Found problems: 6530
2016 Singapore Junior Math Olympiad, 2
Let $a_1,a_2,...,a_9$ be a sequence of numbers satisfying $0 < p \le a_i \le q$ for each $i = 1,2,..., 9$.
Prove that $\frac{a_1}{a_9}+\frac{a_2}{a_8}+...+\frac{a_9}{a_1} \le 1 + \frac{4(p^2+q^2)}{pq}$
1975 Chisinau City MO, 97
Find the smallest value of the expression $(x-1) (x -2) (x -3) (x - 4) + 10$.
2009 Cuba MO, 3
Determine the smallest value of $x^2 + y^2 + z^2$, where $x, y, z$ are real numbers, so that $x^3 + y^3 + z^3 -3xyz = 1.$
2010 Macedonia National Olympiad, 2
Let $a,b,c$ be positive real numbers for which $a+b+c=3$. Prove the inequality
\[\frac{a^3+2}{b+2}+\frac{b^3+2}{c+2}+\frac{c^3+2}{a+2}\ge3\]
2008 VJIMC, Problem 4
The numbers of the set $\{1,2,\ldots,n\}$ are colored with $6$ colors. Let
$$S:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have the same color}\}$$and
$$D:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have three different colors}\}.$$Prove that
$$|D|\le2|S|+\frac{n^2}2.$$
2000 Junior Balkan Team Selection Tests - Moldova, 4
Find the smallest natural number nonzero n so that it exists in real numbers $x_1, x_2,..., x_n$ which simultaneously check the conditions:
1) $x_i \in [1/2 , 2]$ , $i = 1, 2,... , n$
2) $x_1+x_2+...+x_n \ge \frac{7n}{6}$
3) $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$
2015 IMO Shortlist, N8
For every positive integer $n$ with prime factorization $n = \prod_{i = 1}^{k} p_i^{\alpha_i}$, define
\[\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.\]
That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity.
Find all strictly increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ such that
\[\mho(f(a) - f(b)) \le \mho(a - b) \quad \text{for all integers } a \text{ and } b \text{ with } a > b.\]
[i]Proposed by Rodrigo Sanches Angelo, Brazil[/i]
2004 Estonia National Olympiad, 4
Let $a, b, c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that
$$\frac{1}{1+2ab}+\frac{1}{1+2bc}+\frac{1}{1+2ca}\ge 1$$
2006 Croatia Team Selection Test, 2
Assume that $a, b,$ and $c$ are positive real numbers for which $(a+b)(a+c)(b+c) = 1$. Prove that $ab+bc+ca \leq\frac{3 }{4}.$
2020 Saint Petersburg Mathematical Olympiad, 5.
The altitudes $BB_1$ and $CC_1$ of the acute triangle $\triangle ABC$ intersect at $H$. The circle centered at $O_b$ passes through points $A,C_1$, and the midpoint of $BH$. The circle centered at $O_c$ passes through $A,B_1$ and the midpoint of $CH$. Prove that $B_1 O_b +C_1O_c > \frac{BC}{4}$
2020 Sharygin Geometry Olympiad, 13
Let $I$ be the incenter of triangle $ABC$. The excircle with center $I_A$ touches the side $BC$ at point $A'$. The line $l$ passing through $I$ and perpendicular to $BI$ meets $I_AA'$ at point $K$ lying on the medial line parallel to $BC$. Prove that $\angle B \leq 60^\circ$.
1982 Spain Mathematical Olympiad, 4
Determine a polynomial of non-negative real coefficients that satisfies the following two conditions:
$$p(0) = 0, p(|z|) \le x^4 + y^4,$$
being $|z|$ the module of the complex number $z = x + iy$ .
2004 Junior Balkan MO, 1
Prove that the inequality \[ \frac{ x+y}{x^2-xy+y^2 } \leq \frac{ 2\sqrt 2 }{\sqrt{ x^2 +y^2 } } \] holds for all real numbers $x$ and $y$, not both equal to 0.
