Found problems: 6530
2013 Junior Balkan Team Selection Tests - Romania, 2
Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. Show that
$$\frac{1 - a^2}{a + bc} + \frac{1 - b^2}{b + ca} + \frac{1 - c^2}{c + ab} \ge 6$$
2023 CUBRMC, 1
Let $x, y, z$ be positive real numbers. Prove that
$$\sqrt{(z + x)(z + y)} - z \ge \sqrt{xy}.$$
2025 Romania EGMO TST, P2
Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that
$$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$
2010 239 Open Mathematical Olympiad, 6
We have six positive numbers $a_1, a_2, \ldots , a_6$ such that $a_1a_2\ldots a_6 =1$. Prove that:
$$ \frac{1}{a_1(a_2 + 1)} + \frac{1}{a_2(a_3 + 1)} + \ldots + \frac{1}{a_6(a_1 + 1)} \geq 3.$$
2011 Iran MO (3rd Round), 2
For nonnegative real numbers $x,y,z$ and $t$ we know that $|x-y|+|y-z|+|z-t|+|t-x|=4$.
Find the minimum of $x^2+y^2+z^2+t^2$.
[i]proposed by Mohammadmahdi Yazdi, Mohammad Ahmadi[/i]
2008 Argentina Iberoamerican TST, 3
The plane is divided into regions by $ n \ge 3$ lines, no two of which are parallel, and no three of which are concurrent. Some regions are coloured , in such a way that no two coloured regions share a common segment or half-line of their borders. Prove that the number of coloured regions is at most $ \frac{n(n\plus{}1)}{3}$
2012 JBMO ShortLists, 5
Find the largest positive integer $n$ for which the inequality
\[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\]
holds true for all $a, b, c \in [0,1]$. Here we make the convention $\sqrt[1]{abc}=abc$.
Fractal Edition 2, P3
The positive numbers $a$, $b$, and $c$ satisfy $abc = 1$. Show that:
$$
\frac{1}{a^2+a}+\frac{1}{b^2+b}+\frac{1}{c^2+c} \ge \frac{3}{2}.
$$
2007 China Northern MO, 2
Let $ a,\, b,\, c$ be side lengths of a triangle and $ a+b+c = 3$. Find the minimum of
\[ a^{2}+b^{2}+c^{2}+\frac{4abc}{3}\]
2009 China Team Selection Test, 5
Let $ m > 1$ be an integer, $ n$ is an odd number satisfying $ 3\le n < 2m,$ number $ a_{i,j} (i,j\in N, 1\le i\le m, 1\le j\le n)$ satisfies $ (1)$ for any $ 1\le j\le n, a_{1,j},a_{2,j},\cdots,a_{m,j}$ is a permutation of $ 1,2,3,\cdots,m; (2)$ for any $ 1 < i\le m, 1\le j\le n \minus{} 1, |a_{i,j} \minus{} a_{i,{j \plus{} 1}}|\le 1$ holds. Find the minimal value of $ M$, where $ M \equal{} max_{1 < i < m}\sum_{j \equal{} 1}^n{a_{i,j}}.$
1978 IMO Longlists, 18
Given a natural number $n$, prove that the number $M(n)$ of points with integer coordinates inside the circle $(O(0, 0),\sqrt{n})$ satisfies
\[\pi n - 5\sqrt{n} + 1<M(n) < \pi n+ 4\sqrt{n} + 1\]
2017 Kyrgyzstan Regional Olympiad, 1
$a^3 + b^3 + 3abc \ge\ c^3$ prove that where a,b and c are sides of triangle.
IV Soros Olympiad 1997 - 98 (Russia), 11.2
Find the area of a figure consisting of points whose coordinates satisfy the inequality $$(y^3 - arcsin x)(x^3 + arcsin y) \ge 0.$$
2009 Today's Calculation Of Integral, 398
In $ xyz$ space, find the volume of the solid expressed by the sytem of inequality:
$ 0\leqq x\leqq 1,\ 0\leqq y\leqq 1,\ 0\leqq z\leqq 1$
$ x^2 \plus{} y^2 \plus{} z^2 \minus{} 2xy \minus{} 1\geqq 0$
2015 Hanoi Open Mathematics Competitions, 15
Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2+d^2 \le 12$.
Determine the maximum value of $M = 4(a^3 + b^3 + c^3+d^3) - (a^4 + b^4 + c^4+d^4)$
2018 Romania Team Selection Tests, 3
Given an integer $n \geq 2$ determine the integral part of the number
$ \sum_{k=1}^{n-1} \frac {1} {({1+\frac{1} {n}}) \dots ({1+\frac {k} {n})}}$ - $\sum_{k=1}^{n-1} (1-\frac {1} {n}) \dots(1-\frac{k}{n})$
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}$.
2003 Rioplatense Mathematical Olympiad, Level 3, 1
Inside right angle $XAY$, where $A$ is the vertex, is a semicircle $\Gamma$ whose center lies on $AX$ and that is tangent to $AY$ at the point $A$. Describe a ruler-and-compass construction for the tangent to $\Gamma$ such that the triangle enclosed by the tangent and angle $XAY$ has minimum area.
JOM 2015 Shortlist, A7
Given positive reals $ a, b, c $ that satisfy $ a + b + c = 1 $, show that $$ \displaystyle \sum^{}_{cyc}\frac{a^3+bc}{a^2+bc}\ge 2 $$
1973 Czech and Slovak Olympiad III A, 3
Let $\left(a_k\right)_{k=1}^\infty$ be a sequence of real numbers such that \[a_{k-1}+a_{k+1}\ge2a_k\] for all $k>1.$ For $n\ge1$ denote \[A_n=\frac1n\left(a_1+\cdots+a_n\right).\] Show that also the inequality \[A_{n-1}+A_{n+1}\ge2A_n\] holds for every $n>1.$
PEN A Problems, 107
Find four positive integers, each not exceeding $70000$ and each having more than $100$ divisors.
2004 IMC, 3
Let $A_n$ be the set of all the sums $\displaystyle \sum_{k=1}^n \arcsin x_k $, where $n\geq 2$, $x_k \in [0,1]$, and $\displaystyle \sum^n_{k=1} x_k = 1$.
a) Prove that $A_n$ is an interval.
b) Let $a_n$ be the length of the interval $A_n$. Compute $\displaystyle \lim_{n\to \infty} a_n$.
2011 Middle European Mathematical Olympiad, 2
Let $a, b, c$ be positive real numbers such that
\[\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=2.\]
Prove that
\[\frac{\sqrt a + \sqrt b+\sqrt c}{2} \geq \frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}.\]
2017 Balkan MO Shortlist, C3
In the plane, there are $n$ points ($n\ge 4$) where no 3 of them are collinear. Let $A(n)$ be the number of parallelograms whose vertices are those points with area $1$. Prove the following inequality:
$A(n)\leq \frac{n^2-3n}{4}$ for all $n\ge 4$
2015 Romania National Olympiad, 4
Let be a finite set $ A $ of real numbers, and define the sets $ S_{\pm }=\{ x\pm y| x,y\in A \} . $
Show that $ \left| A \right|\cdot\left| S_{-} \right| \le \left| S_{+} \right|^2 . $