Found problems: 6530
2023 SAFEST Olympiad, 2
There are $n!$ empty baskets in a row, labelled $1, 2, . . . , n!$. Caesar
first puts a stone in every basket. Caesar then puts 2 stones in every second basket.
Caesar continues similarly until he has put $n$ stones into every nth basket. In
other words, for each $i = 1, 2, . . . , n,$ Caesar puts $i$ stones into the baskets labelled
$i, 2i, 3i, . . . , n!.$
Let $x_i$ be the number of stones in basket $i$ after all these steps. Show that
$n! \cdot n^2 \leq \sum_{i=1}^{n!} x_i^2 \leq n! \cdot n^2 \cdot \sum_{i=1}^{n} \frac{1}{i} $
2022 Cyprus JBMO TST, 3
If $a,b,c$ are positive real numbers with $abc=1$, prove that
(a) \[2\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\right) \geqslant \frac{9}{ab+bc+ca}\]
(b)\[2\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\right) \geqslant \frac{9}{a^2 b+b^2 c+c^2 a}\]
1949-56 Chisinau City MO, 52
Prove that for any natural number $n$ the following inequality holds $$4^n < (2n+1)C_{2n}^n$$
2000 All-Russian Olympiad, 2
Let $-1 < x_1 < x_2 , \cdots < x_n < 1$ and $x_1^{13} + x_2^{13} + \cdots + x_n^{13} = x_1 + x_2 + \cdots + x_n$. Prove that if $y_1 < y_2 < \cdots < y_n$, then \[ x_1^{13}y_1 + \cdots + x_n^{13}y_n < x_1y_1 + x_2y_2 + \cdots + x_ny_n. \]
2024 Dutch IMO TST, 3
Let $a,b,c$ be real numbers such that $0 \le a \le b \le c$ and $a+b+c=1$. Show that
\[ab\sqrt{b-a}+bc\sqrt{c-b}+ac\sqrt{c-a}<\frac{1}{4}.\]
2009 Turkey Team Selection Test, 2
In a triangle $ ABC$ incircle touches the sides $ AB$, $ AC$ and $ BC$ at $ C_1$, $ B_1$ and $ A_1$ respectively. Prove that $ \sqrt {\frac {AB_1}{AB}} \plus{} \sqrt {\frac {BC_1}{BC}} \plus{} \sqrt {\frac {CA_1}{CA}}\leq\frac {3}{\sqrt {2}}$ is true.
2012 India IMO Training Camp, 2
Let $0<x<y<z<p$ be integers where $p$ is a prime. Prove that the following statements are equivalent:
$(a) x^3\equiv y^3\pmod p\text{ and }x^3\equiv z^3\pmod p$
$(b) y^2\equiv zx\pmod p\text{ and }z^2\equiv xy\pmod p$
2014 China Northern MO, 2
Define a positive number sequence sequence $\{a_n\}$ by \[a_{1}=1,(n^2+1)a^2_{n-1}=(n-1)^2a^2_{n}.\]Prove that\[\frac{1}{a^2_1}+\frac{1}{a^2_2}+\cdots +\frac{1}{a^2_n}\le 1+\sqrt{1-\frac{1}{a^2_n}}
.\]
2024 Regional Olympiad of Mexico West, 6
We say that a triangle of sides $a,b,c$ is [i] virtual[/i] if such measures satisfy
$$\begin{cases}
a^{2024}+b^{2024}> c^{2024},\\
b^{2024}+c^{2024}> a^{2024},\\
c^{2024}+a^{2024}> b^{2024}
\end{cases}$$
Find the number of ordered triples $(a,b,c)$ such that $a,b,c$ are integers between $1$ and $2024$ (inclusive) and $a,b,c$ are the sides of a [i]virtual [/i] triangle.
2022 Indonesia MO, 8
Determine the smallest positive real $K$ such that the inequality
\[ K + \frac{a + b + c}{3} \ge (K + 1) \sqrt{\frac{a^2 + b^2 + c^2}{3}} \]holds for any real numbers $0 \le a,b,c \le 1$.
[i]Proposed by Fajar Yuliawan, Indonesia[/i]
2001 VJIMC, Problem 1
Let $A$ be a set of positive integers such that for any $x,y\in A$,
$$x>y\implies x-y\ge\frac{xy}{25}.$$Find the maximal possible number of elements of the set $A$.
