Found problems: 6530
2011 Saudi Arabia IMO TST, 1
Let $a, b, c$ be real numbers such that $ab + bc + ca = 1$. Prove that
$$\frac{(a + b)^2 + 1}{c^2+2}+\frac{(b + c)^2 + 1}{a^2+2}+ \frac{(c + a)^2 + 1}{b^2+2} \ge 3$$
2010 Contests, 4
If $a,b,c\in (0,1)$ satisfy $a+b+c=2$ , prove that
$\frac{abc}{(1-a)(1-b)(1-c)}\ge 8$
2019 Spain Mathematical Olympiad, 5
We consider all pairs (x, y) of real numbers such that $0\leq x \leq y \leq 1$.Let $M (x,y)$ the maximum value of the set $$A=\{xy, 1-x-y+xy, x+y-2xy\}.$$
Find the minimum value that $M(x,y)$ can take for all these pairs $(x,y)$.
2010 Vietnam Team Selection Test, 1
Let $a,b,c$ be positive integers which satisfy the condition: $16(a+b+c)\geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$.
Prove that
\[\sum_{cyc} \left( \frac{1}{a+b+\sqrt{2a+2c}} \right)^{3}\leq \frac{8}{9}\]
2003 Korea - Final Round, 1
Some computers of a computer room have a following network. Each computers are connected by three cable to three computers. Two arbitrary computers can exchange data directly or indirectly (through other computers). Now let's remove $K$ computers so that there are two computers, which can not exchange data, or there is one computer left. Let $k$ be the minimum value of $K$. Let's remove $L$ cable from original network so that there are two computers, which can not exchange data. Let $l$ be the minimum value of $L$. Show that $k=l$.
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 $n$.
2015 Stars Of Mathematics, 1
Let $a,b,c\ge 0$ be three real numbers such that $$ab+bc+ca+2abc=1.$$ Prove that $\sqrt{a}+\sqrt{b}+\sqrt{c}\ge 2$ and determine equality cases.
Revenge ELMO 2023, 1
In cyclic quadrilateral $ABCD$ with circumcenter $O$ and circumradius $R$, define $X=\overline{AB}\cap\overline{CD}$, $Y=\overline{AC}\cap \overline{BD}$, and $Z=\overline{AD}\cap\overline{BC}$. Prove that
\[OX^2+OY^2+OZ^2\ge 2R^2+2[ABCD].\]
[i]Rohan Bodke[/i]
2003 Croatia National Olympiad, Problem 3
For positive numbers $a_1,a_2,\ldots,a_n$ ($n\ge2$) denote $s=a_1+\ldots+a_n$. Prove that
$$\frac{a_1}{s-a_1}+\ldots+\frac{a_n}{s-a_n}\ge\frac n{n-1}.$$
2021 BMT, 24
Suppose that $a, b, c$, and p are positive integers such that $p$ is a prime number and $$a^2 + b^2 + c^2 = ab + bc + ca + 2021p$$.
Compute the least possible value of $\max \,(a, b, c)$.
2018 Korea National Olympiad, 4
Find all real values of $K$ which satisfies the following.
Let there be a sequence of real numbers $\{a_n\}$ which satisfies the following for all positive integers $n$.
(i). $0 < a_n < n^K$.
(ii). $a_1 + a_2 + \cdots + a_n < \sqrt{n}$.
Then, there exists a positive integer $N$ such that for all integers $n>N$, $$a^{2018}_1 + a^{2018}_2 + \cdots +a^{2018}_n < \frac{n}{2018}$$
2010 Czech And Slovak Olympiad III A, 6
Find the minimum of the expression $\frac{a + b + c}{2} -\frac{[a, b] + [b, c] + [c, a]}{a + b + c}$ where the variables $a, b, c$ are any integers greater than $1$ and $[x, y]$ denotes the least common multiple of numbers $x, y$.
2020 Jozsef Wildt International Math Competition, W25
In the Crelle $[ABCD]$ tetrahedron, we note with $A',B',C',A'',B'',C''$ the tangent points of the hexatangent sphere $\varphi(J,\rho)$, associated with the tetrahedron, with the edges $|BC|,|CA|,|AB|,|DA|,|DB|,|DC|$. Show that these inequalities occur:
a)
$$2\sqrt3R\ge6\rho\ge A'A''+B'B''+C'C''\ge6\sqrt3r$$
b)
$$4R^2\ge12\rho^2\ge(A'A'')^2+(B'B'')^2+(C'C'')^2\ge36r^2$$
c)
$$\frac{8R^3}{3\sqrt3}\ge8\rho^3\ge A'A''\cdot B'B''\cdot C'C''\ge24\sqrt3r^3$$
where $r,R$ is the length of the radius of the sphere inscribed and respectively circumscribed to the tetrahedron.
