Found problems: 6530
2018 Iran MO (3rd Round), 1
For positive real numbers$a,b,c$such that $ab+ac+bc=1$ prove that:
$\prod\limits_{cyc} (\sqrt{bc}+\frac{1}{2a+\sqrt{bc}}) \ge 8abc$
2009 China Girls Math Olympiad, 1
Show that there are only finitely many triples $ (x,y,z)$ of positive integers satisfying the equation $ abc\equal{}2009(a\plus{}b\plus{}c).$
2006 Korea - Final Round, 1
Given three distinct real numbers $a_{1}, a_{2}, a_{3}$ , define $b_{j}= (1+\frac{a_{j}a_{i}}{a_{j}-a_{i}})(1+\frac{a_{j}a_{k}}{a_{j}-a_{k}})$, where $\{i, j, k\}= \{1, 2, 3\}$.
Prove that $1+|a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}| \leq (1+|a_{1}|)(1+|a_{2}|)(1+|a_{3}|)$ and find
the cases of equality.
1999 All-Russian Olympiad Regional Round, 11.5
Are there real numbers $a, b$ and $c$ such that for all real $x$ and $y$ the following inequality holds:
$$|x + a| + |x + y + b| + |y + c| > |x| + |x + y| + |y|?$$
2014 Macedonia National Olympiad, 4
Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that:
\[\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}\]
2002 Iran Team Selection Test, 6
Assume $x_{1},x_{2},\dots,x_{n}\in\mathbb R^{+}$, $\sum_{i=1}^{n}x_{i}^{2}=n$, $\sum_{i=1}^{n}x_{i}\geq s>0$ and $0\leq\lambda\leq1$. Prove that at least $\left\lceil\frac{s^{2}(1-\lambda)^{2}}n\right\rceil$ of these numbers are larger than $\frac{\lambda s}{n}$.
2009 Indonesia TST, 2
For every positive integer $ n$, let $ \phi(n)$ denotes the number of positive integers less than $ n$ that is relatively prime to $ n$ and $ \tau(n)$ denote the sum of all positive divisors of $ n$. Let $ n$ be a positive integer such that $ \phi(n)|n\minus{}1$ and that $ n$ is not a prime number. Prove that $ \tau(n)>2009$.
2022 Greece National Olympiad, 3
The positive real numbers $a,b,c,d$ satisfy the equality
$$a+bc+cd+db+\frac{1}{ab^2c^2d^2}=18.$$
Find the maximum possible value of $a$.
1954 Moscow Mathematical Olympiad, 269
a) Given $100$ numbers $a_1, ..., a_{100}$ such that $\begin{cases}
a_1 - 3a_2 + 2a_3 \ge 0, \\
a_2 - 3a_3 + 2a_4 \ge 0, \\
a_3 - 3a_4 + 2a_5 \ge 0, \\
... \\
a_{99} - 3a_{100} + 2a_1 \ge 0, \\
a_{100} - 3a_1 + 2a_2 \ge 0 \end{cases}$
prove that the numbers are equal.
b) Given numbers $a_1=1, ..., a_{100}$ such that $\begin{cases}
a_1 - 4a_2 + 3a_3 \ge 0, \\
a_2 - 4a_3 + 3a_4 \ge 0, \\
a_3 - 4a_4 + 3a_5 \ge 0, \\
... \\
a_{99} - 4a_{100} + 3a_1 \ge 0, \\
a_{100} - 4a_1 + 3a_2 \ge 0 \end{cases}$
Find $a_2, a_3, ... , a_{100}.$
2021-IMOC qualification, A1
Prove that if positive reals $x,y$ satisfy $x+y= 3$, $x,y \ge 1$ then $$9(x- 1)(y- 1) + (y^2 + y+ 1)(x + 1) + (x^2-x+ 1)(y- 1) \ge 9$$
2016 IOM, 2
Let $a_1, . . . , a_n$ be positive integers satisfying the inequality
$\sum_{i=1}^{n}\frac{1}{a_n}\le \frac{1}{2}$.
