Found problems: 583
1999 Mongolian Mathematical Olympiad, Problem 5
Given $a;b;c$ satisfying $a^{2}+b^{2}+c^{2}=2$ . Prove that:
a) $\left | a+b+c-abc \right |\leqslant 2$ .
b) $\left | a^{3}+b^{3}+c^{3}-3abc \right |\leqslant 2\sqrt{2}$
2012 Polish MO Finals, 6
Show that for any positive real numbers $a, b, c$ true is inequality:
$\left(\frac{a - b}{c}\right)^2 + \left(\frac{b - c}{a}\right)^2 + \left(\frac{c - a}{b}\right)^2 \ge 2\sqrt{2}\left(\frac{a - b}{c} + \frac{b - c}{a} + \frac{c - a}{b} \right)$.
2024 Ukraine National Mathematical Olympiad, Problem 5
For real numbers $a, b, c, d \in [0, 1]$, find the largest possible value of the following expression:
$$a^2+b^2+c^2+d^2-ab-bc-cd-da$$
[i]Proposed by Mykhailo Shtandenko[/i]
2020 DMO Stage 1, 1.
[b]Q.[/b] Find the minimum value of the expression for $x,y,z\in \mathbb{R}^{+}$
$$\sum_{\text{cyc}}\frac{(x+1)^{4}+2(y+1)^{6}-(y+1)^{4}}{(y+1)^{6}}$$
[i]Proposed by Aritra12[/i]
2015 Bosnia And Herzegovina - Regional Olympiad, 2
Let $a$, $b$ and $c$ be positive real numbers such that $abc=2015$. Prove that $$\frac{a+b}{a^2+b^2}+\frac{b+c}{b^2+c^2}+\frac{c+a}{c^2+a^2} \leq \frac{\sqrt{a}+\sqrt{b}+\sqrt{c}}{\sqrt{2015}}$$
2020 Balkan MO Shortlist, A2
Given are positive reals $a, b, c$, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$. Prove that
$\frac{\sqrt{a+\frac{b}{c}}+\sqrt{b+\frac{c}{a}}+\sqrt{c+\frac{a}{b}}}{3}\leq \frac{a+b+c-1}{\sqrt{2}}$.
[i]Albania[/i]
2023 Tuymaada Olympiad, 1
Prove that for $a, b, c \in [0;1]$, $$(1-a)(1+ab)(1+ac)(1-abc) \leq (1+a)(1-ab)(1-ac)(1+abc).$$
2021 Greece Junior Math Olympiad, 1
If positive reals $x,y$ are such that $2(x+y)=1+xy$, find the minimum value of expression $$A=x+\frac{1}{x}+y+\frac{1}{y}$$
1971 IMO Longlists, 21
Let \[ E_n=(a_1-a_2)(a_1-a_3)\ldots(a_1-a_n)+(a_2-a_1)(a_2-a_3)\ldots(a_2-a_n)+\ldots+(a_n-a_1)(a_n-a_2)\ldots(a_n-a_{n-1}). \] Let $S_n$ be the proposition that $E_n\ge0$ for all real $a_i$. Prove that $S_n$ is true for $n=3$ and $5$, but for no other $n>2$.
2016 German National Olympiad, 6
Let \[ f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=x_1x_2x_4+x_2x_3x_5+x_3x_4x_6+x_4x_5x_7+x_5x_6x_1+x_6x_7x_2+x_7x_1x_3 \] be defined for non-negative real numbers $x_1,x_2,\dots,x_7$ with sum $1$.
Prove that $f(x_1,x_2,\dots,x_7)$ has a maximum value and find that value.
2009 Korea Junior Math Olympiad, 6
If positive reals $a,b,c,d$ satisfy $abcd = 1.$ Prove the following inequality $$1<\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{d}{cd+d+1}+\frac{a}{da+a+1}<2.$$
2018 USAJMO, 2
Let \(a,b,c\) be positive real numbers such that \(a+b+c=4\sqrt[3]{abc}\). Prove that \[2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.\]
1967 IMO Longlists, 47
Prove the following inequality:
\[\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1}
x^{n+k-1}_i,\] where $x_i > 0,$ $k \in \mathbb{N}, n \in
\mathbb{N}.$
1971 IMO Longlists, 36
The matrix
\[A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}\]
satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that
\[|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.\]
2024 APMO, 3
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$
2017 Baltic Way, 1
Let $a_0,a_1,a_2,...$ be an infinite sequence of real numbers satisfying $\frac{a_{n-1}+a_{n+1}}{2}\geq a_n$ for all positive integers $n$. Show that $$\frac{a_0+a_{n+1}}{2}\geq \frac{a_1+a_2+...+a_n}{n}$$ holds for all positive integers $n$.
