Let $m,n$ be two positive integers with $m \ge n \ge 2022$. Let $a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n$ be $2n$ real numbers. Prove that the numbers of ordered pairs $(i,j) ~(1 \le i,j \le n)$ such that \[ |a_i+b_j-ij| \le m \] does not exceed $3n\sqrt{m \log n}$.

Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation $$ax+by+cz=n.$$ Prove that there exists constants $\alpha, \beta, \gamma \in \mathbb{R}$ such that for any non-negative integer $n$, $$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$

[b]p1.[/b] Let $a(1), a(2), ..., a(n),...$ be an increasing sequence of positive integers satisfying $a(a(n)) = 3n$ for every positive integer $n$. Compute $a(2019)$. [b]p2.[/b] Consider the function $f(12x - 7) = 18x^3 - 5x + 1$. Then, $f(x)$ can be expressed as $f(x) = ax^3 + bx^2 + cx + d$, for some real numbers $a, b, c$ and $d$. Find the value of $(a + c)(b + d)$. [b]p3.[/b] Let $a, b$ be real numbers such that $\sqrt{5 + 2\sqrt6} = \sqrt{a} +\sqrt{b}$. Find the largest value of the quantity $$X = \dfrac{1}{a +\dfrac{1}{b+ \dfrac{1}{a+...}}}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\mathbb{R}^+$ denote the set of positive real numbers. Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all positive real numbers $x$ and $y$ : \[f(x)f(y+f(x))=f(1+xy)\] [i]Proposed by Otgonbayar Uuye. [/i]

For arbitrary non-constant polynomials $f_1(x),\ldots,f_{2018}(x)\in\mathbb Z[x]$, is it always possible to find a polynomial $g(x)\in\mathbb Z[x]$ such that $$f_1(g(x)),\ldots,f_{2018}(g(x))$$are all reducible.

Given a function $f\colon \mathbb{R}\rightarrow \mathbb{R} $ with $f(x)^2\le f(y)$ for all $x,y\in\mathbb{R} $, $x>y$, prove that $f(x)\in [0,1] $ for all $x\in \mathbb{R}$.

[u]Round 1[/u] [b]p1.[/b] In a chemical lab there are three vials: one that can hold $1$ oz of fluid, another that can hold $2$ oz, and a third that can hold $3$ oz. The first is filled with grape juice, the second with sulfuric acid, and the third with water. There are also $3$ empty vials in the cupboard, also of sizes $1$ oz, $2$ oz, and $3$ oz. In order to save the world with grape-flavored acid, James Bond must make three full bottles, one of each size, filled with a mixture of all three liquids so that each bottle has the same ratio of juice to acid to water. How can he do this, considering he was silly enough not to bring any equipment? [b]p2.[/b] Twelve people, some are knights and some are knaves, are sitting around a table. Knaves always lie and knights always tell the truth. At some point they start up a conversation. The first person says, “There are no knights around this table.” The second says, “There is at most one knight at this table.” The third – “There are at most two knights at the table.” And so on until the $12$th says, “There are at most eleven knights at the table.” How many knights are at the table? Justify your answer. [b]p3.[/b] Aquaman has a barrel divided up into six sections, and he has placed a red herring in each. Aquaman can command any fish of his choice to either ‘jump counterclockwise to the next sector’ or ‘jump clockwise to the next sector.’ Using a sequence of exactly $30$ of these commands, can he relocate all the red herrings to one sector? If yes, show how. If no, explain why not. [img]https://cdn.artofproblemsolving.com/attachments/0/f/956f64e346bae82dee5cbd1326b0d1789100f3.png[/img] [b]p4.[/b] Is it possible to place $13$ integers around a circle so that the sum of any $3$ adjacent numbers is exactly $13$? [b]p5.[/b] Two girls are playing a game. The first player writes the letters $A$ or $B$ in a row, left to right, adding one letter on her turn. The second player switches any two letters after each move by the first player (the letters do not have to be adjacent), or does nothing, which also counts as a move. The game is over when each player has made $2011$ moves. Can the second player plan her moves so that the resulting letters form a palindrome? (A palindrome is a sequence that reads the same forward and backwards, e.g. $AABABAA$.) [u]Round 2[/u] [b]p6.[/b] Eight students participated in a math competition. There were eight problems to solve. Each problem was solved by exactly five people. Show that there are two students who solved all eight problems between them. [b]p7.[/b] There are $3n$ checkers of three different colors: $n$ red, $n$ green and $n$ blue. They were used to randomly fill a board with $3$ rows and $n$ columns so that each square of the board has one checker on it. Prove that it is possible to reshuffle the checkers within each row so that in each column there are checkers of all three colors. Moving checkers to a different row is not allowed. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a finite set $A$ of positive integers and its subset $B$, call $B$ a [i]half subset[/i] of $A$ when it satisfies the equation $\sum_{a\in A}a=2\sum_{b\in B}b$. For example, if $A=\{1,2,3\}$, then $\{1,2\}$ and $\{3\}$ are half subset of $A$. Determine all positive integers $n$ such that there exists a finite set $A$ which has exactly $n$ half subsets.