Every year, the government of Optimistica publishes its Annual Report with n economic indicators. For each $i = 1, . . . , n$,the possible values of the $i-th$ indicator are $1, 2, . . . , a_i$. The Annual Report is said to be optimistic if at least $n - 1$ indicators have higher values than in the previous report. Prove that the government can publish optimistic Annual Reports in an infinitely long sequence.
2010 Contests, 4
Prove that
\[ a^2b^2(a^2+b^2-2) \geq (a+b)(ab-1) \]
for all positive real numbers $a$ and $b.$
2010 Contests, 4
Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that
\[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]
2018 Ramnicean Hope, 1
Show that $ 2/3+\sin 2018^{\circ } >0. $
[i]Costică Ambrinoc[/i]
1976 Czech and Slovak Olympiad III A, 5
Let $\mathbf{P}_1,\mathbf{P}_2$ be convex polygons with perimeters $o_1,o_2,$ respectively. Show that if $\mathbf P_1\subseteq\mathbf P_2,$ then $o_1\le o_2.$
1988 Federal Competition For Advanced Students, P2, 1
If $ a_1,...,a_{1988}$ are positive numbers whose arithmetic mean is $ 1988$, show that:
$ \sqrt[1988]{\displaystyle\prod_{i,j\equal{}1}^{1988} \left( 1\plus{}\frac{a_i}{a_j} \right)} \ge 2^{1988}$
and determine when equality holds.
1953 Moscow Mathematical Olympiad, 257
Let $x_0 = 10^9$, $x_n = \frac{x^2_{n-1}+2}{2x_{n-1}}$ for $n > 0$. Prove that $0 < x_{36} - \sqrt2 < 10^{-9}$.
2005 Morocco TST, 3
Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that \[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \] Prove that $c\geq1$.
2021 Abels Math Contest (Norwegian MO) Final, 2b
If $a_1,\cdots,a_n$ and $b_1,\cdots,b_n$ are real numbers satisfying $a_1^2+\cdots+a_n^2 \le 1$ and $b_1^2+\cdots+b_n^2 \le 1$ , show that:
$$(1-(a_1^2+\cdots+a_n^2))(1-(b_1^2+\cdots+b_n^2)) \le (1-(a_1b_1+\cdots+a_nb_n))^2$$
2019 CMIMC, 6
Across all $x \in \mathbb{R}$, find the maximum value of the expression $$\sin x + \sin 3x + \sin 5x.$$
VI Soros Olympiad 1999 - 2000 (Russia), 8.1
Let $p,q,r$ be prime numbers such that $2p>q$, $q > 2r$ and $q>p+r$. Prove that $p+q+r\ge 20$.
PEN G Problems, 21
Prove that if $ \alpha$ and $ \beta$ are positive irrational numbers satisfying $ \frac{1}{\alpha}\plus{}\frac{1}{\beta}\equal{} 1$, then the sequences
\[ \lfloor\alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 3\alpha\rfloor,\cdots\]
and
\[ \lfloor\beta\rfloor,\lfloor 2\beta\rfloor,\lfloor 3\beta\rfloor,\cdots\]
together include every positive integer exactly once.
1969 IMO Longlists, 15
$(CZS 4)$ Let $K_1,\cdots , K_n$ be nonnegative integers. Prove that $K_1!K_2!\cdots K_n! \ge \left[\frac{K}{n}\right]!^n$, where $K = K_1 + \cdots + K_n$
1996 USAMO, 2
For any nonempty set $S$ of real numbers, let $\sigma(S)$ denote the sum of the elements of $S$. Given a set $A$ of $n$ positive integers, consider the collection of all distinct sums $\sigma(S)$ as $S$ ranges over the nonempty subsets of $A$. Prove that this collection of sums can be partitioned into $n$ classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.
2013 Austria Beginners' Competition, 3
Let $a$ and $ b$ be real numbers with $0\le a, b\le 1$. Prove that $$\frac{a}{b + 1}+\frac{b}{a + 1}\le 1$$ When does equality holds?
(K. Czakler, GRG 21, Vienna)