Found problems: 6530
2015 Junior Balkan Team Selection Tests - Romania, 2
Let $a,b,c>0$ such that $a \geq bc^2$ , $b \geq ca^2$ and $c \geq ab^2$ . Find the maximum value that the expression :
$$E=abc(a-bc^2)(b-ca^2)(c-ab^2)$$ can acheive.
2002 Greece National Olympiad, 3
In a triangle $ABC$ we have $\angle C>10^0$ and $\angle B=\angle C+10^0.$We consider point $E$ on side $AB$ such that $\angle ACE=10^0,$ and point $D$ on side $AC$ such that $\angle DBA=15^0.$ Let $Z\neq A$ be a point of interection of the circumcircles of the triangles $ABD$ and $AEC.$Prove that $\angle ZBA>\angle ZCA.$
BIMO 2022, 1
Given an acute triangle $ABC$, mark $3$ points $X, Y, Z$ in the interior of the triangle. Let $X_1, X_2, X_3$ be the projections of $X$ to $BC, CA, AB$ respectively, and define the points $Y_i, Z_i$ similarly for $i=1, 2, 3$.
a) Suppose that $X_iY_i<X_iZ_i$ for all $i=1,2,3$, prove that $XY<XZ$.
b) Prove that this is not neccesarily true, if triangle $ABC$ is allowed to be obtuse.
[i]Proposed by Ivan Chan Kai Chin[/i]
2015 Iran Team Selection Test, 1
$a,b,c,d$ are positive numbers such that $\sum_{cyc} \frac{1}{ab} =1$. Prove that :
$abcd+16 \geq 8 \sqrt{(a+c)(\frac{1}{a} + \frac{1}{c})}+8\sqrt{(b+d)(\frac{1}{b}+\frac{1}{d})}$
2009 VJIMC, Problem 4
Let $k,m,n$ be positive integers such that $1\le m\le n$ and denote $S=\{1,2,\ldots,n\}$. Suppose that $A_1,A_2,\ldots,A_k$ are $m$-element subsets of $S$ with the following property: for every $i=1,2,\ldots,k$ there exists a partition $S=S_{1,i}\cup S_{2,i}\cup\ldots\cup S_{m,i}$ (into pairwise disjoint subsets) such that
(i) $A_i$ has precisely one element in common with each member of the above partition.
(ii) Every $A_j,j\ne i$ is disjoint from at least one member of the above partition.
Show that $k\le\binom{n-1}{m-1}$.
2011 Estonia Team Selection Test, 4
Let $a,b,c$ be positive real numbers such that $2a^2 +b^2=9c^2$.Prove that $\displaystyle \frac{2c}{a}+\frac cb \ge\sqrt 3$.
2008 Iran Team Selection Test, 4
Let $ P_1,P_2,P_3,P_4$ be points on the unit sphere. Prove that $ \sum_{i\neq j}\frac1{|P_i\minus{}P_j|}$ takes its minimum value if and only if these four points are vertices of a regular pyramid.
2016 Balkan MO Shortlist, A5
Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$.
Find the minimum value and the maximum value of the product $abcd$.
1990 IMO Longlists, 52
Let real numbers $a_1, a_2, \ldots, a_n$ satisfy $0 < a_i \leq a, \ i = 1, 2, \ldots, n$. Prove that
(i) If $n = 4$, then
\[\frac 1a \sum_{i=1}^4 a_i - \frac{a_1a_2 + a_2a_3 + a_3 a_4 + a_4 a_1}{a^2} \leq 2.\]
(ii) If $n = 6$, then
\[\frac 1a \sum_{i=1}^6 a_i - \frac{a_1a_2 + a_2a_3 + \cdots + a_5 a_6 + a_6 a_1}{a^2} \leq 3.\]
1962 All-Soviet Union Olympiad, 13
Given are $a_0,a_1, ... , a_n$, satisfying $a_0=a_n = 0$, and $a_{k-1} - 2a_k+a_{k+1}\ge 0$ for $k=0, 1, ... , n-1$. Prove that all the numbers are negative or zero.
2016 IFYM, Sozopol, 1
Find all functions $f: \mathbb{R}^+\rightarrow \mathbb{R}^+$ with the following property: $a,b,$ and $c$ are lengths of sides of a triangle, if and only if $f(a),f(b),$ and $f(c)$ are lengths of sides of a triangle.