Find all functions $f,g : \mathbb{R} \times \mathbb{R} \mapsto \mathbb{R}^{+}$ such that \[ ( \sum_{j=1}^{n}a_{j}b_{j})^2 \leq (\sum_{j=1}^{n} f({a_{j},b_{j}))(\sum_{j=1}^{n} g({a_{j},b_{j})) \leq (\sum_{j=1}^{n} (a_j)^2 )(\sum_{j=1}^{n} (b_j)^2 ) }}\] for any two sets $a_j$ and $b_j$ of real numbers.

Determine all strictly increasing functions $f : R \to R$ such that for all $x \ne y$ to hold $$\frac{2\left[f(y)-f\left(\frac{x+y}{2}\right) \right]}{f(x)-f(y)}=\frac{f(x)-f(y)}{2\left[f\left(\frac{x+y}{2}\right)-f(x) \right]}$$

Find all the roots $ r_{1}$, $ r_{2}$, $ r_{3}$ y $ r_{4}$ of the equation $ 4x^{4}\minus{}ax^{3}\plus{}bx^{2}\minus{}cx\plus{}5 \equal{} 0$, knowing that they are real, positive and that: \[ \frac{r_{1}}{2}\plus{}\frac{r_{2}}{4}\plus{}\frac{r_{3}}{5}\plus{}\frac{r_{4}}{8}\equal{} 1.\]

Solve the system of equations \[x^2 + y^2 + z^2 + t^2 = 50;\] \[x^2 - y^2 + z^2 - t^2 = -24;\] \[xy = zt;\] \[x - y + z - t = 0.\]

A rectangle has an area of $16$ and a perimeter of $18$; determine the length of the diagonal of the rectangle. [i]2015 CCA Math Bonanza Individual Round #8[/i]

Let $x, y, z$ be nonzero real numbers with $$\frac{x + y}{z}=\frac{y + z}{x}=\frac{z + x}{y}.$$ Determine all possible values of $$\frac{(x + y)(y + z)(z + x)}{xyz}.$$ [i](Walther Janous)[/i]

From an infinite arithmetic progression $ 1,a_1,a_2,\dots$ of real numbers some terms are deleted, obtaining an infinite geometric progression $ 1,b_1,b_2,\dots$ whose ratio is $ q$. Find all the possible values of $ q$.

