Found problems: 6530
2011 Mediterranean Mathematics Olympiad, 2
Let $A$ be a finite set of positive reals, let $B = \{x/y\mid x,y\in A\}$ and let $C = \{xy\mid x,y\in A\}$.
Show that $|A|\cdot|B|\le|C|^2$.
[i](Proposed by Gerhard Woeginger, Austria)[/i]
KoMaL A Problems 2017/2018, A. 721
Let $n\ge 2$ be a positive integer, and suppose $a_1,a_2,\cdots ,a_n$ are positive real numbers whose sum is $1$ and whose squares add up to $S$. Prove that if $b_i=\tfrac{a^2_i}{S} \;(i=1,\cdots ,n)$, then for every $r>0$, we have $$\sum_{i=1}^n \frac{a_i}{{(1-a_i)}^r}\le \sum_{i=1}^n
\frac{b_i}{{(1-b_i)}^r}.$$
1992 India Regional Mathematical Olympiad, 6
Prove that \[ 1 < \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001} < 1 \frac{1}{3}. \]
1970 Czech and Slovak Olympiad III A, 3
Let $p>0$ be a given parameter. Determine all real $x$ such that \[\frac{1}{\,x+\sqrt{p-x^2\,}\,}+\frac{1}{\,x-\sqrt{p-x^2\,}\,}\ge\frac{1}{\,p\,}.\]
2024 South Africa National Olympiad, 2
Determine which of the following is larger:
\[ \sqrt{2+\sqrt[3]{5}}\qquad \text{or}\qquad \sqrt[3]{5+\sqrt{2}}.\]
Fully explain your reasoning.
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]
1997 Iran MO (2nd round), 1
Let $x_1,x_2,x_3,x_4$ be positive reals such that $x_1x_2x_3x_4=1$. Prove that:
\[ \sum_{i=1}^{4}{x_i^3}\geq\max\{ \sum_{i=1}^{4}{x_i},\sum_{i=1}^{4}{\frac{1}{x_i}} \}. \]
1966 IMO Longlists, 2
Given $n$ positive numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ such that $a_{1}\cdot a_{2}\cdot ...\cdot a_{n}=1.$ Prove \[ \left( 1+a_{1}\right) \left( 1+a_{2}\right) ...\left(1+a_{n}\right) \geq 2^{n}.\]
2011 Saudi Arabia Pre-TST, 2.2
Prove that for any positive real numbers $a, b, c$, $$2(a^3 + b^3 + c^3 + abc) \ge (a+b)(b + c)(c + a)$$.
1993 AMC 12/AHSME, 26
Find the largest positive value attained by the function
\[ f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48}, \qquad x\ \text{a real number} \]
$ \textbf{(A)}\ \sqrt{7}-1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2\sqrt{3} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{55}-\sqrt{5} $
2006 MOP Homework, 2
Prove that $\frac{a}{(a + 1)(b + 1)} +\frac{ b}{(b + 1)(c + 1)} + \frac{c}{(c + 1)(a + 1)} \ge \frac34$ where $a, b$ and $c$ are positive real numbers satisfying $abc = 1$.
2018 Taiwan TST Round 1, 1
Let $ a,b,c,d $ be four non-negative reals such that $ a+b+c+d = 4 $. Prove that $$ a\sqrt{3a+b+c}+b\sqrt{3b+c+d}+c\sqrt{3c+d+a}+d\sqrt{3d+a+b} \ge 4\sqrt{5} $$
2022 Kyiv City MO Round 2, Problem 1
Positive reals $x, y, z$ satisfy $$\frac{xy+1}{x+1} = \frac{yz+1}{y+1} = \frac{zx+1}{z+1}$$
Do they all have to be equal?
[i](Proposed by Oleksii Masalitin)[/i]
2011 India IMO Training Camp, 2
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
2013 Today's Calculation Of Integral, 868
In the coordinate space, define a square $S$, defined by the inequality $|x|\leq 1,\ |y|\leq 1$ on the $xy$-plane, with four vertices $A(-1,\ 1,\ 0),\ B(1,\ 1,\ 0),\ C(1,-1,\ 0), D(-1,-1,\ 0)$. Let $V_1$ be the solid by a rotation of the square $S$ about the line $BD$ as the axis of rotation, and let $V_2$ be the solid by a rotation of the square $S$ about the line $AC$ as the axis of rotation.
(1) For a real number $t$ such that $0\leq t<1$, find the area of cross section of $V_1$ cut by the plane $x=t$.
(2) Find the volume of the common part of $V_1$ and $V_2$.
2009 IberoAmerican Olympiad For University Students, 3
Let $a, b, c, d, e \in \mathbb{R}^+$ and $f:\{(x, y) \in (\mathbb{R}^+)^2|c-dx-ey > 0\}\to \mathbb{R}^+$ be given by $f(x, y) = (ax)(by)(c- dx- ey)$.
Find the maximum value of $f$.
2022 Auckland Mathematical Olympiad, 8
Find the least value of the expression $(x+y)(y+z)$, under the conditionthat $x,y,z$ are positive numbers satisfying the equation $xyz(x + y + z) = 1$.
2014 Taiwan TST Round 1, 2
A triangle has side lengths $a$, $b$, $c$, and the altitudes have lengths $h_a$, $h_b$, $h_c$. Prove that
\[ \left( \frac{a}{h_a} \right)^2
+ \left( \frac{b}{h_b} \right)^2
+ \left( \frac{c}{h_c} \right)^2 \ge 4. \]
2008 China Team Selection Test, 1
Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.
1981 Czech and Slovak Olympiad III A, 5
Let $n$ be a positive integer. Determine the maximum of the sum $x_1+\cdots+x_n$ where $x_1,\ldots,x_n$ are non-negative integers satisfying the condition \[x_1^3+\cdots+x_n^3\le7n.\]
1969 Swedish Mathematical Competition, 4
Define $g(x)$ as the largest value of$ |y^2 - xy|$ for $y$ in $[0, 1]$. Find the minimum value of $g$ (for real $x$).
2023 CIIM, 6
Let $n$ be a positive integer. We define $f(n)$ as the number of finite sequences $(a_1, a_2, \ldots , a_k)$ of positive integers such that $a_1 < a_2 < a_3 < \cdots < a_k$ and $$a_1+a_2^2+a_3^3+\cdots + a_k^k \leq n.$$ Determine the positive constants $\alpha$ and $C$ such that $$\lim\limits_{n\rightarrow \infty} \frac{f(n)}{n^\alpha}=C.$$
1992 Canada National Olympiad, 2
For $ x,y,z \geq 0,$ establish the inequality
\[ x(x\minus{}z)^2 \plus{} y(y\minus{}z)^2 \geq (x\minus{}z)(y\minus{}z)(x\plus{}y\minus{}z)\]
and determine when equality holds.
2012 China Second Round Olympiad, 4
Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that for any real numbers $a,b,0\le a\le b\le 1$, there exist infinite many $n\in\mathbb{N}$ such that
\[a<S_n-[S_n]<b\]
where $[x]$ represents the largest integer not exceeding $x$.
2023 China Northern MO, 2
Let $ a,b,c \in (0,1) $ and $ab+bc+ca=4abc .$ Prove that $$\sqrt{a+b+c}\geq \sqrt{1-a}+\sqrt{1-b}+\sqrt{1-c}$$