A polynomial $P(x)$ of degree $n \ge 5$ with integer coefficients has $n$ distinct integer roots, one of which is $0$. Find all integer roots of the polynomial $P(P(x))$.

Let $x, y, z$ be non-negative numbers, so that $x + y + z = 1$. Prove that $$\sqrt{x+\frac{(y-z)^2}{12}}+\sqrt{y+\frac{(x-z)^2}{12}}+\sqrt{z+\frac{(x-y)^2}{12}}\le \sqrt{3}$$

Find the magnitude of the product of all complex numbers $c$ such that the recurrence defined by $x_1 = 1$, $x_2 = c^2 - 4c + 7$, and $x_{n+1} = (c^2 - 2c)^2 x_n x_{n-1} + 2x_n - x_{n-1}$ also satisfies $x_{1006} = 2011$. [i]Author: Alex Zhu[/i]

Prove the following inequalities: (1) $ x\minus{}\sin x\leq \tan x\minus{}x\ \ \left(0\leq x<\frac{\pi}{2}\right)$ (2) $ \int_0^x \cos (\tan t\minus{}t)\ dt\leq \sin (\sin x)\plus{}\frac 12 \left(x\minus{}\frac{\sin 2x}{2}\right)\ \left(0\leq x\leq \frac{\pi}{3}\right)$

The graph of $y=f(x)$, where $f(x)$ is a polynomial of degree $3$, contains points $A(2,4)$, $B(3,9)$, and $C(4,16)$. Lines $AB$, $AC$, and $BC$ intersect the graph again at points $D$, $E$, and $F$, respectively, and the sum of the $x$-coordinates of $D$, $E$, and $F$ is $24$. What is $f(0)$? $\textbf{(A) } -2 \qquad \textbf{(B) } 0 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } \frac{24}{5} \qquad \textbf{(E) } 8$

Rational numbers $a$, $b$, $c$ satisfy the equation \[(a+b+c)(a+b-c)=c^2\,.\] Show that $a+b=c=0$.

Let $f : R \to R$, and let $P$ be a nonzero polynomial with degree no more than $2015$. For any nonnegative integer $n$, $f^{(n)}(x)$ denotes the function defined as $f$ composed with itself $n$ times. For example, $f^{(0)}(x) = x$, $f^{(1)}(x) = f(x)$, $f^{(2)}(x) = f(f(x))$, etc. Show that there always exists a real number $q$ such that $$f^{((2017^{2017})!)(q)} \ne (q + 2017)(qP(q) - 1).$$

Denote by $\mathbb{N}$ the positive integers. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a function such that, for any $w,x,y,z \in \mathbb{N}$, \[ f(f(f(z)))f(wxf(yf(z)))=z^{2}f(xf(y))f(w). \] Show that $f(n!) \ge n!$ for every positive integer $n$. [i]Pakawut Jiradilok[/i]

Let $\theta_{i}\in(0,\frac{\pi}{2})(i=1,2,\cdots,n)$. Prove: $$(\sum_{i=1}^n\tan\theta_{i})(\sum_{i=1}^n\cot\theta_{i})\geq(\sum_{i=1}^n\sin\theta_{i})^2+(\sum_{i=1}^n\cos\theta_{i})^2.$$

If $a,b,c$ are three real numbers and \[ a + \dfrac{1}{b} = b + \dfrac{1}{c} = c + \dfrac{1}{a} = t \] for some real number $t$, prove that $abc + t = 0 .$

Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all functions $f:\mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that for all $x,y\in \mathbb{R}_{>0}$ we have \[f(x) = f(f(f(x)) + y) + f(xf(y)) f(x+y).\]

Solve in real number the system $x^3=\frac{z}{y}-\frac{2y}{z}, y^3=\frac{x}{z}-\frac{2z}{x}, z^3=\frac{y}{x}-\frac{2x}{y}$

Show that there are an infinity of integer numbers $m,n$, with $gcd(m,n)=1$ such that the equation $(x+m)^{3}=nx$ has 3 different integer sollutions.

