Found problems: 6530
2003 Alexandru Myller, 3
Let be a nonnegative integer $ n. $ Prove that there exists an increasing and finite sequence of positive real numbers, $
\left( a_k \right)_{0\le k\le n} , $ that satisfy the equality
$$ a_0/0! +a_1/1! +a_2/2! +\cdots +a_n/n! =1/n! , $$
and the inequality
$$ a_0+a_1+a_2+\cdots +a_n<\frac{3}{2^n} . $$
[i]Dorin Andrica[/i]
2014 Irish Math Olympiad, 5
Suppose $a_1,a_2,\ldots,a_n>0 $, where $n>1$ and $\sum_{i=1}^{n}a_i=1$.
For each $i=1,2,\ldots,n $, let $b_i=\frac{a^2_i}{\sum\limits_{j=1}^{n}a^2_j}$. Prove that \[\sum_{i=1}^{n}\frac{a_i}{1-a_i}\le \sum_{i=1}^{n}\frac{b_i}{1-b_i} .\]
When does equality occur ?
2005 South East Mathematical Olympiad, 8
Let $0 < \alpha, \beta, \gamma < \frac{\pi}{2}$ and $\sin^{3} \alpha + \sin^{3} \beta + \sin^3 \gamma = 1$. Prove that
\[ \tan^{2} \alpha + \tan^{2} \beta + \tan^{2} \gamma \geq \frac{3 \sqrt{3}}{2} . \]
2023 Romania EGMO TST, P4
Let $n\geqslant 3$ be an integer and $a_1,\ldots,a_n$ be nonzero real numbers, with sum $S{}$. Prove that \[\sum_{i=1}^n\left|\frac{S-a_i}{a_i}\right|\geqslant\frac{n-1}{n-2}.\]
MathLinks Contest 3rd, 2
Let n be a positive integer and let $a_1, a_2, ..., a_n, b_1, b_2, ... , b_n, c_2, c_3, ... , c_{2n}$ be $4n - 1$ positive real numbers such that $c^2_{i+j} \ge a_ib_j $, for all $1 \le i, j \le n$. Also let $m = \max_{2 \le i\le 2n} c_i$.
Prove that $$\left(\frac{m + c_2 + c_3 +... + c_{2n}}{2n} \right)^2 \ge \left(\frac{a_1+a_2 + ... +a_n}{n}\right)\left(\frac{
b_1 + b_2 + ...+ b_n}{n}\right)$$
2010 IMO Shortlist, 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]
2003 China Western Mathematical Olympiad, 2
Let $ a_1, a_2, \ldots, a_{2n}$ be $ 2n$ real numbers satisfying the condition $ \sum_{i \equal{} 1}^{2n \minus{} 1} (a_{i \plus{} 1} \minus{} a_i)^2 \equal{} 1$. Find the greatest possible value of $ (a_{n \plus{} 1} \plus{} a_{n \plus{} 2} \plus{} \ldots \plus{} a_{2n}) \minus{} (a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n)$.
2016 China Team Selection Test, 2
Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.
1995 Austrian-Polish Competition, 8
Consider the cube with the vertices at the points $(\pm 1, \pm 1, \pm 1)$. Let $V_1,...,V_{95}$ be arbitrary points within this cube. Denote $v_i = \overrightarrow{OV_i}$, where $O = (0,0,0)$ is the origin. Consider the $2^{95}$ vectors of the form $s_1v_1 + s_2v_2 +...+ s_{95}v_{95}$, where $s_i = \pm 1$.
(a) If $d = 48$, prove that among these vectors there is a vector $w = (a, b, c)$ such that $a^2 + b^2 + c^2 \le 48$.
(b) Find a smaller $d$ (the smaller, the better) with the same property.
1996 Canada National Olympiad, 2
Find all real solutions to the following system of equations. Carefully justify your answer.