2023 Mongolian Mathematical Olympiad, 1
Let $u, v$ be arbitrary positive real numbers. Prove that \[\min{(u, \frac{100}{v}, v+\frac{2023}{u})} \leq \sqrt{2123}.\]
2012 ELMO Shortlist, 1
Let $x_1,x_2,x_3,y_1,y_2,y_3$ be nonzero real numbers satisfying $x_1+x_2+x_3=0, y_1+y_2+y_3=0$. Prove that
\[\frac{x_1x_2+y_1y_2}{\sqrt{(x_1^2+y_1^2)(x_2^2+y_2^2)}}+\frac{x_2x_3+y_2y_3}{\sqrt{(x_2^2+y_2^2)(x_3^2+y_3^2)}}+\frac{x_3x_1+y_3y_1}{\sqrt{(x_3^2+y_3^2)(x_1^2+y_1^2)}} \ge -\frac32.\]
[i]Ray Li, Max Schindler.[/i]
2019 New Zealand MO, 3
Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 3$. Prove that $$a^a + b^b + c^c \ge 3$$
2022 Stanford Mathematics Tournament, 1
If $x$, $y$, and $z$ are real numbers such that $x^2+2y^2+3z^2=96$, what is the maximum possible value of $x+2y+3z$?
2011 Uzbekistan National Olympiad, 2
Prove that $ \forall n\in\mathbb{N}$,$ \exists a,b,c\in$$\bigcup_{k\in\mathbb{N}}(k^{2},k^{2}+k+3\sqrt 3) $ such that $n=\frac{ab}{c}$.
2004 China Team Selection Test, 2
Let $ k$ be a positive integer. Set $ A \subseteq \mathbb{Z}$ is called a $ \textbf{k \minus{} set}$ if there exists $ x_1, x_2, \cdots, x_k \in \mathbb{Z}$ such that for any $ i \neq j$, $ (x_i \plus{} A) \cap (x_j \plus{} A) \equal{} \emptyset$, where $ x \plus{} A \equal{} \{ x \plus{} a \mid a \in A \}$. Prove that if $ A_i$ is $ \textbf{k}_i\textbf{ \minus{} set}$($ i \equal{} 1,2, \cdots, t$), and $ A_1 \cup A_2 \cup \cdots \cup A_t \equal{} \mathbb{Z}$, then $ \displaystyle \frac {1}{k_1} \plus{} \frac {1}{k_2} \plus{} \cdots \plus{} \frac {1}{k_t} \geq 1$.
2003 Romania Team Selection Test, 10
Let $\mathcal{P}$ be the set of all primes, and let $M$ be a subset of $\mathcal{P}$, having at least three elements, and such that for any proper subset $A$ of $M$ all of the prime factors of the number $ -1+\prod_{p\in A}p$ are found in $M$. Prove that $M= \mathcal{P}$.
[i]Valentin Vornicu[/i]
2008 Greece National Olympiad, 4
If $a_1, a_2, \ldots , a_n$ are positive integers and $k = \max\{a_1, \ldots, a_n\}$, $t = \min\{a_1,\ldots, a_n\}$, prove the inequality
\[\left(\frac{a_1^2+a_2^2+\cdots+a_n^2}{a_1+a_2+\cdots+a_n}\right)^{\frac{kn}{t}} \geq a_1a_2\cdots a_n.\]
When does equality hold?
2006 QEDMO 3rd, 11
Guess I should stop proposing problems at 2:00 AM, as this can lead to ones like this here:
Let $a$, $b$, $c$ be three positive reals. Prove the inequality
$\frac{a^2+2b^2}{b+c}+\frac{b^2+2c^2}{c+a}+\frac{c^2+2a^2}{a+b}\geq\frac32\left(a+b+c\right)$.
2018 Pan African, 5
Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd.
$$
Show that
$$
\frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12
$$
and that $-12$ is the maximum.
2023 Indonesia TST, 2
Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that
\[2^{j-i}x_ix_j>2^{s-3}.\]
2019 Junior Balkan Team Selection Tests - Romania, 2
Let $a, b, c, d \ge 0$ such that $a^2 + b^2 + c^2 + d^2 = 4$. Prove that $$\frac{a + b + c + d}{2} \ge 1 + \sqrt{abcd}$$ When does the equality hold?
Leonard Giugiuc and Valmir B. Krasniqi
2017 Singapore MO Open, 2
Let $a_1,a_2,...,a_n,b_1,b_2,...,b_n,p$ be real numbers with $p >- 1$. Prove that
$$\sum_{i=1}^{n}(a_i-b_i)\left(a_i (a_1^2+a_2^2+...+a_n^2)^{p/2}-b_i (b_1^2+b_2^2+...+b_n^2)^{p/2}\right)\ge 0$$
2003 Baltic Way, 4
Let $a,b,c$ be positive real numbers. Prove that
\[ \frac{2a}{a^{2}+bc}+\frac{2b}{b^{2}+ca}+\frac{2c}{c^{2}+ab}\leq\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab} \]