[b]6.1[/b] The capacities of cubic vessels are in the ratio 1:8:27 and the volumes of liquid poured into them are 1: 2: 3. After this, from the first to a certain amount of liquid was poured into the second vessel, and then from the second in the third so that in all three vessels the liquid level became the same. After this, 128 4/7 liters were poured from the first vessel into the second, and from the second in the first back so much that the height of the liquid column in the first vessel became twice as large as in the second. It turned out that in the first vessel there were 100 fewer liters than at first. How much liquid was initially in each vessel? [b]6.2[/b] How many times a day do all three hands on a clock coincide, including the second hand? [b]6.3.[/b] Prove that in Leningrad there are two people who have the same number of familiar Leningraders. [b]6.4 / 7.4[/b] Each of the eight given different natural numbers less than $16$. Prove that among their pairwise differences there is at least at least three are the same. [b]6.5 / 7.6[/b] The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

Let $a_1, a_2, \dots, a_{2020}$ and $b_1, b_2, \dots, b_{2020}$ be real numbers(not necessarily distinct). Suppose that the set of positive integers $n$ for which the following equation: $|a_1|x-b_1|+a_2|x-b_2|+\dots+a_{2020}|x-b_{2020}||=n$ (1) has exactly two real solutions, is a finite set. Prove that the set of positive integers $n$ for which the equation (1) has at least one real solution, is also a finite set.

Find all polynomials $p(x)$ such that $p'(x)^2 = c p(x) p''(x)$ for some constant $c$.

Consider a sequence $\{a_n\}_{n\geq 0}$ such that $a_{n+1}=a_n-\lfloor{\sqrt{a_n}}\rfloor\ (n\geq 0),\ a_0\geq 0$. (1) If $a_0=24$, then find the smallest $n$ such that $a_n=0$. (2) If $a_0=m^2\ (m=2,\ 3,\ \cdots)$, then for $j$ with $1\leq j\leq m$, express $a_{2j-1},\ a_{2j}$ in terms of $j,\ m$. (3) Let $m\geq 2$ be integer and for integer $p$ with $1\leq p\leq m-1$, let $a\0=m^2-p$. Find $k$ such that $a_k=(m-p)^2$, then find the smallest $n$ such that $a_n=0$.

Let $m<n$ be two positive integers and $x_m<x_{m+1}<\cdots<x_n$ be a sequence of rational numbers. Suppose that $kx_k$ is an integer for all integers $k$ which $m\leq k\leq n$. Prove that $$x_n-x_m\geq \frac{1}{m}-\frac{1}{n}.$$

Solve the equation [b]a)[/b] $ \log_2^2 +(x-1)\log_2 x =6-2x $ in $ \mathbb{R} . $ [b]b)[/b] $ 2^{x+1}+3^{x+1} +2^{1/x^2}+3^{1/x^2}=18 $ in $ (0,\infty ) . $ [i]Cristinel Mortici[/i]

Let $S = \{-17, -16, ..., 16, 17\}$. We call a subset $T$ of $S$ a good set if $-x \in T$ for all $x \in T$ and if $x, y, z \in T (x, y, z$ may be equal) then $x + y + z \ne 0$. Find the largest number of elements in a good set.

$x$, $y$, and $z$ are real numbers such that $xyz=10$. What is the maximum possible value of $x^3y^3z^3-3x^4-12y^2-12z^4$?

Let $a$ and $b$ be positive numbers. Prove the inequality $$\sqrt[3]{\frac ab}+\sqrt[3]{\frac ba}\le\sqrt[3]{2(a+b)\left(\frac1a+\frac1b\right)}.$$

Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ that satisfy \[ (x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2) \]

Find all functions $f :\mathbb{R}\to\mathbb{R}$, satisfying the condition $f(f(x)+x+y)=2x+f(y)$ for any real $x$ and $y$.

A quadratic polynomial $p(x)$ has positive real coefficients with sum $1$. Show that given any positive real numbers with product $1$, the product of their values under $p$ is at least $1$.