Found problems: 6530
2017 Thailand Mathematical Olympiad, 8
Let $a, b, c$ be side lengths of a right triangle. Determine the minimum possible value of $\frac{a^3 + b^3 + c^3}{abc}$.
2014 Harvard-MIT Mathematics Tournament, 7
Find the largest real number $c$ such that \[\sum_{i=1}^{101}x_i^2\geq cM^2\] whenever $x_1,\ldots,x_{101}$ are real numbers such that $x_1+\cdots+x_{101}=0$ and $M$ is the median of $x_1,\ldots,x_{101}$.
2013 Romania Team Selection Test, 1
Let $a$ and $b$ be two square-free, distinct natural numbers. Show that there exist $c>0$ such that
\[
\left | \{n\sqrt{a}\}-\{n\sqrt{b}\} \right |>\frac{c}{n^3}\]
for every positive integer $n$.
1981 All Soviet Union Mathematical Olympiad, 319
Positive numbers $x,y$ satisfy equality $$x^3+y^3=x-y$$ Prove that $$x^2+y^2<1$$
2015 Balkan MO, 1
If ${a, b}$ and $c$ are positive real numbers, prove that
\begin{align*}
a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 &\ge{ abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right)}.
\end{align*}
[i](Montenegro).[/i]
1984 National High School Mathematics League, 5
$x_1,x_2,\cdots,x_n$ are positive real numbers. Prove that
$$\frac{x_1^2}{x_2}+\frac{x_2^2}{x_3}+\cdots+\frac{x_n^2}{x_1}\geq x_1+x_2+\cdots x_n.$$
2021 Israel TST, 1
Which is greater: \[\frac{1^{-3}-2^{-3}}{1^{-2}-2^{-2}}-\frac{2^{-3}-3^{-3}}{2^{-2}-3^{-2}}+\frac{3^{-3}-4^{-3}}{3^{-2}-4^{-2}}-\cdots +\frac{2019^{-3}-2020^{-3}}{2019^{-2}-2020^{-2}}\]
or \[1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots +\frac{1}{5781}?\]
2012 Korea - Final Round, 1
Let $ x, y, z $ be positive real numbers. Prove that
\[ \frac{2x^2 + xy}{(y+ \sqrt{zx} + z )^2} + \frac{2y^2 + yz}{(z+ \sqrt{xy} + x )^2} + \frac{2z^2 + zx}{(x+ \sqrt{yz} +y )^2} \ge 1 \]
1979 IMO Longlists, 63
Let the sequence $\{a_i\}$ of $n$ positive reals denote the lengths of the sides of an arbitrary $n$-gon. Let $s=\sum_{i=1}^{n}{a_i}$. Prove that $2\ge \sum_{i=1}^{n}{\frac{a_i}{s-a_i}}\ge \frac{n}{n-1}$.
1987 IMO Longlists, 4
Let $a_1, a_2, a_3, b_1, b_2, b_3$ be positive real numbers. Prove that
\[(a_1b_2 + a_2b_1 + a_1b_3 + a_3b_1 + a_2b_3 + a_3b_2)^2 \geq 4(a_1a_2 + a_2a_3 + a_3a_1)(b_1b_2 + b_2b_3 + b_3b_1)\]
and show that the two sides of the inequality are equal if and only if $\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}.$
1990 IMO Longlists, 7
Let $S$ be the incenter of triangle $ABC$. $A_1, B_1, C_1$ are the intersections of $AS, BS, CS$ with the circumcircle of triangle $ABC$ respectively. Prove that $SA_1 + SB_1 + SC_1 \geq SA + SB + SC.$
2010 Saudi Arabia Pre-TST, 3.1
Let $a \ge b \ge c > 0$. Prove that $$(a-b+c)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right) \ge 1$$
2015 Caucasus Mathematical Olympiad, 4
The sum of the numbers $a,b$ and $c$ is zero, and their product is negative.
Prove that the number $\frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}$ is positive.
2017 Saudi Arabia IMO TST, 2
Denote by $\{x\}$ the fractional part of a real number $x$, that is $\{x\} = x - \rfloor x \lfloor $ where $\rfloor x \lfloor $ is the maximum integer not greater than$ x$ . Prove that
a) For every integer $n$, we have $\{n\sqrt{17}\}> \frac{1}{2\sqrt{17} n}$
b) The value $\frac{1}{2\sqrt{17} }$ is the largest constant $c$ such that the inequality $\{n\sqrt{17}\}> c n $ holds for all positive integers $n$
2006 Bulgaria Team Selection Test, 3
[b]Problem 3.[/b] Let $n\geq 3$ is given natural number, and $M$ is the set of the first $n$ primes. For any nonempty subset $X$ of $M$ with $P(X)$ denote the product of its elements. Let $N$ be a set of the kind $\ds\frac{P(A)}{P(B)}$, $A\subset M, B\subset M, A\cap B=\emptyset$ such that the product of any 7 elements of $N$ is integer. What is the maximal number of elements of $N$?