[i]Proposed by Marius Olteanu[/i]
2013 IMC, 4
Let $\displaystyle{n \geqslant 3}$ and let $\displaystyle{{x_1},{x_2},...,{x_n}}$ be nonnegative real numbers. Define $\displaystyle{A = \sum\limits_{i = 1}^n {{x_i}} ,B = \sum\limits_{i = 1}^n {x_i^2} ,C = \sum\limits_{i = 1}^n {x_i^3} }$. Prove that:
\[\displaystyle{\left( {n + 1} \right){A^2}B + \left( {n - 2} \right){B^2} \geqslant {A^4} + \left( {2n - 2} \right)AC}.\]
[i]Proposed by Géza Kós, Eötvös University, Budapest.[/i]
VI Soros Olympiad 1999 - 2000 (Russia), 10.1
For real numbers $x,y, \in [1,2]$, prove the inequality $3(x + y)\ge 2xy + 4$
MathLinks Contest 5th, 3.3
Let $x_1, x_2,... x_n$ be positive numbers such that $S = x_1+x_2+...+x_n =\frac{1}{x_1}+...+\frac{1}{x_n}$
Prove that $$\sum_{i=1}^{n} \frac{1}{n - 1 + x_i} \ge \sum_{i=1}^{n} \frac{1}{1+S - x_i}$$
2020 Jozsef Wildt International Math Competition, W11
If $a,b,c\in\mathbb N\setminus\{0,1,2,3\}$ then prove:
$$b^2\cdot\sqrt[a]a+c^2\cdot\sqrt[b]b+a^2\cdot\sqrt[c]c\ge48\sqrt2$$
[i]Proposed by Daniel Sitaru[/i]
1976 IMO Longlists, 24
Let $0 \le x_1 \le x_2\le\cdots\le x_n \le 1$. Prove that for all $A \ge 1$, there exists an interval $I$ of length $2\sqrt[n]{A}$ such that for all $x \in I$,
\[|(x - x_1)(x - x_2) \cdots (x -x_n)| \le A.\]
2020 Brazil Team Selection Test, 8
Let $a_1, a_2,\dots$ be an infinite sequence of positive real numbers such that for each positive integer $n$ we have \[\frac{a_1+a_2+\cdots+a_n}n\geq\sqrt{\frac{a_1^2+a_2^2+\cdots+a_{n+1}^2}{n+1}}.\]
Prove that the sequence $a_1,a_2,\dots$ is constant.
[i]Proposed by Alex Zhai[/i]
2013 AMC 12/AHSME, 17
Let $a,b,$ and $c$ be real numbers such that \begin{align*}
a+b+c &= 2, \text{ and} \\
a^2+b^2+c^2&= 12
\end{align*}
What is the difference between the maximum and minimum possible values of $c$?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ \frac{10}{3}\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ \frac{16}{3}\qquad\textbf{(E)}\ \frac{20}{3} $
2012 Turkey MO (2nd round), 4
For all positive real numbers $x, y, z$, show that $ \frac{x(2x-y)}{y(2z+x)}+\frac{y(2y-z)}{z(2x+y)}+\frac{z(2z-x)}{x(2y+z)} \geq 1$ is true.
1982 IMO Longlists, 21
Al[u][b]l[/b][/u] edges and all diagonals of regular hexagon $A_1A_2A_3A_4A_5A_6$ are colored blue or red such that each triangle $A_jA_kA_m, 1 \leq j < k < m\leq 6$ has at least one red edge. Let $R_k$ be the number of red segments $A_kA_j, (j \neq k)$. Prove the inequality
\[\sum_{k=1}^6 (2R_k-7)^2 \leq 54.\]
2006 All-Russian Olympiad Regional Round, 10.5
Prove that for every $x$ such that $\sin x \ne 0$, there is such natural $n$, which $$ | \sin nx| \ge \frac{\sqrt3}{2}.$$
2005 Germany Team Selection Test, 2
Let n be a positive integer, and let $a_1$, $a_2$, ..., $a_n$, $b_1$, $b_2$, ..., $b_n$ be positive real numbers such that $a_1\geq a_2\geq ...\geq a_n$ and $b_1\geq a_1$, $b_1b_2\geq a_1a_2$, $b_1b_2b_3\geq a_1a_2a_3$, ..., $b_1b_2...b_n\geq a_1a_2...a_n$.
Prove that $b_1+b_2+...+b_n\geq a_1+a_2+...+a_n$.
2007 Hong Kong TST, 3
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url]
Problem 3
Let $A$, $B$ and $C$ be real numbers such that
(i) $\sin A \cos B+|\cos A \sin B|=\sin A |\cos A|+|\sin B|\cos B$,
(ii) $\tan C$ and $\cot C$ are defined.
Find the minimum value of $(\tan C-\sin A)^{2}+(\cot C-\cos B)^{2}$.