Found problems: 6530
2008 Sharygin Geometry Olympiad, 3
(R.Pirkuliev) Prove the inequality
\[ \frac1{\sqrt {2\sin A}} \plus{} \frac1{\sqrt {2\sin B}} \plus{} \frac1{\sqrt {2\sin C}}\leq\sqrt {\frac {p}{r}},
\]
where $ p$ and $ r$ are the semiperimeter and the inradius of triangle $ ABC$.
2017 AMC 12/AHSME, 2
Real numbers $x$, $y$, and $z$ satisfy the inequalities
$$0<x<1,\qquad-1<y<0,\qquad\text{and}\qquad1<z<2.$$
Which of the following numbers is nessecarily positive?
$\textbf{(A) } y+x^2 \qquad \textbf{(B) } y+xz \qquad \textbf{(C) }y+y^2 \qquad \textbf{(D) }y+2y^2 \qquad\\
\textbf{(E) } y+z$
1979 IMO Longlists, 59
Determine the maximum value of $x^2 y^2 z^2 w$ for $\{x,y,z,w\}\in\mathbb{R}^{+} \cup\{0\}$ and $2x+xy+z+yzw=1$.
1993 Poland - First Round, 10
Given positive real numbers $p,q$ with $p+q=1$. Prove that for all positive integers $m,n$ the following inequality holds
$(1-p^m)^n+(1-q^n)^m \geq 1$.
2010 Silk Road, 3
For positive real numbers $a, b, c, d,$ satisfying the following conditions:
$a(c^2 - 1)=b(b^2+c^2)$ and $d \leq 1$, prove that : $d(a \sqrt{1-d^2} + b^2 \sqrt{1+d^2}) \leq \frac{(a+b)c}{2}$
1977 Swedish Mathematical Competition, 6
Show that there are positive reals $a$, $b$, $c$ such that
\[\left\{ \begin{array}{l}
a^2 + b^2 + c^2 > 2 \\
a^3 + b^3 + c^3 <2 \\
a^4 + b^4 + c^4 > 2 \\
\end{array} \right.
\]
1999 Croatia National Olympiad, Problem 2
How do I prove that, for every $a, b, c$ positive real numbers such that $a+b+c = 1$ the following inequality holds: $\frac{a^3}{a^2+b^2} +\frac{b^3}{b^2+c^2} +\frac {c^3}{c^2+a^2} \geq \frac{1}{2}$?
2016 Stars of Mathematics, 3
Let $ n $ be a natural number, and $ 2n $ nonnegative real numbers $ a_1,a_2,\ldots ,a_{2n} $ such that $ a_1a_2\cdots a_{2n}=1. $ Show that
$$ 2^{n+1} +\left( a_1^2+a_2^2 \right)\left( a_3^2+a_4^2 \right)\cdots\left( a_{2n-1}^2+a_{2n}^2 \right) \ge 3\left( a_1+a_2 \right)\left( a_3+a_4 \right)\cdots\left( a_{2n-1}+a_{2n} \right) , $$
and specify in which circumstances equality happens.
