Found problems: 6530
2008 IberoAmerican Olympiad For University Students, 3
Prove that $x+\frac{1}{x^x}<2$ for $0<x<1$.
2022 Serbia JBMO TST, 1
Prove that for all positive real numbers $a$, $b$ the following inequality holds:
\begin{align*}
\sqrt{\frac{a^2+b^2}{2}}+\frac{2ab}{a+b}\ge \frac{a+b}{2}+ \sqrt{ab}
\end{align*}
When does equality hold?
2018 VJIMC, 2
Let $n$ be a positive integer and let $a_1\le a_2 \le \dots \le a_n$ be real numbers such that
\[a_1+2a_2+\dots+na_n=0.\]
Prove that
\[a_1[x]+a_2[2x]+\dots+a_n[nx] \ge 0\]
for every real number $x$. (Here $[t]$ denotes the integer satisfying $[t] \le t<[t]+1$.)
2007 All-Russian Olympiad, 7
Given a tetrahedron $ T$. Valentin wants to find two its edges $ a,b$ with no common vertices so that $ T$ is covered by balls with diameters $ a,b$. Can he always find such a pair?
[i]A. Zaslavsky[/i]
2013 Bosnia Herzegovina Team Selection Test, 4
Find all primes $p,q$ such that $p$ divides $30q-1$ and $q$ divides $30p-1$.
2013 Miklós Schweitzer, 9
Prove that there is a function ${f: (0,\infty) \rightarrow (0,\infty)}$ which is nowhere continuous and for all ${x,y \in (0,\infty)}$ and any rational ${\alpha}$ we have
\[ \displaystyle f\left( \left(\frac{x^\alpha+y^\alpha}{2}\right)^{\frac{1}{\alpha}}\right)\leq \left(\frac{f(x)^\alpha +f(y)^\alpha }{2}\right)^{\frac{1}{\alpha}}. \]
Is there such a function if instead the above relation holds for every ${x,y \in (0,\infty)}$ and for every irrational ${\alpha}?$
[i]Proposed by Maksa Gyula and Zsolt Páles[/i]
2004 Purple Comet Problems, 13
A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer two layers of unit cubes are removed from the block, more than half the original unit cubes will still remain?
2012 Switzerland - Final Round, 9
Let $a, b, c > 0$ be real numbers with $abc = 1$. Show
$$1 + ab + bc + ca \ge \min \left\{ \frac{(a + b)^2}{ab} , \frac{(b+c)^2}{bc} , \frac{(c + a)^2}{ca}\right\}.$$
When does equality holds?
2019 Jozsef Wildt International Math Competition, W. 61
If $a$, $b$, $c \in \mathbb{R}$ then$$\sum \limits_{cyc} \sqrt{(c+a)^2b^2+c^2a^2}+\sqrt{5}\left |\sum \limits_{cyc} \sqrt{ab}\right |\geq \sum \limits_{cyc}\sqrt{(ab+2bc+ca)^2+(b+c)^2a^2}$$
1998 Baltic Way, 11
If $a,b,c$ be the lengths of the sides of a triangle. Let $R$ denote its circumradius. Prove that
\[ R\ge \frac{a^2+b^2}{2\sqrt{2a^2+2b^2-c^2}}\]
When does equality hold?
2010 All-Russian Olympiad Regional Round, 11.5
The angles of the triangle $\alpha, \beta, \gamma$ satisfy the inequalities $$\sin \alpha > \cos \beta, \sin \beta > \cos \gamma, \sin \gamma > \cos \alpha. $$Prove that the trαiangle is acute-angled.
2022 Pan-African, 3
Let $n$ be a positive integer, and $a_1, a_2, \dots, a_{2n}$ be a sequence of positive real numbers whose product is equal to $2$. For $k = 1, 2, \dots, 2n$, set $a_{2n + k} = a_k$, and define
$$
A_k = \frac{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + n - 2}}{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + 2n - 2}}.
$$
Suppose that $A_1, A_2, \dots, A_{2n}$ are pairwise distinct; show that exactly half of them are less than $\sqrt{2} - 1$.