[b]p1.[/b] Evaluate $1+3+5+··· +2019$. [b]p2.[/b] Evaluate $1^2 -2^2 +3^2 -4^2 +...· +99^2 -100^2$. [b]p3. [/b]Find the sum of all solutions to $|2018+|x -2018|| = 2018$. [b]p4.[/b] The angles in a triangle form a geometric series with common ratio $\frac12$ . Find the smallest angle in the triangle. [b]p5.[/b] Compute the number of ordered pairs $(a,b,c,d)$ of positive integers $1 \le a,b,c,d \le 6$ such that $ab +cd$ is a multiple of seven. [b]p6.[/b] How many ways are there to arrange three birch trees, four maple, and five oak trees in a row if trees of the same species are considered indistinguishable. [b]p7.[/b] How many ways are there for Mr. Paul to climb a flight of 9 stairs, taking steps of either two or three at a time? [b]p8.[/b] Find the largest natural number $x$ for which $x^x$ divides $17!$ [b]p9.[/b] How many positive integers less than or equal to $2018$ have an odd number of factors? [b]p10.[/b] Square $MAIL$ and equilateral triangle $LIT$ share side $IL$ and point $T$ is on the interior of the square. What is the measure of angle $LMT$? [b]p11.[/b] The product of all divisors of $2018^3$ can be written in the form $2^a \cdot 2018^b$ for positive integers $a$ and $b$. Find $a +b$. [b]p12.[/b] Find the sum all four digit palindromes. (A number is said to be palindromic if its digits read the same forwards and backwards. [b]p13.[/b] How ways are there for an ant to travel from point $(0,0)$ to $(5,5)$ in the coordinate plane if it may only move one unit in the positive x or y directions each step, and may not pass through the point $(1, 1)$ or $(4, 4)$? [b]p14.[/b] A certain square has area $6$. A triangle is constructed such that each vertex is a point on the perimeter of the square. What is the maximum possible area of the triangle? [b]p15.[/b] Find the value of ab if positive integers $a,b$ satisfy $9a^2 -12ab +2b^2 +36b = 162$. [b]p16.[/b] $\vartriangle ABC$ is an equilateral triangle with side length $3$. Point $D$ lies on the segment $BC$ such that $BD = 1$ and $E$ lies on $AC$ such that $AE = AD$. Compute the area of $\vartriangle ADE$. [b]p17[/b]. Let $A_1, A_2,..., A_{10}$ be $10$ points evenly spaced out on a line, in that order. Points $B_1$ and $B_2$ lie on opposite sides of the perpendicular bisector of $A_1A_{10}$ and are equidistant to $l$. Lines $B_1A_1,...,B_1A_{10}$ and $B_2A_1,...· ,B_2A_{10}$ are drawn. How many triangles of any size are present? [b]p18.[/b] Let $T_n = 1+2+3··· +n$ be the $n$th triangular number. Determine the value of the infinite sum $\sum_{k\ge 1} \frac{T_k}{2^k}$. [b]p19.[/b] An infinitely large bag of coins is such that for every $0.5 < p \le 1$, there is exactly one coin in the bag with probability $p$ of landing on heads and probability $1- p$ of landing on tails. There are no other coins besides these in the bag. A coin is pulled out of the bag at random and when flipped lands on heads. Find the probability that the coin lands on heads when flipped again. [b]p20.[/b] The sequence $\{x_n\}_{n\ge 1}$ satisfies $x1 = 1$ and $(4+ x_1 + x_2 +··· + x_n)(x_1 + x_2 +··· + x_{n+1}) = 1$ for all $n \ge 1$. Compute $\left \lfloor \frac{x_{2018}}{x_{2019}} \right \rfloor$. PS. You had better use hide for answers.

A infinite sequence $\{ a_n \}_{n \ge 0}$ of real numbers satisfy $a_n \ge n^2$. Suppose that for each $i, j \ge 0$ there exist $k, l$ with $(i,j) \neq (k,l)$, $l - k = j - i$, and $a_l - a_k = a_j - a_i$. Prove that $a_n \ge (n + 2016)^2$ for some $n$.

All russian olympiad 2016,Day 2 ,grade 9,P8 : Let $a, b, c, d$ be are positive numbers such that $a+b+c+d=3$ .Prove that$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}\le\frac{1}{a^2b^2c^2d^2}$$ All russian olympiad 2016,Day 2,grade 11,P7 : Let $a, b, c, d$ be are positive numbers such that $a+b+c+d=3$ .Prove that $$\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\le\frac{1}{a^3b^3c^3d^3}$$ Russia national 2016

The teacher wrote on the blackboard quadratic polynomial $x^2 + 10x + 20$. Then in turn each student in the class either increased or decreased by $1$ either the coefficient of $x$ or the constant term. At the end the quadratic polynomial became $x^2+20x+10$. Is it true that at certain moment a quadratic polynomial with integer roots was on the board?

Determine all pairs of positive real numbers $(a, b)$ for which there exists a function $ f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}} $ satisfying for all positive real numbers $x$ the equation $ f(f(x))=af(x)- bx $

Let $n\geq 3$ be an odd integer. Determine how many real solutions there are to the set of $n$ equations \[\left\{\begin{array}{cc}x_1(x_1+1)=x_2(x_2-1)\\x_2(x_2+1)=x_3(x_3-1)\\ \vdots \\ x_n(x_n+1) = x_1(x_1-1)\end{array}\right.\]

Prove that if the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are all distinct, then the polynomial \[(x-a_{1})^{2}(x-a_{2})^{2}\cdots (x-a_{n})^{2}+1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

Determine the general term of the sequence ($a_n$) given by $a_0 =\alpha > 0$ and $a_{n+1} =\frac{a_n}{1+a_n}$ .

Prove that for all positive integers $n$, all complex roots $r$ of the polynomial \[P(x) = (2n)x^{2n} + (2n-1)x^{2n-1} + \dots + (n+1)x^{n+1} + nx^n + (n+1)x^{n-1} + \dots + (2n-1)x + 2n\] lie on the unit circle (i.e. $|r| = 1$).

For all valid values ​​of $a, b$, and $c$, solve the equation $$\frac{a (x-b) (x-c) }{(a-b) (a-c)} + \frac{b (x-c) (x-a)}{(b-c) (b-a)} +\frac{c (x-a) (x-b) }{(c-a ) (c-b)} = x^2$$