Found problems: 6530
2012 District Olympiad, 3
Let $a, b$, and $c$ be positive real numbers. Find the largest integer $n$ such that $$\frac{1}{ax + b + c}
+\frac{1}{a + bx + c}+\frac{1}{a + b + cx} \ge \frac{n}{a + b + c},$$
for all $ x \in [0, 1]$ .
2016 JBMO TST - Turkey, 6
Prove that
\[ (x^4+y)(y^4+z)(z^4+x) \geq (x+y^2)(y+z^2)(z+x^2) \]
for all positive real numbers $x, y, z$ satisfying $xyz \geq 1$.
2005 China Second Round Olympiad, 2
Assume that positive numbers $a, b, c, x, y, z$ satisfy $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function \[ f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}. \]
2014 Baltic Way, 3
Positive real numbers $a, b, c$ satisfy $\frac{1}{a} +\frac{1}{b} +\frac{1}{c} = 3.$ Prove the inequality \[\frac{1}{\sqrt{a^3+ b}}+\frac{1}{\sqrt{b^3 + c}}+\frac{1}{\sqrt{c^3 + a}}\leq \frac{3}{\sqrt{2}}.\]
2009 Iran Team Selection Test, 2
Let $ a$ be a fix natural number . Prove that the set of prime divisors of $ 2^{2^{n}} \plus{} a$ for $ n \equal{} 1,2,\cdots$ is infinite
2010 Kyrgyzstan National Olympiad, 1
Given that $a,b,c > 0$ and $a + b + c = 1$. Prove that $\sqrt {\frac{{ab}}{{ab + c}}} + \sqrt {\frac{{bc}}{{bc + a}}} + \sqrt {\frac{{ca}}{{ca + b}}} \leqslant \frac{3}{2}$.
1981 Putnam, B2
Determine the minimum value of
$$(r-1)^2 + \left(\frac{s}{r}-1 \right)^2 + \left(\frac{t}{s}-1 \right)^{2} + \left( \frac{4}{t} -1 \right)^2$$
for all real numbers $1\leq r \leq s \leq t \leq 4.$
1991 Cono Sur Olympiad, 3
Given a positive integrer number $n$ ($n\ne 0$), let $f(n)$ be the average of all the positive divisors of $n$. For example, $f(3)=\frac{1+3}{2}=2$, and $f(12)=\frac{1+2+3+4+6+12}{6}=\frac{14}{3}$.
[b]a[/b] Prove that $\frac{n+1}{2} \ge f(n)\ge \sqrt{n}$.
[b]b[/b] Find all $n$ such that $f(n)=\frac{91}{9}$.
OMMC POTM, 2024 9
For all positive reals $x,y$ and $z$, prove that $$x^x+y^y+z^z \ge x^y+y^z+z^x.$$
2011 Iran MO (3rd Round), 4
For positive real numbers $a,b$ and $c$ we have $a+b+c=3$. Prove
$\frac{a}{1+(b+c)^2}+\frac{b}{1+(a+c)^2}+\frac{c}{1+(a+b)^2}\le \frac{3(a^2+b^2+c^2)}{a^2+b^2+c^2+12abc}$.
[i]proposed by Mohammad Ahmadi[/i]
2003 China Team Selection Test, 3
Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define
\[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.
2018 SIMO, Q2
Let $x_1, x_2, x_3, y_1, y_2, y_3$ be real numbers in $[-1, 1]$. Find the maximum value of
\[(x_1y_2-x_2y_1)(x_2y_3-x_3y_2)(x_3y_1-x_1y_3).\]
2011 Saudi Arabia BMO TST, 3
Let $a, b, c$ be positive real numbers. Prove that $$\frac{1}{a+b+\frac{1}{abc}+1}+\frac{1}{b+c+\frac{1}{abc}+1}+\frac{1}{c+a+\frac{1}{abc}+1}\le \frac{a + b + c}{a+b+c+1}$$
2012 India IMO Training Camp, 2
Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that
\[\text{max}_{|z|=1}\{|P(z)|\}\ge\sqrt{2|a_ma_n|+\sum_{k=m}^{n} |a_k|^2}\]
2013 USAMTS Problems, 4
Bunbury the bunny is hopping on the positive integers. First, he is told a positive integer $n$. Then Bunbury chooses positive integers $a,d$ and hops on all of the spaces $a,a+d,a+2d,\dots,a+2013d$. However, Bunbury must make these choices so that the number of every space that he hops on is less than $n$ and relatively prime to $n$.
A positive integer $n$ is called [i]bunny-unfriendly[/i] if, when given that $n$, Bunbury is unable to find positive integers $a,d$ that allow him to perform the hops he wants. Find the maximum bunny-unfriendly integer, or prove that no such maximum exists.
2022 Thailand TSTST, 2
Let $a,b,c>0$ satisfy $a\geq b\geq c$. Prove that
$$\frac{4}{a^2(b+c)}+\frac{4}{b^2(c+a)}+\frac{4}{c^2(a+b)} \leq \left(\sum_{cyc} \frac{a^2+1} {b^2} \right)\left(\sum_{cyc} \frac{b^3}{a^2(a^3+2b^3)}\right).$$
1985 Canada National Olympiad, 3
Let $P_1$ and $P_2$ be regular polygons of 1985 sides and perimeters $x$ and $y$ respectively. Each side of $P_1$ is tangent to a given circle of circumference $c$ and this circle passes through each vertex of $P_2$. Prove $x + y \ge 2c$. (You may assume that $\tan \theta \ge \theta$ for $0 \le \theta < \frac{\pi}{2}$.)
1953 Moscow Mathematical Olympiad, 257
Let $x_0 = 10^9$, $x_n = \frac{x^2_{n-1}+2}{2x_{n-1}}$ for $n > 0$. Prove that $0 < x_{36} - \sqrt2 < 10^{-9}$.
2014 ELMO Shortlist, 9
Let $a$, $b$, $c$ be positive reals. Prove that \[ \sqrt{\frac{a^2(bc+a^2)}{b^2+c^2}}+\sqrt{\frac{b^2(ca+b^2)}{c^2+a^2}}+\sqrt{\frac{c^2(ab+c^2)}{a^2+b^2}}\ge a+b+c. \][i]Proposed by Robin Park[/i]
2011 All-Russian Olympiad Regional Round, 9.4
$x$, $y$ and $z$ are positive real numbers. Prove the inequality
\[\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x}{y}+\frac{y}{z}+\frac{z}{x}.\]
(Authors: A. Khrabrov, B. Trushin)
I Soros Olympiad 1994-95 (Rus + Ukr), 9.8
Let $f(x) =x^2-2x$. Find all $x$ for which $f(f(x))<3$.
2021 Cyprus JBMO TST, 1
Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2=3$. Prove that
\[ xyz(x+y+z)+2021\geqslant 2024xyz\]
2017 Turkey MO (2nd round), 3
Denote the sequence $a_{i,j}$ in positive reals such that $a_{i,j}$.$a_{j,i}=1$. Let $c_i=\sum_{k=1}^{n}a_{k,i}$. Prove that $1\ge$$\sum_{i=1}^{n}\dfrac {1}{c_i}$
2016 Philippine MO, 3
Let \(n\) be any positive integer. Prove that \[\sum^n_{i=1} \frac{1}{(i^2+i)^{3/4}} > 2-\frac{2}{\sqrt{n+1}}\].
1991 India Regional Mathematical Olympiad, 2
If $a,b,c,d$ be any four positive real numbers, then prove that \[ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \geq 4. \]