A$ be the set of all nonenagative integers. It is given function such that $f:\mathbb{A}\rightarrow\mathbb{A}$ with $f(1) = 1$ and for every element $n$ od set $A$ following holds: [b]1)[/b] $3 f(n) \cdot f(2n+1) = f(2n) \cdot (1+3 \cdot f(n))$; [b]2)[/b] $f(2n) < 6f(n)$, Find all solutions of $f(k)+f(l) = 293$, $k<l$.

[b]p1.[/b] Ben and his dog are walking on a path around a lake. The path is a loop $500$ meters around. Suddenly the dog runs away with velocity $10$ km/hour. Ben runs after it with velocity $8$ km/hour. At the moment when the dog is $250$ meters ahead of him, Ben turns around and runs at the same speed in the opposite direction until he meets the dog. For how many minutes does Ben run? [b]p2.[/b] The six interior angles in two triangles are measured. One triangle is obtuse (i.e. has an angle larger than $90^o$) and the other is acute (all angles less than $90^o$). Four angles measure $120^o$, $80^o$, $55^o$ and $10^o$. What is the measure of the smallest angle of the acute triangle? [b]p3.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img] [b]p4.[/b] Two three-digit whole numbers are called relatives if they are not the same, but are written using the same triple of digits. For instance, $244$ and $424$ are relatives. What is the minimal number of relatives that a three-digit whole number can have if the sum of its digits is $10$? [b]p5.[/b] Three girls, Ann, Kelly, and Kathy came to a birthday party. One of the girls wore a red dress, another wore a blue dress, and the last wore a white dress. When asked the next day, one girl said that Kelly wore a red dress, another said that Ann did not wear a red dress, the last said that Kathy did not wear a blue dress. One of the girls was truthful, while the other two lied. Which statement was true? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The integers $a$ and $b$ have the property that for every nonnegative integer $n$ the number of $2^n{a}+b$ is the square of an integer. Show that $a=0$.

Given is the function $f(x)=-x+\sqrt{(x+a)(x+b)}$, where $a$, $b$ are distinct given positive real numbers. Prove that for all real numbers $s\in (0,1)$ there exist only one positive real number $\alpha$ such that \[ f(\alpha)=\sqrt [s]{\frac{a^s+b^s}{2}} . \]

Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$, and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.

Let $ a_{0},a_{1},\ldots,a_{n}$ be integers, one of which is nonzero, and all of the numbers are not less than $ \minus{} 1$. Prove that if \[ a_{0} \plus{} 2a_{1} \plus{} 2^{2}a_{2} \plus{} \cdots \plus{} 2^{n}a_{n} \equal{} 0,\] then $ a_{0} \plus{} a_{1} \plus{} \cdots \plus{} a_{n} > 0$.

Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$

Find the number of arithmetic sequences $a_1,a_2,a_3$ of three nonzero integers such that the sum of the terms in the sequence is equal to the product of the terms in the sequence. [i]Proposed by Sammy Charney[/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 $a \le b + 3$.

Determine all polynomials $f (x)$ with real coeffcients that satisfy \[f (x^{2}-2x) = f^{2}(x-2)\] for all $x.$

Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?

For real numbers $a, b, c$ with $a + b + c = 3$, prove that $$a^2(b - c)^2 + b^2(c - a)^2 + c^2(a - b)^2 \ge \frac9 2 abc(1 - abc)$$ and state when equality is reached.