Found problems: 6530
1961 Putnam, B5
Let $k$ be a positive integer, and $n$ a positive integer greater than $2$. Define
$$f_{1}(n)=n,\;\; f_{2}(n)=n^{f_{1}(n)},\;\ldots\;, f_{j+1}(n)=n^{f_{j}(n)}.$$
Prove either part of the inequality
$$f_{k}(n) < n!! \cdots ! < f_{k+1}(n),$$
where the middle term has $k$ factorial symbols.
1960 Czech and Slovak Olympiad III A, 1
Determine all real $x$ satisfying $$\frac{1}{\sin^2 x} -\frac{1}{\cos^2x} \ge \frac83.$$
2023 Vietnam National Olympiad, 3
Find the maximum value of the positive real number $k$ such that the inequality
$$\frac{1}{kab+c^2} +\frac{1} {kbc+a^2} +\frac{1} {kca+b^2} \geq \frac{k+3}{a^2+b^2+c^2} $$holds for all positive real numbers $a,b,c$ such that $a^2+b^2+c^2=2(ab+bc+ca).$
2012 Junior Balkan Team Selection Tests - Moldova, 1
Let $ 1\leq a,b,c,d,e,f,g,h,k \leq 9 $ and $ a,b,c,d,e,f,g,h,k $ are different integers, find the minimum value of the expression $ E = a*b*c+d*e*f+g*h*k $ and prove that it is minimum.
2013 IMC, 1
Let $\displaystyle{z}$ be a complex number with $\displaystyle{\left| {z + 1} \right| > 2}$. Prove that $\displaystyle{\left| {{z^3} + 1} \right| > 1}$.
[i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]
1988 IMO Shortlist, 12
In a triangle $ ABC,$ choose any points $ K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $ F \in KL.$ If $ E_1, E_2, E_3, E_4, E_5, E_6$ and $ E$ denote the areas of the triangles $ AMR, CKR, BKF, ALF, BNM, CLN$ and $ ABC$ respectively, show that
\[ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.
\]
2004 AMC 10, 15
Given that $ \minus{} 4\le x\le \minus{} 2$ and $ 2\le y\le 4$, what is the largest possible value of $ (x \plus{} y)/x$?
$ \textbf{(A)}\ \minus{}\!1\qquad
\textbf{(B)}\ \minus{}\!\frac {1}{2}\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ \frac {1}{2}\qquad
\textbf{(E)}\ 1$
2006 China Team Selection Test, 2
Given three positive real numbers $ x$, $ y$, $ z$ such that $ x \plus{} y \plus{} z \equal{} 1$, prove that
$ \frac {xy}{\sqrt {xy \plus{} yz}} \plus{} \frac {yz}{\sqrt {yz \plus{} zx}} \plus{} \frac {zx}{\sqrt {zx \plus{} xy}} \le \frac {\sqrt {2}}{2}$.
2024 Baltic Way, 4
Find the largest real number $\alpha$ such that, for all non-negative real numbers $x$, $y$ and $z$, the following inequality holds:
\[
(x+y+z)^3 + \alpha (x^2z + y^2x + z^2y) \geq \alpha (x^2y + y^2z + z^2x).
\]
2013 Romania Team Selection Test, 3
Determine all injective functions defined on the set of positive integers into itself satisfying the following condition: If $S$ is a finite set of positive integers such that $\sum\limits_{s\in S}\frac{1}{s}$ is an integer, then $\sum\limits_{s\in S}\frac{1}{f\left( s\right) }$ is also an integer.
2010 Danube Mathematical Olympiad, 5
Let $n\ge3$ be a positive integer. Find the real numbers $x_1\ge0,\ldots,x_n\ge 0$, with $x_1+x_2+\ldots +x_n=n$, for which the expression \[(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n\] takes a minimal value.