\[ \left\{ \begin{array}{c} \displaystyle\frac{4x^2}{1+4x^2} = y \\ \\ \displaystyle\frac{4y^2}{1+4y^2} = z \\ \\ \displaystyle\frac{4z^2}{1+4z^2} = x \end{array} \right. \]
1996 Iran MO (2nd round), 1
Let $a, b, c$ be real numbers. Prove that there exists a triangle with side lengths $a, b, c$ if and only if
\[2(a^4 + b^4 + c^4) < (a^2 + b^2 + c^2)^2.\]
2010 Korea National Olympiad, 1
$ x, y, z $ are positive real numbers such that $ x+y+z=1 $. Prove that
\[ \sqrt{ \frac{x}{1-x} } + \sqrt{ \frac{y}{1-y} } + \sqrt{ \frac{z}{1-z} } > 2 \]
2021 Indonesia MO, 4
Let $x,y$ and $z$ be positive reals such that $x + y + z = 3$. Prove that
\[ 2 \sqrt{x + \sqrt{y}} + 2 \sqrt{y + \sqrt{z}} + 2 \sqrt{z + \sqrt{x}} \le \sqrt{8 + x - y} + \sqrt{8 + y - z} + \sqrt{8 + z - x} \]
2002 China Team Selection Test, 1
Given $ n \geq 3$, $ n$ is a integer. Prove that:
\[ (2^n \minus{} 2) \cdot \sqrt{2i\minus{}1} \geq \left( \sum_{j\equal{}0}^{i\minus{}1}C_n^j \plus{} C_{n\minus{}1}^{i\minus{}1} \right) \cdot \sqrt{n}\]
where if $ n$ is even, then $ \displaystyle 1 \leq i \leq \frac{n}{2}$; if $ n$ is odd, then $ \displaystyle 1 \leq i \leq \frac{n\minus{}1}{2}$.
1993 IMO Shortlist, 6
For three points $A,B,C$ in the plane, we define $m(ABC)$ to be the smallest length of the three heights of the triangle $ABC$, where in the case $A$, $B$, $C$ are collinear, we set $m(ABC) = 0$. Let $A$, $B$, $C$ be given points in the plane. Prove that for any point $X$ in the plane,
\[ m(ABC) \leq m(ABX) + m(AXC) + m(XBC). \]
2011 Junior Balkan MO, 1
Let $a,b,c$ be positive real numbers such that $abc = 1$. Prove that:
$\displaystyle\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)$
2021 Kyiv City MO Round 1, 10.1
Prove the following inequality:
$$\sin{1} + \sin{3} + \ldots + \sin{2021} > \frac{2\sin{1011}^2}{\sqrt{3}}$$
[i]Proposed by Oleksii Masalitin[/i]
2012 Junior Balkan Team Selection Tests - Romania, 1
Let $a, b, c, d$ be distinct non-zero real numbers satisfying the following two conditions:
$ac = bd$ and $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 4$.
Determine the largest possible value of the expression $\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}$.
2025 India National Olympiad, P4
Let $n\ge 3$ be a positive integer. Find the largest real number $t_n$ as a function of $n$ such that the inequality
\[\max\left(|a_1+a_2|, |a_2+a_3|, \dots ,|a_{n-1}+a_{n}| , |a_n+a_1|\right) \ge t_n \cdot \max(|a_1|,|a_2|, \dots ,|a_n|)\]
holds for all real numbers $a_1, a_2, \dots , a_n$ .
[i]Proposed by Rohan Goyal and Rijul Saini[/i]
2008 Moldova National Olympiad, 9.8
Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.
2017 District Olympiad, 3
Let $ a $ be a positive real number. Prove that
$$ a^{\sin x}\cdot (a+1)^{\cos x}\ge a,\quad\forall x\in \left[ 0,\frac{\pi }{2} \right] . $$
2009 IMC, 2
Suppose $f:\mathbb{R}\to \mathbb{R}$ is a two times differentiable function satisfying $f(0)=1,f^{\prime}(0)=0$ and for all $x\in [0,\infty)$, it satisfies
\[ f^{\prime \prime}(x)-5f^{\prime}(x)+6f(x)\ge 0 \]
Prove that, for all $x\in [0,\infty)$,
\[ f(x)\ge 3e^{2x}-2e^{3x} \]
2012 Irish Math Olympiad, 3
Suppose $a,b,c$ are positive numbers. Prove that $$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\right)^2\ge (2a+b+c) \left(\frac{2}{a}+\frac{1}{b}+\frac{1}{c}\right)$$ with equality if and only if $a=b=c$.
2017 China Second Round Olympiad, 10
Let $x_1,x_2,x_3\geq 0$ and $x_1+x_2+x_3=1$. Find the minimum value and the maximum value of $(x_1+3x_2+5x_3)\left(x_1+\frac{x_2}{3}+\frac{x_3}{5}\right).$
2016 Turkey EGMO TST, 1
Prove that
\[ x^4y+y^4z+z^4x+xyz(x^3+y^3+z^3) \geq (x+y+z)(3xyz-1) \]
for all positive real numbers $x, y, z$.