2021 Science ON all problems, 2
$a,b,c$ are nonnegative integers that satisfy $a^2+b^2+c^2=3$. Find the minimum and maximum value the sum
$$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a}$$
may achieve and find all $a,b,c$ for which equality occurs.\\ \\
[i](Andrei Bâra)[/i]
KoMaL A Problems 2023/2024, A. 869
Let $A$ and $B$ be given real numbers. Let the sum of real numbers $0\le x_1\le x_2\le\ldots \le x_n$ be $A$, and let the sum of real numbers $0\le y_1\le y_2\le \ldots\le y_n$ be $B$. Find the largest possible value of
\[\sum_{i=1}^n (x_i-y_i)^2.\]
[i]Proposed by Péter Csikvári, Budapest[/i]
2023 China Northern MO, 2
Let $ a,b,c \in (0,1) $ and $ab+bc+ca=4abc .$ Prove that $$\sqrt{a+b+c}\geq \sqrt{1-a}+\sqrt{1-b}+\sqrt{1-c}$$
2024 Indonesia MO, 6
Suppose $A_1 A_2 \ldots A_n$ is an $n$-sided polygon with $n \geq 3$ and $\angle A_j \leq 180^{\circ}$ for each $j$ (in other words, the polygon is convex or has fewer than $n$ distinct sides).
For each $i \leq n$, suppose $\alpha_i$ is the smallest possible value of $\angle{A_i A_j A_{i+1}}$ where $j$ is neither $i$ nor $i+1$. (Here, we define $A_{n+1} = A_1$.) Prove that
\[ \alpha_1 + \alpha_2 + \cdots + \alpha_n \leq 180^{\circ} \] and determine all equality cases.
2009 Bosnia and Herzegovina Junior BMO TST, 2
Let $a$ , $b$, $c$ and $d$ be positive real numbers such that $a+b+c+d=8$. Prove that $\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d}\geq8$
2008 Germany Team Selection Test, 1
Let $ A_0 \equal{} (a_1,\dots,a_n)$ be a finite sequence of real numbers. For each $ k\geq 0$, from the sequence $ A_k \equal{} (x_1,\dots,x_k)$ we construct a new sequence $ A_{k \plus{} 1}$ in the following way.
1. We choose a partition $ \{1,\dots,n\} \equal{} I\cup J$, where $ I$ and $ J$ are two disjoint sets, such that the expression
\[ \left|\sum_{i\in I}x_i \minus{} \sum_{j\in J}x_j\right|
\]
attains the smallest value. (We allow $ I$ or $ J$ to be empty; in this case the corresponding sum is 0.) If there are several such partitions, one is chosen arbitrarily.
2. We set $ A_{k \plus{} 1} \equal{} (y_1,\dots,y_n)$ where $ y_i \equal{} x_i \plus{} 1$ if $ i\in I$, and $ y_i \equal{} x_i \minus{} 1$ if $ i\in J$.
Prove that for some $ k$, the sequence $ A_k$ contains an element $ x$ such that $ |x|\geq\frac n2$.
[i]Author: Omid Hatami, Iran[/i]
2021 Science ON grade IX, 3
Real numbers $a,b,c$ with $0\le a,b,c\le 1$ satisfy the condition
$$a+b+c=1+\sqrt{2(1-a)(1-b)(1-c)}.$$
Prove that
$$\sqrt{1-a^2}+\sqrt{1-b^2}+\sqrt{1-c^2}\le \frac{3\sqrt 3}{2}.$$
[i] (Nora Gavrea)[/i]
2024 Sharygin Geometry Olympiad, 10.3
Let $BE$ and $CF$ be the bisectors of a triangle $ABC$. Prove that $2EF \leq BF + CE$.
2009 India National Olympiad, 6
Let $ a,b,c$ be positive real numbers such that $ a^3 \plus{} b^3 \equal{} c^3$.Prove that:
$ a^2 \plus{} b^2 \minus{} c^2 > 6(c \minus{} a)(c \minus{} b)$.