Found problems: 6530
1951 Polish MO Finals, 3
Prove that if $ a > 0 $, $ b > 0 $, $ c > 0 $, then the inequality holds
$$ ab (a + b) + bc (b + c) + ca (c + a) \geq 6abc.$$
2018 Taiwan TST Round 2, 1
Given positive integers $a_1,a_2,\ldots, a_n$ with $a_1<a_2<\cdots<a_n)$, and a positive real $k$ with $k\geq 1$. Prove that
\[\sum_{i=1}^{n}a_i^{2k+1}\geq \left(\sum_{i=1}^{n}a_i^k\right)^2.\]
1997 Federal Competition For Advanced Students, Part 2, 1
Determine all quadruples $(a, b, c, d)$ of real numbers satisfying the equation
\[256a^3b^3c^3d^3 = (a^6+b^2+c^2+d^2)(a^2+b^6+c^2+d^2)(a^2+b^2+c^6+d^2)(a^2+b^2+c^2+d^6).\]
2021 Indonesia MO, 5
Let $P(x) = x^2 + rx + s$ be a polynomial with real coefficients. Suppose $P(x)$ has two distinct real roots, both of which are less than $-1$ and the difference between the two is less than $2$. Prove that $P(P(x)) > 0$ for all real $x$.
2007 Romania National Olympiad, 4
Let $ m,n$ be two natural numbers with $ m > 1$ and $ 2^{2m \plus{} 1} \minus{} n^2\geq 0$. Prove that:
\[ 2^{2m \plus{} 1} \minus{} n^2\geq 7 .\]
2012 AIME Problems, 6
Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$.
Find $c+d$.
2000 VJIMC, Problem 3
Prove that if m,n are nonnegative integers and 0<=x<=1 then
$(1-x^n)^m + (1-(1-x)^m)^n \ge 1$
2012 Balkan MO, 4
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold:
(i) $f(n!)=f(n)!$ for every positive integer $n$,
(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.
2019 Benelux, 1
[list=a]
[*]Let $a,b,c,d$ be real numbers with $0\leqslant a,b,c,d\leqslant 1$. Prove that
$$ab(a-b)+bc(b-c)+cd(c-d)+da(d-a)\leqslant \frac{8}{27}.$$[/*]
[*]Find all quadruples $(a,b,c,d)$ of real numbers with $0\leqslant a,b,c,d\leqslant 1$ for which equality holds in the above inequality.
[/list]
2012 China National Olympiad, 1
Let $f(x)=(x + a)(x + b)$ where $a,b>0$. For any reals $x_1,x_2,\ldots ,x_n\geqslant 0$ satisfying $x_1+x_2+\ldots +x_n =1$, find the maximum of $F=\sum\limits_{1 \leqslant i < j \leqslant n} {\min \left\{ {f({x_i}),f({x_j})} \right\}} $.
1998 USAMTS Problems, 2
Determine the smallest rational number $\frac{r}{s}$ such that $\frac{1}{k}+\frac{1}{m}+\frac{1}{n}\leq \frac{r}{s}$ whenever $k, m,$ and $n$ are positive integers that satisfy the inequality $\frac{1}{k}+\frac{1}{m}+\frac{1}{n} < 1$.
2016 Korea USCM, 4
Suppose a continuous function $f:[-\frac{\pi}{4},\frac{\pi}{4}]\to[-1,1]$ and differentiable on $(-\frac{\pi}{4},\frac{\pi}{4})$. Then, there exists a point $x_0\in (-\frac{\pi}{4},\frac{\pi}{4})$ such that
$$|f'(x_0)|\leq 1+f(x_0)^2$$
2011 Serbia National Math Olympiad, 3
Set $T$ consists of $66$ points in plane, and $P$ consists of $16$ lines in plane. Pair $(A,l)$ is [i]good[/i] if $A \in T$, $l \in P$ and $A \in l$. Prove that maximum number of good pairs is no greater than $159$, and prove that there exits configuration with exactly $159$ good pairs.
2013 South East Mathematical Olympiad, 1
Let $a,b$ be real numbers such that the equation $x^3-ax^2+bx-a=0$ has three positive real roots . Find the minimum of $\frac{2a^3-3ab+3a}{b+1}$.
2010 Contests, 1
Let $a_1,a_2,\cdots, a_n$ and $b_1,b_2,\cdots, b_n$ be two permutations of the numbers $1,2,\cdots, n$. Show that
\[\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2\]
2012 Singapore Senior Math Olympiad, 5
For $a,b,c,d \geq 0$ with $a + b = c + d = 2$, prove
\[(a^2 + c^2)(a^2 + d^2)(b^2 + c^2)(b^2 + d^2) \leq 25\]
2008 Ukraine Team Selection Test, 3
For positive $ a, b, c, d$ prove that
$ (a \plus{} b)(b \plus{} c)(c \plus{} d)(d \plus{} a)(1 \plus{} \sqrt [4]{abcd})^{4}\geq16abcd(1 \plus{} a)(1 \plus{} b)(1 \plus{} c)(1 \plus{} d)$
1973 IMO, 1
Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1.
1967 IMO Longlists, 37
Prove that for arbitrary positive numbers the following inequality holds
\[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]
2022 Korea National Olympiad, 7
Suppose that the sequence $\{a_n\}$ of positive reals satisfies the following conditions:
[list]
[*]$a_i \leq a_j$ for every positive integers $i <j$.
[*]For any positive integer $k \geq 3$, the following inequality holds:
$$(a_1+a_2)(a_2+a_3)\cdots(a_{k-1}+a_k)(a_k+a_1)\leq (2^k+2022)a_1a_2\cdots a_k$$
[/list]
Prove that $\{a_n\}$ is constant.
2016 Iran Team Selection Test, 4
Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.
2022 Taiwan TST Round 1, A
Let $a_1, a_2, a_3, \ldots$ be a sequence of reals such that there exists $N\in\mathbb{N}$ so that $a_n=1$ for all $n\geq N$, and for all $n\geq 2$ we have
\[a_{n}\leq a_{n-1}+2^{-n}a_{2n}.\]
Show that $a_k>1-2^{-k}$ for all $k\in\mathbb{N}$.
[i]
Proposed by usjl[/i]
2007 Greece National Olympiad, 2
Let $a,b,c$ be sides of a triangle, show that
\[\frac{(c+a-b)^{4}}{a(a+b-c)}+\frac{(a+b-c)^{4}}{b(b+c-a)}+\frac{(b+c-a)^{4}}{c(c+a-b)}\geq ab+bc+ca.\]
1998 Iran MO (3rd Round), 2
Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE$ and $EF = FA$. Prove that
\[\frac{AB}{BE}+\frac{CD}{AD}+\frac{EF}{CF} \geq \frac{3}{2}.\]
1995 IMO Shortlist, 6
Let $ n$ be an integer,$ n \geq 3.$ Let $ x_1, x_2, \ldots, x_n$ be real numbers such that $ x_i < x_{i\plus{}1}$ for $ 1 \leq i \leq n \minus{} 1$. Prove that
\[ \frac{n(n\minus{}1)}{2} \sum_{i < j} x_ix_j > \left(\sum^{n\minus{}1}_{i\equal{}1} (n\minus{}i)\cdot x_i \right) \cdot \left(\sum^{n}_{j\equal{}2} (j\minus{}1) \cdot x_j \right)\]