[i]Alexandar Ivanov[/i]
1990 China Team Selection Test, 1
Given a triangle $ ABC$ with angle $ C \geq 60^{\circ}$. Prove that:
$ \left(a \plus{} b\right) \cdot \left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \right) \geq 4 \plus{} \frac {1}{\sin\left(\frac {C}{2}\right)}.$
1966 IMO Longlists, 5
Prove the inequality
\[\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1\]
for any $x, \alpha$ with $0 \leq x \leq \frac{\pi }{2}$ and $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$
1991 China Team Selection Test, 3
All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.
Gheorghe Țițeica 2025, P2
Let $k\geq 2$ be a positive integer and $x_1,x_2,\dots ,x_k\in (0,1)$. Also, let $m_1,m_2,\dots ,m_k$ and $n_1,n_2,\dots ,n_k$ be integers. Define $$A=x_1^{m_1}x_2^{m_2}\dots x_k^{m_k},\quad B=x_1^{n_1}x_2^{n_2}\dots x_k^{n_k}.$$ Let $$C=x_1^{\min(m_1,n_1)}x_2^{\min(m_2,n_2)}\dots x_k^{\min(m_k,n_k)}$$ $$D=x_1^{\max(m_1,n_1)}x_2^{\max(m_2,n_2)}\dots x_k^{\max(m_k,n_k)}.$$ Prove that $A+B\leq C+D$. When does equality hold?
[i]Dorel Miheț[/i]
2008 AIME Problems, 10
Let $ ABCD$ be an isosceles trapezoid with $ \overline{AD}\parallel{}\overline{BC}$ whose angle at the longer base $ \overline{AD}$ is $ \dfrac{\pi}{3}$. The diagonals have length $ 10\sqrt {21}$, and point $ E$ is at distances $ 10\sqrt {7}$ and $ 30\sqrt {7}$ from vertices $ A$ and $ D$, respectively. Let $ F$ be the foot of the altitude from $ C$ to $ \overline{AD}$. The distance $ EF$ can be expressed in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.
2021 Spain Mathematical Olympiad, 4
Let $a,b,c,d$ real numbers such that:
$$
a+b+c+d=0 \text{ and } a^2+b^2+c^2+d^2 = 12
$$
Find the minimum and maximum possible values for $abcd$, and determine for which values of $a,b,c,d$ the minimum and maximum are attained.
2003 Polish MO Finals, 2
Let $0 < a < 1$ be a real number. Prove that for all finite, strictly increasing sequences $k_1, k_2, \ldots , k_n$ of non-negative integers we have the inequality
\[\biggl( \sum_{i=1}^n a^{k_i} \biggr)^2 < \frac{1+a}{1-a} \sum_{i=1}^n a^{2k_i}.\]
2018 Azerbaijan BMO TST, 1
Problem Shortlist BMO 2017
Let $ a $,$ b$,$ c$, be positive real numbers such that $abc= 1 $. Prove that
$$\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+b^{5}+b^{2}}\leq 1 . $$
2019 EGMO, 5
Let $n\ge 2$ be an integer, and let $a_1, a_2, \cdots , a_n$ be positive integers. Show that there exist positive integers $b_1, b_2, \cdots, b_n$ satisfying the following three conditions:
$\text{(A)} \ a_i\le b_i$ for $i=1, 2, \cdots , n;$
$\text{(B)} \ $ the remainders of $b_1, b_2, \cdots, b_n$ on division by $n$ are pairwise different; and
$\text{(C)} \ $ $b_1+b_2+\cdots b_n \le n\left(\frac{n-1}{2}+\left\lfloor \frac{a_1+a_2+\cdots a_n}{n}\right \rfloor \right)$
(Here, $\lfloor x \rfloor$ denotes the integer part of real number $x$, that is, the largest integer that does not exceed $x$.)
2016 Israel Team Selection Test, 1
Let $a,b,c$ be positive numbers satisfying $ab+bc+ca+2abc=1$. Prove that $4a+b+c \geq 2$.