Found problems: 426
2010 IMC, 5
Suppose that $a,b,c$ are real numbers in the interval $[-1,1]$ such that $1 + 2abc \geq a^2+b^2+c^2$. Prove that
$1+2(abc)^n \geq a^{2n} + b^{2n} + c^{2n}$ for all positive integers $n$.
2006 Regional Competition For Advanced Students, 1
Let $ 0 < x <y$ be real numbers. Let
$ H\equal{}\frac{2xy}{x\plus{}y}$ , $ G\equal{}\sqrt{xy}$ , $ A\equal{}\frac{x\plus{}y}{2}$ , $ Q\equal{}\sqrt{\frac{x^2\plus{}y^2}{2}}$
be the harmonic, geometric, arithmetic and root mean square (quadratic mean) of $ x$ and $ y$. As generally known $ H<G<A<Q$. Arrange the intervals $ [H,G]$ , $ [G,A]$ and $ [A,Q]$ in ascending order by their length.
1999 Romania Team Selection Test, 4
Show that for all positive real numbers $x_1,x_2,\ldots,x_n$ with product 1, the following inequality holds
\[ \frac 1{n-1+x_1 } +\frac 1{n-1+x_2} + \cdots + \frac 1{n-1+x_n} \leq 1. \]
2007 Mediterranean Mathematics Olympiad, 1
Let $x \geq y \geq z$ be real numbers such that $xy + yz + zx = 1$. Prove that $xz < \frac 12.$ Is it possible to improve the value of constant $\frac 12 \ ?$
1981 Vietnam National Olympiad, 2
Let $p, q$ be real numbers with $0 < p < q$ and let $t_1, t_2, \cdots, t_n$ be real numbers in the interval $[p, q]$. Denote by $A$ and $B$ the arithmetic means of $t_1, t_2, \cdots, t_n$ and of $t_1^2, t_2^2,\cdots , t_n^2$, respectively. Prove that
\[\frac{A^2}{B}\ge\frac{4pq}{(p + q)^2}.\]
2010 Czech-Polish-Slovak Match, 2
Let $x$, $y$, $z$ be positive real numbers satisfying $x+y+z\ge 6$. Find, with proof, the minimum value of \[ x^2+y^2+z^2+\frac{x}{y^2+z+1}+\frac{y}{z^2+x+1}+\frac{z}{x^2+y+1}. \]
1998 Greece JBMO TST, 1
If $x,y,z > 0, k>2$ and $a=x+ky+kz, b=kx+y+kz, c=kx+ky+z$, show that $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} \ge \frac{3}{2k+1}$.
1982 Putnam, B6
Denote by $S(a,b,c)$ the area of a triangle whose lengthes of three sides are $a,b,c$
Prove that for any positive real numbers $a_{1},b_{1},c_{1}$ and $a_{2},b_{2},c_{2}$ which can serve as the lengthes of three sides of two triangles respectively ,we have
$ \sqrt{S(a_{1},b_{1},c_{1})}+\sqrt{S(a_{2},b_{2},c_{2})}\le\sqrt{S(a_{1}+a_{2},b_{1}+b_{2},c_{1}+c_{2})}$
2009 Silk Road, 1
Prove that, abc≤1 and a,b,c>0 \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq 1+ \frac{6}{a+b+c} \]
1998 China National Olympiad, 3
Let $x_1,x_2,\ldots ,x_n$ be real numbers, where $n\ge 2$, satisfying $\sum_{i=1}^{n}x^2_i+ \sum_{i=1}^{n-1}x_ix_{i+1}=1$ . For each $k$, find the maximal value of $|x_k|$.
2006 Turkey MO (2nd round), 1
$x_{1},...,x_{n}$ are positive reals such that their sum and their squares' sum are equal to $t$. Prove that $\sum_{i\neq{j}}\frac{x_{i}}{x_{j}}\ge\frac{(n-1)^{2}\cdot{t}}{t-1}$
1989 Vietnam National Olympiad, 2
The sequence of polynomials $ \left\{P_n(x)\right\}_{n\equal{}0}^{\plus{}\infty}$ is defined inductively by $ P_0(x) \equal{} 0$ and $ P_{n\plus{}1}(x) \equal{} P_n(x)\plus{}\frac{x \minus{} P_n^2(x)}{2}$. Prove that for any $ x \in [0, 1]$ and any natural number $ n$ it holds that $ 0\le\sqrt x\minus{} P_n(x)\le\frac{2}{n \plus{} 1}$.
