Found problems: 592
2018 Azerbaijan Junior NMO, 3
$a;b\in\mathbb{R^+}$. Prove the following inequality: $$\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{a}}\leq\sqrt[3]{2(a+b)(\frac1{a}+\frac1{b})}$$
2018 ITAMO, 3
Let $x_1,x_2, ... , x_n$ be positive integers,Asumme that in their decimal representations no $x_i$ "prolongs" $x_j$.For instance , $123$ prolongs $12$ , $459$ prolongs $4$ , but $124$ does not prolog $123$.
Prove that :
$\frac {1}{x_1}+\frac {1}{x_2}+...+\frac {1}{x_n} < 3$.
2024 Korea National Olympiad, 5
Find the smallest real number $M$ such that
$$\sum_{k = 1}^{99}\frac{a_{k+1}}{a_k+a_{k+1}+a_{k+2}} < M$$
for all positive real numbers $a_1, a_2, \dots, a_{99}$. ($a_{100} = a_1, a_{101} = a_2$)
1985 IMO Shortlist, 7
The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$
Russian TST 2022, P3
Let $n\geqslant 3$ be an integer and $x_1>x_2>\cdots>x_n$ be real numbers. Suppose that $x_k>0\geqslant x_{k+1}$ for an index $k{}$. Prove that \[\sum_{i=1}^k\left(x_i^{n-2}\prod_{j\neq i}\frac{1}{x_i-x_j}\right)\geqslant 0.\]
2009 IMO Shortlist, 2
Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that:
\[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\]
[i]Proposed by Juhan Aru, Estonia[/i]
2010 Kosovo National Mathematical Olympiad, 5
Let $x,y$ be positive real numbers such that $x+y=1$. Prove that
$\left(1+\frac {1}{x}\right)\left(1+\frac {1}{y}\right)\geq 9$.
2018 South East Mathematical Olympiad, 1
Assume $c$ is a real number. If there exists $x\in[1,2]$ such that $\max\left\{\left |x+\frac cx\right |, \left |x+\frac cx + 2\right |\right\}\geq 5$, please find the value range of $c$.
2007 Korea Junior Math Olympiad, 5
For all positive real numbers $a, b,c.$ Prove the folllowing inequality$$\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq\frac{1}{2}.$$
2020 Junior Balkan Team Selection Tests - Moldova, 2
The positive real numbers $a, b, c$ satisfy the equation $a+b+c=1$.
Prove the identity: $\sqrt{\frac{(a+bc)(b+ca)}{c+ab}}+\sqrt{\frac{(b+ca)(c+ab)}{a+bc}}+\sqrt{\frac{(c+ab)(a+bc)}{b+ca}} = 2$
1970 IMO Longlists, 52
The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$.
[b]a.)[/b] Prove that $0\le b_n<2$.
[b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.
2018 JBMO Shortlist, A6
For $a,b,c$ positive real numbers such that $ab+bc+ca=3$, prove:
$ \frac{a}{\sqrt{a^3+5}}+\frac{b}{\sqrt{b^3+5}}+\frac{c}{\sqrt{c^3+5}} \leq \frac{\sqrt{6}}{2}$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2020 Turkey Team Selection Test, 4
Let $Z^+$ be positive integers set. $f:\mathbb{Z^+}\to\mathbb{Z^+}$ is a function and
we show $ f \circ f \circ ...\circ f $ with $f_l$ for all $l\in \mathbb{Z^+}$ where $f$ is repeated $l$ times. Find all $f:\mathbb{Z^+}\to\mathbb{Z^+}$ functions such that $$ (n-1)^{2020}< \prod _{l=1}^{2020} {f_l}(n)< n^{2020}+n^{2019} $$
for all $n\in \mathbb{Z^+}$
2014 JBMO Shortlist, 3
For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$
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 $x_i > 0,$ $k \in \mathbb{N}, n \in
\mathbb{N}.$
2007 USAMO, 1
Let $n$ be a positive integer. Define a sequence by setting $a_{1}= n$ and, for each $k > 1$, letting $a_{k}$ be the unique integer in the range $0\leq a_{k}\leq k-1$ for which $a_{1}+a_{2}+...+a_{k}$ is divisible by $k$. For instance, when $n = 9$ the obtained sequence is $9,1,2,0,3,3,3,...$. Prove that for any $n$ the sequence $a_{1},a_{2},...$ eventually becomes constant.
2022 Thailand Mathematical Olympiad, 2
Define a function $f:\mathbb{N}\times \mathbb{N}\to\{-1,1\}$ such that
$$f(m,n)=\begin{cases} 1 &\text{if }m,n\text{ have the same parity, and} \\ -1 &\text{if }m,n\text{ have different parity}\end{cases}$$
for every positive integers $m,n$. Determine the minimum possible value of
$$\sum_{1\leq i<j\leq 2565} ijf(x_i,x_j)$$
across all permutations $x_1,x_2,x_3,\dots,x_{2565}$ of $1,2,\dots,2565$.
1967 IMO Longlists, 58
A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$
\[|l(z)| \leq M \rho,\]
where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$
1982 IMO Longlists, 19
Show that
\[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\]
holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.
1968 IMO Shortlist, 12
If $a$ and $b$ are arbitrary positive real numbers and $m$ an integer, prove that
\[\Bigr( 1+\frac ab \Bigl)^m +\Bigr( 1+\frac ba \Bigl)^m \geq 2^{m+1}.\]
JOM 2013, 1.
Determine the minimum value of $\dfrac{m^m}{1\cdot 3\cdot 5\cdot \ldots \cdot(2m-1)}$ for positive integers $m$.
1971 IMO Shortlist, 5
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 $n>2$.
2014 Contests, 1
Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.
1978 IMO Shortlist, 6
Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$
2019 Moldova Team Selection Test, 6
Let $a,b,c \ge 0$ such that $a+b+c=1$ and $s \ge 5$.
Prove that $s(a^2+b^2+c^2) \le 3(s-3)(a^3+b^3+c^3)+1$