2012 Balkan MO Shortlist, A5
Let $f, g:\mathbb{Z}\rightarrow [0,\infty )$ be two functions such that $f(n)=g(n)=0$ with the exception of finitely many integers $n$. Define $h:\mathbb{Z}\rightarrow [0,\infty )$ by \[h(n)=\max \{f(n-k)g(k): k\in\mathbb{Z}\}.\] Let $p$ and $q$ be two positive reals such that $1/p+1/q=1$. Prove that \[ \sum_{n\in\mathbb{Z}}h(n)\geq \Bigg(\sum_{n\in\mathbb{Z}}f(n)^p\Bigg)^{1/p}\Bigg(\sum_{n\in\mathbb{Z}}g(n)^q\Bigg)^{1/q}.\]
2013 Swedish Mathematical Competition, 6
Let $a, b, c$, be real numbers such that $$a^2b^2 + 18 abc > 4b^3+4a^3c+27c^2 .$$
Prove that $a^2>3b$.
2013 Online Math Open Problems, 44
Suppose tetrahedron $PABC$ has volume $420$ and satisfies $AB = 13$, $BC = 14$, and $CA = 15$. The minimum possible surface area of $PABC$ can be written as $m+n\sqrt{k}$, where $m,n,k$ are positive integers and $k$ is not divisible by the square of any prime. Compute $m+n+k$.
[i]Ray Li[/i]
2011 Brazil Team Selection Test, 3
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
2005 France Team Selection Test, 4
Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.
2014 India Regional Mathematical Olympiad, 4
let $ABC$ be a right angled triangle with inradius $1$
find the minimum area of triangle $ABC$
1986 IMO Longlists, 66
One hundred red points and one hundred blue points are chosen in the plane, no three of them lying on a line. Show that these points can be connected pairwise, red ones with blue ones, by disjoint line segments.
2005 Alexandru Myller, 3
Let be three positive real numbers $ a,b,c $ whose sum is $ 1. $ Prove:
$$ 0\le\sum_{\text{cyc}} \log_a\frac{(abc)^a}{a^2+b^2+c^2} $$
2005 Hong kong National Olympiad, 2
Suppose there are $4n$ line segments of unit length inside a circle of radius $n$. Furthermore, a straight line $L$ is given. Prove that there exists a straight line $L'$ that is either parallel or perpendicular to $L$ and that $L'$ cuts at least two of the given line segments.
1986 Traian Lălescu, 2.3
Let $ f:[0,2]\longrightarrow \mathbb{R} $ a differentiable function having a continuous derivative and satisfying $ f(0)=f(2)=1 $ and $ |f’|\le 1. $ Show that
$$ \left| \int_0^2 f(t) dt\right| >1. $$
2003 Vietnam Team Selection Test, 2
Let $A$ be the set of all permutations $a = (a_1, a_2, \ldots, a_{2003})$ of the 2003 first positive integers such that each permutation satisfies the condition: there is no proper subset $S$ of the set $\{1, 2, \ldots, 2003\}$ such that $\{a_k | k \in S\} = S.$
For each $a = (a_1, a_2, \ldots, a_{2003}) \in A$, let $d(a) = \sum^{2003}_{k=1} \left(a_k - k \right)^2.$
[b]I.[/b] Find the least value of $d(a)$. Denote this least value by $d_0$.
[b]II.[/b] Find all permutations $a \in A$ such that $d(a) = d_0$.
2013 Mexico National Olympiad, 1
All the prime numbers are written in order, $p_1 = 2, p_2 = 3, p_3 = 5, ...$
Find all pairs of positive integers $a$ and $b$ with $a - b \geq 2$, such that $p_a - p_b$ divides $2(a-b)$.
1978 IMO Longlists, 45
If $r > s >0$ and $a > b > c$, prove that
\[a^rb^s + b^rc^s + c^ra^s \ge a^sb^r + b^sc^r + c^sa^r.\]
2013 Mediterranean Mathematics Olympiad, 3
Let $x,y,z$ be positive reals for which:
$\sum (xy)^{2}=6xyz$
Prove that:
$\sum \sqrt{\frac{x}{x+yz}}\geq \sqrt{3}$.