1992 China Team Selection Test, 3
For any $n,T \geq 2, n, T \in \mathbb{N}$, find all $a \in \mathbb{N}$ such that $\forall a_i > 0, i = 1, 2, \ldots, n$, we have
\[\sum^n_{k=1} \frac{a \cdot k + \frac{a^2}{4}}{S_k} < T^2 \cdot \sum^n_{k=1} \frac{1}{a_k},\] where $S_k = \sum^k_{i=1} a_i.$
1993 All-Russian Olympiad Regional Round, 9.1
If $a$ and $b$ are positive numbers, prove the inequality
$$a^2 +ab+b^2\ge 3(a+b-1).$$
1997 Akdeniz University MO, 2
Let $x,y,z,t$ be real numbers such that, $1 \leq x \leq y \leq z \leq t \leq 100$. Find minimum value of
$$\frac{x}{y}+\frac{z}{t}$$
2017 Singapore MO Open, 4
Let $n > 3$ be an integer. Prove that there exist positive integers $x_1,..., x_n$ in geometric progression and positive integers $y_1,..., y_n$ in arithmetic progression such that $x_1<y_1<x_2<y_2<...<x_n<y_n$
2003 China Team Selection Test, 3
Let $a_{1},a_{2},...,a_{n}$ be positive real number $(n \geq 2)$,not all equal,such that $\sum_{k=1}^n a_{k}^{-2n}=1$,prove that:
$\sum_{k=1}^n a_{k}^{2n}-n^2.\sum_{1 \leq i<j \leq n}(\frac{a_{i}}{a_{j}}-\frac{a_{j}}{a_{i}})^2 >n^2$
2006 Romania Team Selection Test, 4
Let $p$, $q$ be two integers, $q\geq p\geq 0$. Let $n \geq 2$ be an integer and $a_0=0, a_1 \geq 0, a_2, \ldots, a_{n-1},a_n = 1$ be real numbers such that \[ a_{k} \leq \frac{ a_{k-1} + a_{k+1} } 2 , \ \forall \ k=1,2,\ldots, n-1 . \] Prove that \[ (p+1) \sum_{k=1}^{n-1} a_k^p \geq (q+1) \sum_{k=1}^{n-1} a_k^q . \]
2012 Silk Road, 4
Prove that for any positive integer $n$, the arithmetic mean of $\sqrt[1]{1},\sqrt[2]{2},\sqrt[3]{3},\ldots ,\sqrt[n]{n}$ lies in $\left[ 1,1+\frac{2\sqrt{2}}{\sqrt{n}} \right]$ .
2012 ELMO Shortlist, 4
A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles.
[i]Calvin Deng.[/i]
2004 Purple Comet Problems, 13
A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer two layers of unit cubes are removed from the block, more than half the original unit cubes will still remain?
1979 Yugoslav Team Selection Test, Problem 1
Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$
2014 JBMO Shortlist, 8
Let $\displaystyle {x, y, z}$ be positive real numbers such that $\displaystyle {xyz = 1}$. Prove the inequality:$$\displaystyle{\dfrac{1}{x\left(ay+b\right)}+\dfrac{1}{y\left(az+b\right)}+\dfrac{1}{z\left(ax+b\right)}\geq 3}$$
if:
(A) $\displaystyle {a = 0, b = 1}$
(B) $\displaystyle {a = 1, b = 0}$
(C) $\displaystyle {a + b = 1, \; a, b> 0}$
When the equality holds?
2008 Switzerland - Final Round, 2
Determine all functions $f : R^+ \to R^+$, so that for all $x, y > 0$:
$$f(xy) \le \frac{xf(y) + yf(x)}{2}$$
2008 Singapore MO Open, 4
let $0<a,b<\pi/2$. Show that
$\frac{5}{cos^2(a)}+\frac{5}{sin^2(a)sin^2(b)cos^2(b)} \geq 27cos(a)+36sin(a) $
2000 Brazil Team Selection Test, Problem 4
[b]Problem:[/b]For a positive integer $ n$,let $ V(n; b)$ be the number of decompositions of $ n$ into a
product of one or more positive integers greater than $ b$. For example,$ 36 \equal{} 6.6 \equal{}4.9 \equal{} 3.12 \equal{} 3 .3. 4$, so that $ V(36; 2) \equal{} 5$.Prove that for all positive integers $ n$; b it holds that $ V(n;b)<\frac{n}{b}$. :)
2010 China Team Selection Test, 3
Given integer $n\geq 2$ and real numbers $x_1,x_2,\cdots, x_n$ in the interval $[0,1]$. Prove that there exist real numbers $a_0,a_1,\cdots,a_n$ satisfying the following conditions:
(1) $a_0+a_n=0$;
(2) $|a_i|\leq 1$, for $i=0,1,\cdots,n$;
(3) $|a_i-a_{i-1}|=x_i$, for $i=1,2,\cdots,n$.
2010 Romania Team Selection Test, 1
Let $n$ be a positive integer and let $x_1, x_2, \ldots, x_n$ be positive real numbers such that $x_1x_2 \cdots x_n = 1$. Prove that \[\displaystyle\sum_{i=1}^n x_i^n (1 + x_i) \geq \dfrac{n}{2^{n-1}} \prod_{i=1}^n (1 + x_i).\]
[i]IMO Shortlist[/i]
1960 IMO Shortlist, 2
For what values of the variable $x$ does the following inequality hold: \[ \dfrac{4x^2}{(1-\sqrt{2x+1})^2}<2x+9 \ ? \]