2021 Vietnam National Olympiad, 5
Let the polynomial $P(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots +a_1x+a_0$ where $1011\leq a_i\leq 2021$ for all $i=0,1,2,...,21.$ Given that $P(x)$ has an integer root and there exists an positive real number$c$ such that $|a_{k+2}-a_k|\leq c$ for all $k=0,1,...,19.$
a) Prove that $P(x)$ has an only integer root.
b) Prove that $$\sum_{k=0}^{10}(a_{2k+1}-a_{2k})^2\leq 440c^2.$$
1997 Singapore Senior Math Olympiad, 1
Let $x_1,x_2,x_3,x_4, x_5,x_6$ be positive real numbers. Show that
$$\left( \frac{x_2}{x_1} \right)^5+\left( \frac{x_4}{x_2} \right)^5+\left( \frac{x_6}{x_3} \right)^5+\left( \frac{x_1}{x_4} \right)^5+\left( \frac{x_3}{x_5} \right)^5+\left( \frac{x_5}{x_6} \right)^5 \ge \frac{x_1}{x_2}+\frac{x_2}{x_4}+\frac{x_3}{x_6}+\frac{x_4}{x_1}+\frac{x_5}{x_3}+\frac{x_6}{x_5}$$
2014 ISI Entrance Examination, 7
Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$.
\begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge (z-y)\int_{x}^{z}f(u)\,\mathrm{du} \end{align*}
2008 Swedish Mathematical Competition, 3
The function $f(x)$ has the property that $\frac{f(x)}{x}$ is increasing for $x>0$. Show that
\[
f(x)+f(y) \leq f(x+y) \qquad , \qquad \text{for all } x,y>0
\]
2018 India IMO Training Camp, 3
Let $a_n, b_n$ be sequences of positive reals such that,$$a_{n+1}= a_n + \frac{1}{2b_n}$$ $$b_{n+1}= b_n + \frac{1}{2a_n}$$ for all $n\in\mathbb N$.
Prove that, $\text{max}\left(a_{2018}, b_{2018}\right) >44$.
2023 Bangladesh Mathematical Olympiad, P10
Let all possible $2023$-degree real polynomials: $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$,
where $P(0)+P(1)=0$, and the polynomial has 2023 real roots $r_1, r_2,\cdots r_{2023}$ [not necessarily distinct] so that $0\leq r_1,r_2,\cdots r_{2023}\leq1$. What is the maximum value of $r_1 \cdot r_2 \cdots r_{2023}?$
2006 IMO Shortlist, 5
If $a,b,c$ are the sides of a triangle, prove that
\[\frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\frac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\leq 3 \]
[i]Proposed by Hojoo Lee, Korea[/i]
2018 Korea USCM, 6
Suppose a continuous function $f:[0,1]\to\mathbb{R}$ is differentiable on $(0,1)$ and $f(0)=1$, $f(1)=0$. Then, there exists $0<x_0<1$ such that
$$|f'(x_0)| \geq 2018 f(x_0)^{2018}$$
1988 Poland - Second Round, 2
Given real numbers $ x_i $, $ y_i $ ($ i = 1, 2, \ldots, n $) such that $$ \qquad x_1 \geq x_2 \geq \ldots \geq x_n \geq 0, \ \ y_1 > y_2 > \ldots > y_n \geq 0,$$
and $$ \prod_{i=1}^k x_i \geq \prod_{i=1}^k y_i, \ \ \text{ for } \ \ k=1,2,\ldots, n.$$
Prove that
$$
\sum_{i=1}^n x_i > \sum_{i=1}^n y_i.$$
2002 Moldova National Olympiad, 2
Let $ a,b,c\in \mathbb R$ such that $ a\ge b\ge c > 1$. Prove the inequality:
$ \log_c\log_c b \plus{} \log_b\log_b a \plus{} \log_a\log_a c\geq 0$
1949-56 Chisinau City MO, 50
Prove the inequality: $ctg \frac{a}{2}> 1 + ctg a$ for $0 <a <\frac{\pi}{2}$
2011 JBMO Shortlist, 4
$\boxed{\text{A4}}$ Let $x,y$ be positive reals satisfying the condition $x^3+y^3\leq x^2+y^2$.Find the maximum value of $xy$.