1983 Vietnam National Olympiad, 2
$(a)$ Prove that $\sqrt{2}(\sin t + \cos t) \ge 2\sqrt[4]{\sin 2t}$ for $0 \le t \le\frac{\pi}{2}.$
$(b)$ Find all $y, 0 < y < \pi$, such that $1 +\frac{2 \cot 2y}{\cot y} \ge \frac{\tan 2y}{\tan y}$.
.
2005 Hong kong National Olympiad, 4
Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=1$. Prove that\[ 6(a^3+b^3+c^3+d^3)\ge(a^2+b^2+c^2+d^2)+\frac{1}{8} \]
2005 China Team Selection Test, 3
Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds.
Prove that $\sum_{j=1}^n |a_j| \leq 3$.
1999 Tuymaada Olympiad, 4
Prove the inequality
\[
{x\over y^2-z}+{y\over z^2-x}+{z\over x^2-y} > 1,
\]
where $2 < x, y, z < 4.$
[i]Proposed by A. Golovanov[/i]
2011 India IMO Training Camp, 1
Let $ABC$ be an acute-angled triangle. Let $AD,BE,CF$ be internal bisectors with $D, E, F$ on $BC, CA, AB$ respectively. Prove that
\[\frac{EF}{BC}+\frac{FD}{CA}+\frac{DE}{AB}\geq 1+\frac{r}{R}\]
2009 Polish MO Finals, 4
Let $ x_1,x_2,..,x_n$ be non-negative numbers whose sum is $ 1$ . Show that there exist numbers $ a_1,a_2,\ldots ,a_n$ chosen from amongst $ 0,1,2,3,4$ such that $ a_1,a_2,\ldots ,a_n$ are different from $ 2,2,\ldots ,2$ and $ 2\leq a_1x_1\plus{}a_2x_2\plus{}\ldots\plus{}a_nx_n\leq 2\plus{}\frac{2}{3^n\minus{}1}$.
1997 Bulgaria National Olympiad, 1
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc=1$.
Prove that
$ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c}$.
2003 China Team Selection Test, 1
$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of:
\[ x^7(yz-1)+y^7(zx-1)+z^7(xy-1) \]
1980 Vietnam National Olympiad, 1
Let $\alpha_{1}, \alpha_{2}, \cdots , \alpha_{ n}$ be numbers in the interval $[0, 2\pi]$ such that the number $\displaystyle\sum_{i=1}^n (1 + \cos \alpha_{ i})$ is an odd integer. Prove that
\[\displaystyle\sum_{i=1}^n \sin \alpha_i \ge 1\]
2013 Moldova Team Selection Test, 3
Consider the obtuse-angled triangle $\triangle ABC$ and its side lengths $a,b,c$. Prove that $a^3\cos\angle A +b^3\cos\angle B + c^3\cos\angle C < abc$.
1988 India National Olympiad, 4
If $ a$ and $ b$ are positive and $ a \plus{} b \equal{} 1$, prove that
\[ \left(a\plus{}\frac{1}{a}\right)^2\plus{}\left(b\plus{}\frac{1}{b}\right)^2 \geq \frac{25}{2}\]
2005 Uzbekistan National Olympiad, 1
Given a,b c are lenth of a triangle (If ABC is a triangle then AC=b, BC=a, AC=b) and $a+b+c=2$.
Prove that $1+abc<ab+bc+ca\leq \frac{28}{27}+abc$
2005 Taiwan TST Round 2, 2
Find all positive integers $n \ge 3$ such that there exists a positive constant $M_n$ satisfying the following inequality for any $n$ positive reals $a_1, a_2,\dots\>,a_n$: \[\displaystyle \frac{a_1+a_2+\cdots\>+a_n}{\sqrt[n]{a_1a_2\cdots\>a_n}} \le M_n \biggl( \frac{a_2}{a_1} + \frac{a_3}{a_2} +\cdots\>+ \frac{a_n}{a_{n-1}} + \frac {a_1}{a_n} \biggr).\] Moreover, find the minimum value of $M_n$ for such $n$.
The difficulty is finding $M_n$...