2024 New Zealand MO, 2
Prove the following inequality $$\dfrac{6}{2024^3} < \left(1-\dfrac{3}{4}\right)\left(1-\dfrac{3}{5}\right)\left(1-\dfrac{3}{6}\right)\left(1-\dfrac{3}{7}\right)\ldots\left(1-\dfrac{3}{2025}\right).$$
2020 Dutch IMO TST, 1
Given are real numbers $a_1, a_2,..., a_{2020}$, not necessarily different.
For every $n \ge 2020$, define $a_{n + 1}$ as the smallest real zero of the polynomial $$P_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n$$, if it exists. Assume that $a_{n + 1}$ exists for all $n \ge 2020$.
Prove that $a_{n + 1} \le a_n$ for all $n \ge 2021$.
2018 Korea National Olympiad, 4
Find all real values of $K$ which satisfies the following.
Let there be a sequence of real numbers $\{a_n\}$ which satisfies the following for all positive integers $n$.
(i). $0 < a_n < n^K$.
(ii). $a_1 + a_2 + \cdots + a_n < \sqrt{n}$.
Then, there exists a positive integer $N$ such that for all integers $n>N$, $$a^{2018}_1 + a^{2018}_2 + \cdots +a^{2018}_n < \frac{n}{2018}$$
2008 Vietnam National Olympiad, 6
Let $ x, y, z$ be distinct non-negative real numbers. Prove that
\[ \frac{1}{(x\minus{}y)^2} \plus{} \frac{1}{(y\minus{}z)^2} \plus{} \frac{1}{(z\minus{}x)^2} \geq \frac{4}{xy \plus{} yz \plus{} zx}.\]
When does the equality hold?
2006 Bulgaria Team Selection Test, 2
Prove that if $a,b,c>0,$ then \[ \frac{ab}{3a+4b+5c}+\frac{bc}{3b+4c+5a}+\frac{ca}{3c+4a+5b}\le \frac{a+b+c}{12}. \]
[i] Nikolai Nikolov[/i]
the 11th XMO, 2
Suppose $a,b,c>0$ and $abc=64$, show that
$$\sum_{cyc}\frac{a^2}{\sqrt{a^3+8}\sqrt{b^3+8}}\ge\frac{2}{3}$$
1974 IMO Longlists, 26
Let $g(k)$ be the number of partitions of a $k$-element set $M$, i.e., the number of families $\{ A_1,A_2,\ldots ,A_s\}$ of nonempty subsets of $M$ such that $A_i\cap A_j=\emptyset$ for $i\not= j$ and $\bigcup_{i=1}^n A_i=M$. Prove that, for every $n$,
\[n^n\le g(2n)\le (2n)^{2n}\]
2007 Baltic Way, 4
Let $a_1,a_2,\ldots ,a_n$ be positive real numbers, and let $S=a_1+a_2 +\ldots +a_n$ . Prove that
\[(2S+n)(2S+a_1a_2+a_2a_3+\ldots +a_na_1)\ge 9(\sqrt{a_1a_2}+\sqrt{a_2a_3}+\ldots +\sqrt{a_na_1})^2 \]
1993 All-Russian Olympiad, 1
The lengths of the sides of a triangle are prime numbers of centimeters. Prove that its area cannot be an integer number of square centimeters.
2010 AIME Problems, 9
Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 \minus{} xyz \equal{} 2$, $ y^3 \minus{} xyz \equal{} 6$, $ z^3 \minus{} xyz \equal{} 20$. The greatest possible value of $ a^3 \plus{} b^3 \plus{} c^3$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
1973 Putnam, B4
(a) On $[0, 1]$, let $f(x)$ have a continuous derivative satisfying $0 <f'(x) \leq1$. Also suppose that $f(0) = 0.$ Prove that
$$ \left( \int_{0}^{1} f(x)\; dx \right)^{2} \geq \int_{0}^{1} f(x)^{3}\; dx.$$
(b) Show an example in which equality occurs.
2003 Rioplatense Mathematical Olympiad, Level 3, 1
Inside right angle $XAY$, where $A$ is the vertex, is a semicircle $\Gamma$ whose center lies on $AX$ and that is tangent to $AY$ at the point $A$. Describe a ruler-and-compass construction for the tangent to $\Gamma$ such that the triangle enclosed by the tangent and angle $XAY$ has minimum area.
2014 Denmark MO - Mohr Contest, 5
Let $x_0, x_1, . . . , x_{2014}$ be a sequence of real numbers, which for all $i < j$ satisfy $x_i + x_j \le 2j$. Determine the largest possible value of the sum $x_0 + x_1 + · · · + x_{2014}$.
2001 Brazil National Olympiad, 4
A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.)