Prove there exist two relatively prime polynomials $P(x),Q(x)$ having integer coefficients and a real number $u>0$ such that if for positive integers $a,b,c,d$ we have: $$|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}$$ $$| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}$$ Then we have : $$bP(\frac{a}{c})=dQ(\frac{a}{c})$$ (Two polynomials are relatively prime if they don't have a common root) Proposed by [i]Navid Safaii[/i] and [i]Alireza Haghi[/i]

Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}^{+}$ such that for all positive real numbers $x$ and $y$, the following equation holds: \[(x+y)f(f(x)y)=x^2f(f(x)+f(y)).\]

Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$

The real numbers $a, b, c, d$ are such that $a\geq b\geq c\geq d>0$ and $a+b+c+d=1$. Prove that \[(a+2b+3c+4d)a^ab^bc^cd^d<1\] [i]Proposed by Stijn Cambie, Belgium[/i]

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x, y, z,$ \[f(x+y^2+z)=f(f(x))+yf(y)+f(z). \]

Let $ n, k$ be positive integers and suppose that the polynomial $ x^{2k}\minus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$. Prove that $ x^{2k}\plus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$.

Let $a, b, c, d, e$ be real numbers such that $a<b<c<d<e.$ The least possible value of the function $f: \mathbb R \to \mathbb R$ with $f(x) = |x-a| + |x - b|+ |x - c| + |x - d|+ |x - e|$ is $$ \mathrm a. ~ e+d+c+b+a\qquad \mathrm b.~e+d+c-b-a\qquad \mathrm c. ~e+d+|c|-b-a \qquad \mathrm d. ~e+d+b-a \qquad \mathrm e. ~e+d-b-a $$

Let $a_1, a_2, . . . , a_{2019}$ be any positive real numbers such that $\frac{1}{a_1 + 2019}+\frac{1}{a_2 + 2019}+ ... +\frac{1}{a_{2019} + 2019}=\frac{1}{2019}$. Find the minimum value of $a_1a_2... a_{2019}$ and determine for which values of $a_1, a_2, . . . , a_{2019}$ this minimum occurs

Find all pairs $(m,n)$ of natural numbers with $m<n$ such that $m^2+1$ is a multiple of $n$ and $n^2+1$ is a multiple of $m$.

Find minimal value of $a \in \mathbb{R}$ such that system $$\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}=a-1$$ $$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}=a+1$$ has solution in set of real numbers

Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$

Given are $a_0,a_1, ... , a_n$, satisfying $a_0=a_n = 0$, and $a_{k-1} - 2a_k+a_{k+1}\ge 0$ for $k=0, 1, ... , n-1$. Prove that all the numbers are negative or zero.

Solve equation $$x^2-\sqrt{a-x}=a$$ where $x$ is real number and $a$ is real parameter

Simultaneously from the same point of a circular route and in the same direction for two hours two bodies move evenly. The first body performs a complete rotation three minutes faster than the second body and exceeds it every $9$ minutes and $20$ seconds. Whenever the first body will overtake the other the second exactly at the starting point?

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

Find all pairs $(a,b,c)$ of positive integers such that $$\frac{a}{2^a}=\frac{b}{2^b}+\frac{c}{2^c}$$

Positive numbers $a_1,a_2,...,a_n, b_1, b_2,...,b_n$ satisfy the condition $a_1+a_2+...+a_n=b_1+ b_2+...+b_n=1$. Find the smallest possible value of the sum $$\frac{a_1^2}{a_1+b_1}+\frac{a_2^2}{a_2+b_2}+...+\frac{a_n^2}{a_n+b_n}$$ (V.Kolbun)

Let $a_1, a_2,...,a_{2n}$ be positive real numbers such that $a_i + a_{n+i} = 1$, for all $i = 1,...,n$. Prove that there exist two different integers $1 \le j, k \le 2n$ for which $$\sqrt{a^2_j-a^2_k} < \frac{1}{\sqrt{n} +\sqrt{n - 1}}$$

[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 5 The sequence $\{a_{n}\}$ is defined by $a_{1}=0$ and $(n+1)^{3}a_{n+1}=2n^{2}(2n+1)a_{n}+2(3n+1)$ for all integers $\geq 1$. Show that infintely many members of the sequence are positive integers.

Let $D$ be the set of real numbers excluding $-1$. Find all functions $f: D \to D$ such that for all $x,y \in D$ satisfying $x \neq 0$ and $y \neq -x$, the equality $$(f(f(x))+y)f \left(\frac{y}{x} \right)+f(f(y))=x$$ holds.

Let $p(x)=x^4-5773x^3-46464x^2-5773x+46$. Determine the sum of $\arctan$-s of its real roots.

Consider all finite sequences of positive real numbers each of whose terms is at most $3$ and the sum of whose terms is more than $100$. For each such sequence, let $S$ denote the sum of the subsequence whose sum is the closest to $100$, and define the [i]defect[/i] of this sequence to be the value $|S-100|$. Find the maximum possible value of the defect.

[b]p1.[/b] There are $16$ students in a class. Each month the teacher divides the class into two groups. What is the minimum number of months that must pass for any two students to be in different groups in at least one of the months? [b]p2.[/b] Find all functions $f(x)$ defined for all real $x$ that satisfy the equation $2f(x) + f(1 - x) = x^2$. [b]p3.[/b] Arrange the digits from $1$ to $9$ in a row (each digit only once) so that every two consecutive digits form a two-digit number that is divisible by $7$ or $13$. [b]p4.[/b] Prove that $\cos 1^o$ is irrational. [b]p5.[/b] Consider $2n$ distinct positive Integers $a_1,a_2,...,a_{2n}$ not exceeding $n^2$ ($n>2$). Prove that some three of the differences $a_i- a_j$ are equal . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ n \geq 2$ be a natural number and $ p(x)$ a real polynomial of degree at most $ n$ for which \[ \max _{ \minus{}1 \leq x \leq 1} |p(x)| \leq 1, \; p(\minus{}1)\equal{}p(1)\equal{}0 \ .\] Prove that then \[ |p'(x)| \leq \frac{n \cos \frac{\pi}{2n}}{\sqrt{1\minus{}x^2 \cos^2 \frac{\pi}{2n}}} \;\;\;\;\; \left( \minus{}\frac{1}{\cos \frac{\pi}{2n}} < x < \frac{1}{\cos \frac{\pi}{2n}} \\\\\ \right).\] [i]J. Szabados[/i]

Let $\mathbb{Q}^+$ be the set of all positive rational numbers. Find all functions $f:\mathbb{Q}^+\rightarrow \mathbb{Q}^+$ satisfying $f(1)=1$ and \[ f(x+n)=f(x)+nf(\frac{1}{x}) \forall n\in\mathbb{N},x\in\mathbb{Q}^+\]