Found problems: 583
1969 IMO Longlists, 69
$(YUG 1)$ Suppose that positive real numbers $x_1, x_2, x_3$ satisfy
$x_1x_2x_3 > 1, x_1 + x_2 + x_3 <\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}$
Prove that:
$(a)$ None of $x_1, x_2, x_3$ equals $1$.
$(b)$ Exactly one of these numbers is less than $1.$
2019 India Regional Mathematical Olympiad, 3
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that
$$\frac{a}{a^2+b^3+c^3}+\frac{b}{b^2+a^3+c^3}+\frac{c}{c^2+a^3+b^3}\leq\frac{1}{5abc}$$
2018 Tajikistan Team Selection Test, 4
Problem 4. Let a,b be positive real numbers and let x,y be positive real numbers less than 1, such that:
a/(1-x)+b/(1-y)=1
Prove that:
∛ay+∛bx≤1.
1980 Bulgaria National Olympiad, Problem 4
Let $a $, $b $, and $c $ be non-negative reals. Prove that $a^3+b^3+c^3+6abc\ge \frac{(a+b+c)^3}{4} $.
1991 IMO Shortlist, 26
Let $ n \geq 2, n \in \mathbb{N}$ and let $ p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R}$ satisfying $ \frac{1}{2} \leq p \leq 1,$ $ 0 \leq a_i,$ $ 0 \leq b_i \leq p,$ $ i \equal{} 1, \ldots, n,$ and \[ \sum^n_{i\equal{}1} a_i \equal{} \sum^n_{i\equal{}1} b_i.\] Prove the inequality: \[ \sum^n_{i\equal{}1} b_i \prod^n_{j \equal{} 1, j \neq i} a_j \leq \frac{p}{(n\minus{}1)^{n\minus{}1}}.\]
1983 IMO Longlists, 66
Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that
\[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0.
\]
Determine when equality occurs.
2024 Korea - Final Round, P3
Find the smallest real number $p(\leq 1)$ that satisfies the following condition.
(Condition) For real numbers $x_1, x_2, \dots, x_{2024}, y_1, y_2, \dots, y_{2024}$, if
[list]
[*] $0 \leq x_1 \leq x_2 \leq \dots \leq x_{2024} \leq 1$,
[*] $0 \leq y_1 \leq y_2 \leq \dots \leq y_{2024} \leq 1$,
[*] $\displaystyle \sum_{i=1}^{2024}x_i = \displaystyle \sum_{i=1}^{2024}y_i = 2024p$,
[/list]
then the inequality $\displaystyle \sum_{i=1}^{2024}x_i(y_{2025-i}-y_{2024-i}) \geq 1 - p$ holds.
1996 French Mathematical Olympiad, Problem 4
(a) A function $f$ is defined by $f(x)=x^x$ for all $x>0$. Find the minimum value of $f$.
(b) If $x$ and $y$ are two positive real numbers, show that $x^y+y^x>1$.
Russian TST 2022, P3
Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]
1967 IMO Shortlist, 2
Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]
1969 IMO Shortlist, 35
$(HUN 2)$ Prove that $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{5}{4}$
1976 IMO Shortlist, 2
Let $a_0, a_1, \ldots, a_n, a_{n+1}$ be a sequence of real numbers satisfying the following conditions:
\[a_0 = a_{n+1 }= 0,\]\[ |a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).\]
Prove that $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$
2009 German National Olympiad, 4
Let $a$ and $b$ be two fixed positive real numbers. Find all real numbers $x$, such that inequality holds $$\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{a+b-x}} < \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$$
2017 Junior Balkan Team Selection Tests - Romania, 2
a) Find :
$A=\{(a,b,c) \in \mathbb{R}^{3} | a+b+c=3 , (6a+b^2+c^2)(6b+c^2+a^2)(6c+a^2+b^2) \neq 0\}$
b) Prove that for any $(a,b,c) \in A$ next inequality hold :
\begin{align*}
\frac{a}{6a+b^2+c^2}+\frac{b}{6b+c^2+a^2}+\frac{c}{6c+a^2+b^2} \le \frac{3}{8}
\end{align*}
2022 Indonesia TST, A
Let $a$ and $b$ be two positive reals such that the following inequality
\[ ax^3 + by^2 \geq xy - 1 \] is satisfied for any positive reals $x, y \geq 1$. Determine the smallest possible value of $a^2 + b$.
[i]Proposed by Fajar Yuliawan[/i]
1979 IMO Longlists, 33
Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$
2022 Grosman Mathematical Olympiad, P4
Along a circle-shaped path are $100$ boys and $100$ girls. The distance between two points on the path is defined as the length of the smaller arc through which it is possible to get from one point to the other.
Prove that the sum of distances between pairs of the same gender is always less than or equal to the sum of distances between all pairs of a boy and a girl.
1981 USAMO, 5
If $x$ is a positive real number, and $n$ is a positive integer, prove that
\[[ nx] > \frac{[ x]}1 + \frac{[ 2x]}2 +\frac{[ 3x]}3 + \cdots + \frac{[ nx]}n,\]
where $[t]$ denotes the greatest integer less than or equal to $t$. For example, $[ \pi] = 3$ and $\left[\sqrt2\right] = 1$.
2020 Junior Macedonian National Olympiad, 2
Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 27$. Prove that
$x + y + z \ge \sqrt{3xyz}$.
When does equality hold?
1985 Yugoslav Team Selection Test, Problem 3
1)
proove for positive $a, b, c, d$
$ \frac{a}{b+c} + \frac{b}{c+d} + \frac{c}{d+a} + \frac{d}{a+b} \ge 2$
2025 Kosovo National Mathematical Olympiad`, P2
Let $x$ and $y$ be real numbers where at least one of them is bigger than $2$ and $xy+4 > 2(x+y)$ holds.
Show that $xy>x+y$.
2022 Iran MO (3rd Round), 6
Prove that among any $9$ distinct real numbers, there exist $4$ distinct numbers $a,b,c,d$ such that
$$(ac+bd)^2\ge\frac{9}{10}(a^2+b^2)(c^2+d^2)$$
1985 IMO Shortlist, 6
Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that
\[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]
2016 Junior Balkan Team Selection Tests - Romania, 2
a,b,c>0 and $abc\ge 1$.Prove that:
$\dfrac{1}{a^3+2b^3+6}+\dfrac{1}{b^3+2c^3+6}+\dfrac{1}{c^3+2a^3+6} \le \dfrac{1}{3}$
1969 IMO Shortlist, 56
Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$