2008 China Team Selection Test, 3
Let $ 0 < x_{1}\leq\frac {x_{2}}{2}\leq\cdots\leq\frac {x_{n}}{n}, 0 < y_{n}\leq y_{n \minus{} 1}\leq\cdots\leq y_{1},$ Prove that $ (\sum_{k \equal{} 1}^{n}x_{k}y_{k})^2\leq(\sum_{k \equal{} 1}^{n}y_{k})(\sum_{k \equal{} 1}^{n}(x_{k}^2 \minus{} \frac {1}{4}x_{k}x_{k \minus{} 1})y_{k}).$ where $ x_{0} \equal{} 0.$
2000 JBMO ShortLists, 16
Find all the triples $(x,y,z)$ of real numbers such that
\[2x\sqrt{y-1}+2y\sqrt{z-1}+2z\sqrt{x-1} \ge xy+yz+zx \]
1968 German National Olympiad, 5
Prove that for all real numbers $x$ of the interval $0 < x <\pi$ the inequality
$$\sin x +\frac12 \sin 2x +\frac13 \sin 3x > 0$$
holds.
1948 Moscow Mathematical Olympiad, 145
Without tables and such, prove that $\frac{1}{\log_2 \pi}+\frac{1}{\log_5 \pi} >2$
2008 JBMO Shortlist, 2
Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\
ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]
2013 District Olympiad, 3
Let $n\in {{\mathbb{N}}^{*}}$ and ${{a}_{1}},{{a}_{2}},...,{{a}_{n}}\in \mathbb{R}$ so ${{a}_{1}}+{{a}_{2}}+...+{{a}_{k}}\le k,\left( \forall \right)k\in \left\{ 1,2,...,n \right\}.$Prove that
$\frac{{{a}_{1}}}{1}+\frac{{{a}_{2}}}{2}+...+\frac{{{a}_{n}}}{n}\le \frac{1}{1}+\frac{1}{2}+...+\frac{1}{n}$
1995 Poland - Second Round, 4
Positive real numbers $x_1,x_2,...,x_n$ satisfy the condition $\sum_{i=1}^n x_i \le \sum_{i=1}^n x_i ^2$ .
Prove the inequality $\sum_{i=1}^n x_i^t \le \sum_{i=1}^n x_i ^{t+1}$ for all real numbers $t > 1$.
2008 Thailand Mathematical Olympiad, 3
For each positive integer $n$, define $a_n = n(n + 1)$. Prove that
$$n^{1/a_1} + n^{1/a_3} + n^{1/a_5} + ...+ n^{1/a_{2n-1}} \ge n^{a_{3n+2}/a_{3n+1}}$$
.
1961 Kurschak Competition, 2
$x, y, z$ are positive reals less than $1$. Show that at least one of $(1 - x)y$, $(1 - y)z$ and $(1 - z)x$ does not exceed $\frac14$ .
1999 Turkey MO (2nd round), 5
In an acute triangle $\vartriangle ABC$ with circumradius $R$, altitudes $\overline{AD},\overline{BE},\overline{CF}$ have lengths ${{h}_{1}},{{h}_{2}},{{h}_{3}}$, respectively. If ${{t}_{1}},{{t}_{2}},{{t}_{3}}$ are lengths of the tangents from $A,B,C$, respectively, to the circumcircle of triangle $\vartriangle DEF$, prove that
$\sum\limits_{i=1}^{3}{{{\left( \frac{t{}_{i}}{\sqrt{h{}_{i}}} \right)}^{2}}\le }\frac{3}{2}R$.
2022 China Team Selection Test, 3
Let $a_1, a_2, \ldots, a_n$ be $n$ positive integers that are not divisible by each other, i.e. for any $i \neq j$, $a_i$ is not divisible by $a_j$. Show that
\[ a_1+a_2+\cdots+a_n \ge 1.1n^2-2n. \]
[i]Note:[/i] A proof of the inequality when $n$ is sufficient large will be awarded points depending on your results.
2002 JBMO ShortLists, 10
Let $ ABC$ be a triangle with area $ S$ and points $ D,E,F$ on the sides $ BC,CA,AB$. Perpendiculars at points $ D,E,F$ to the $ BC,CA,AB$ cut circumcircle of the triangle $ ABC$ at points $ (D_1,D_2), (E_1,E_2), (F_1,F_2)$. Prove that:
$ |D_1B\cdot D_1C \minus{} D_2B\cdot D_2C| \plus{} |E_1A\cdot E_1C \minus{} E_2A\cdot E_2C| \plus{} |F_1B\cdot F_1A \minus{} F_2B\cdot F_2A| > 4S$
2014 Math Prize For Girls Problems, 6
There are $N$ students in a class. Each possible nonempty group of students selected a positive integer. All of these integers are distinct and add up to 2014. Compute the greatest possible value of $N$.