Found problems: 6530
VMEO III 2006 Shortlist, A3
For positive real numbers $x,y,z$ that satisfy $ xy + yz + zx + xyz=4$, prove that
$$\frac{x+y+z}{xy+yz+zx}\le 1+\frac{5}{247}\cdot \left( (x-y)^2+(y-z)^2+(z-x)^2\right)$$
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}}.$
2001 Mongolian Mathematical Olympiad, Problem 2
For positive real numbers $b_1,b_2,\ldots,b_n$ define
$$a_1=\frac{b_1}{b_1+b_2+\ldots+b_n}\enspace\text{ and }\enspace a_k=\frac{b_1+\ldots+b_k}{b_1+\ldots+b_{k-1}}\text{ for }k>1.$$Prove that $a_1+a_2+\ldots+a_n\le\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_n}$
1985 IMO Shortlist, 6
Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that
\[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]
2021 China Team Selection Test, 5
Let $n$ be a positive integer and $a_1,a_2,\ldots a_{2n+1}$ be positive reals. For $k=1,2,\ldots ,2n+1$, denote $b_k = \max_{0\le m\le n}\left(\frac{1}{2m+1} \sum_{i=k-m}^{k+m} a_i \right)$, where indices are taken modulo $2n+1$. Prove that the number of indices $k$ satisfying $b_k\ge 1$ does not exceed $2\sum_{i=1}^{2n+1} a_i$.
2019 Iran Team Selection Test, 6
$x,y$ and $z$ are real numbers such that $x+y+z=xy+yz+zx$. Prove that
$$\frac{x}{\sqrt{x^4+x^2+1}}+\frac{y}{\sqrt{y^4+y^2+1}}+\frac{z}{\sqrt{z^4+z^2+1}}\geq \frac{-1}{\sqrt{3}}.$$
[i]Proposed by Navid Safaei[/i]
1999 German National Olympiad, 2
Determine all real numbers $x$ for which $1+\frac{x}{2} -\frac{x^2}{8} \le \sqrt{1+x} \le 1+\frac{x}{2}$
2015 Costa Rica - Final Round, A2
Determine, if they exist, the real values of $x$ and $y$ that satisfy that $$\frac{x^2}{y^2} +\frac{y^2}{x^2} +\frac{x}{y}+\frac{y}{x} = 0$$ such that $x + y <0.$
1997 Vietnam Team Selection Test, 2
There are $ 25$ towns in a country. Find the smallest $ k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:
1) from each town there are exactly $ k$ direct routes to $ k$ other towns;
2) if two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.
2002 Tuymaada Olympiad, 5
Prove that for all $ x, y \in \[0, 1\] $ the inequality $ 5 (x^2+ y^2) ^2 \leq 4 + (x +y) ^4$ holds.
2004 Baltic Way, 3
Let $p, q, r$ be positive real numbers and $n$ a natural number. Show that if $pqr = 1$, then \[ \frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+1} \leq 1. \]
2015 India IMO Training Camp, 3
Prove that for any triangle $ABC$, the inequality $\displaystyle\sum_{\text{cyclic}}\cos A\le\sum_{\text{cyclic}}\sin (A/2)$ holds.
2012 Today's Calculation Of Integral, 805
Prove the following inequalities:
(1) For $0\leq x\leq 1$,
\[1-\frac 13x\leq \frac{1}{\sqrt{1+x^2}}\leq 1.\]
(2) $\frac{\pi}{3}-\frac 16\leq \int_0^{\frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1-x^4}}dx\leq \frac{\pi}{3}.$
2012 Thailand Mathematical Olympiad, 9
Let $n$ be a positive integer and let $P(x) = x^n + a_{n-1}x^{n-1} +... + a_1x + 1$ be a polynomial with positive real coefficients. Under the assumption that the roots of $P$ are all real, show that $P(x) \ge (x + 1)^n$ for all $x > 0$.
2021 Olympic Revenge, 1
Let $a$, $b$, $c$, $k$ be positive reals such that $ab+bc+ca \leq 1$ and $0 < k \leq \frac{9}{2}$. Prove that:
\[\sqrt[3]{ \frac{k}{a} + (9-3k)b} + \sqrt[3]{\frac{k}{b} + (9-3k)c} + \sqrt[3]{\frac{k}{c} + (9-3k)a } \leq \frac{1}{abc}.\]
[i]Proposed by Zhang Yanzong and Song Qing[/i]
1997 Israel National Olympiad, 3
Let $n?$ denote the product of all primes smaller than $n$.
Prove that $n? > n$ holds for any natural number $n > 3$.
PEN S Problems, 13
The sum of the digits of a natural number $n$ is denoted by $S(n)$. Prove that $S(8n) \ge \frac{1}{8} S(n)$ for each $n$.
1911 Eotvos Mathematical Competition, 1
Show that, if the real numbers $a, b, c, A, B, C$ satisfy
$$aC -2bB + cA = 0 \ \ and \ \ ac - b^2 > 0,$$
then $$AC - B^2 \le 0.$$
2013 Balkan MO Shortlist, A2
Let $a, b, c$ and $d$ are positive real numbers so that $abcd = \frac14$. Prove that holds
$$\left( 16ac +\frac{a}{c^2b}+\frac{16c}{a^2d}+\frac{4}{ac}\right)\left( bd +\frac{b}{256d^2c}+\frac{d}{b^2a}+\frac{1}{64bd}\right) \ge \frac{81}{4}$$
When does the equality hold?
2006 France Team Selection Test, 2
Let $a,b,c$ be three positive real numbers such that $abc=1$. Show that:
\[ \displaystyle \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+ \frac{c}{(c+1)(a+1)} \geq \frac{3}{4}. \]
When is there equality?
2005 Serbia Team Selection Test, 4
Let $T$ be the centroid of triangle $ABC$. Prove that \[ \frac 1{\sin \angle TAC} + \frac 1{\sin \angle TBC} \geq 4 \]
2020 South East Mathematical Olympiad, 1
Let $f(x)=a(3a+2c)x^2-2b(2a+c)x+b^2+(c+a)^2$ $(a,b,c\in R, a(3a+2c)\neq 0).$ If $$f(x)\leq 1$$for any real $x$, find the maximum of $|ab|.$
1969 Swedish Mathematical Competition, 3
$a_1 \le a_2 \le ... \le a_n$ is a sequence of reals $b_1, _b2, b_3,..., b_n$ is any rearrangement of the sequence $B_1 \le B_2 \le ...\le B_n$. Show that $\sum a_ib_i \le \sum a_i B_i$.
2017 JBMO Shortlist, A4
Let $x,y,z$ be positive integers such that $x\neq y\neq z \neq x$ .Prove that $$(x+y+z)(xy+yz+zx-2)\geq 9xyz.$$
When does the equality hold?
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2004 Bundeswettbewerb Mathematik, 4
A cube is decomposed in a finite number of rectangular parallelepipeds such that the volume of the cube's circum sphere volume equals the sum of the volumes of all parallelepipeds' circum spheres. Prove that all these parallelepipeds are cubes.