Found problems: 6530
2006 USAMO, 1
Let $p$ be a prime number and let $s$ be an integer with $0 < s < p.$ Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and
\[ \left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p} \]
if and only if $s$ is not a divisor of $p-1$.
Note: For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of x.
2019 Jozsef Wildt International Math Competition, W. 66
If $0 < a \leq b$ then$$\frac{2}{\sqrt{3}}\tan^{-1}\left(\frac{2(b^2 - a^2)}{(a^2+2)(b^2+2)}\right)\leq \int \limits_a^b \frac{(x^2+1)(x^2+x+1)}{(x^3 + x^2 + 1) (x^3 + x + 1)}dx\leq \frac{4}{\sqrt{3}}\tan^{-1}\left(\frac{(b-a)\sqrt{3}}{a+b+2(1+ab)}\right)$$
1995 IMO Shortlist, 3
Let $ n$ be an integer, $ n \geq 3.$ Let $ a_1, a_2, \ldots, a_n$ be real numbers such that $ 2 \leq a_i \leq 3$ for $ i \equal{} 1, 2, \ldots, n.$ If $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n,$ prove that
\[ \frac{a^2_1 \plus{} a^2_2 \minus{} a^2_3}{a_1 \plus{} a_2 \minus{} a_3} \plus{} \frac{a^2_2 \plus{} a^2_3 \minus{} a^2_4}{a_2 \plus{} a_3 \minus{} a_4} \plus{} \ldots \plus{} \frac{a^2_n \plus{} a^2_1 \minus{} a^2_2}{a_n \plus{} a_1 \minus{} a_2} \leq 2s \minus{} 2n.\]
2023 Romania National Olympiad, 1
We consider real positive numbers $a,b,c$ such that $a + b + c = 3.$
Prove that $a^2 + b^2 + c^2 + a^2b + b^2 c + c^2 a \ge 6.$
2024 Polish MO Finals, 5
We are given an integer $n \ge 2024$ and a sequence $a_1,a_2,\dots,a_{n^2}$ of real numbers satisfying
\[\vert a_k-a_{k-1}\vert \le \frac{1}{k} \quad \text{and} \quad \vert a_1+a_2+\dots+a_k\vert \le 1\]
for $k=2,3,\dots,n^2$. Show that $\vert a_{n(n-1)}\vert \le \frac{2}{n}$.
[i]Note: Proving $\vert a_{n(n-1)}\vert \le \frac{75}{n}$ will be rewarded partial points.[/i]
2025 Poland - Second Round, 6
Let $1\le k\le n$. Suppose that the sequence $a_1, a_2, \ldots, a_n$ satisfies $0\le a_1 \le a_2 \le \ldots \le a_k$ and $0 \le a_n \le a_{n-1} \le \ldots \le a_k$. The sequence $b_1, b_2, \ldots, b_n$ is the nondecreasing permutation of $a_1, a_2, \ldots, a_n$. Prove that
\[\sum_{i=1}^n \sum_{j=1}^n (j-i)^2a_ia_j \le \sum_{i=1}^n \sum_{j=1}^n (j-i)^2b_ib_j \]
2001 Austrian-Polish Competition, 5
The fields of the $8\times 8$ chessboard are numbered from $1$ to $64$ in the following manner: For $i=1,2,\cdots,63$ the field numbered by $i+1$ can be reached from the field numbered by $i$ by one move of the knight. Let us choose positive real numbers $x_{1},x_{2},\cdots,x_{64}$. For each white field numbered by $i$ define the number $y_{i}=1+x_{i}^{2}-\sqrt[3]{x_{i-1}^{2}x_{i+1}}$ and for each black field numbered by $j$ define the number $y_{j}=1+x_{j}^{2}-\sqrt[3]{x_{j-1}x_{j+1}^{2}}$ where $x_{0}=x_{64}$ and $x_{1}=x_{65}$. Prove that \[\sum_{i=1}^{64}y_{i}\geq 48\]
2014 PUMaC Algebra A, 7
$x$, $y$, and $z$ are positive real numbers that satisfy $x^3+2y^3+6z^3=1$. Let $k$ be the maximum possible value of $2x+y+3z$. Let $n$ be the smallest positive integer such that $k^n$ is an integer. Find the value of $k^n+n$.
2022 Kyiv City MO Round 1, Problem 2
For any reals $x, y$, show the following inequality:
$$\sqrt{(x+4)^2 + (y+2)^2} + \sqrt{(x-5)^2 + (y+4)^2} \le \sqrt{(x-2)^2 + (y-6)^2} + \sqrt{(x-5)^2 + (y-6)^2} + 20$$
[i](Proposed by Bogdan Rublov)[/i]
PEN H Problems, 66
Let $b$ be a positive integer. Determine all $2002$-tuples of non-negative integers $(a_{1}, a_{2}, \cdots, a_{2002})$ satisfying \[\sum^{2002}_{j=1}{a_{j}}^{a_{j}}=2002{b}^{b}.\]
OMMC POTM, 2023 11
Consider an infinite strictly increasing sequence of positive integers $a_1$, $a_2$,$...$ where for any real number $C$, there exists an integer $N$ where $a_k >Ck$ for any $k >N$. Do there necessarily exist inifinite many indices $k$ where $2a_k <a_{k-1}+a_{k+1}$ for any $0<i<k$?
2015 Thailand TSTST, 3
Let $a, b, c$ be positive real numbers. Prove that
$$\frac {3(ab + bc + ca)}{2(a^2b^2+b^2c^2+c^2a^2)}\leq \frac1{a^2 + bc} + \frac1{b^2 + ca} + \frac1{c^2 + ab}\leq\frac{a+b+c}{2abc}.$$
2018 Oral Moscow Geometry Olympiad, 5
Two ants sit on the surface of a tetrahedron. Prove that they can meet by breaking the sum of a distance not exceeding the diameter of a circle is circumscribed around the edge of a tetrahedron.
1993 Balkan MO, 2
A positive integer given in decimal representation $\overline{ a_na_{n-1} \ldots a_1a_0 }$ is called [i]monotone[/i] if $a_n\leq a_{n-1} \leq \cdots \leq a_0$. Determine the number of monotone positive integers with at most 1993 digits.
2012 Romania Team Selection Test, 2
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 \[ \sum_{n\in\mathbb{Z}}h(n)\geq \Bigg(\sum_{n\in\mathbb{Z}}f(n)^p\Bigg)^{1/p}\Bigg(\sum_{n\in\mathbb{Z}}g(n)^q\Bigg)^{1/q}.\]
2010 AMC 12/AHSME, 20
A geometric sequence $ (a_n)$ has $ a_1\equal{}\sin{x}, a_2\equal{}\cos{x},$ and $ a_3\equal{}\tan{x}$ for some real number $ x$. For what value of $ n$ does $ a_n\equal{}1\plus{}\cos{x}$?
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$
1996 Tournament Of Towns, (488) 1
Prove that if $a, b$ and $c$ are positive numbers such that
$$a^2 + b^2 - ab = c^2,$$
then $(a - c)(b - c) < 0.$
(A Egorov)
1963 Dutch Mathematical Olympiad, 3
Twenty numbers $a_1,a_2,..,a_{20}$ satisfy:
$$a_k \ge 7k \,\,\,\,\, for \,\,\,\,\, k = 1,2,..., 20$$
$$a_1+a_2+...+a_{20}=1518$$
Prove that among the numbers $k = 1,2,... ,20$ there are no more than seventeen, for which $a_k \ge 20k -2k^2$.
2021 Taiwan TST Round 2, A
Prove that if non-zero complex numbers $\alpha_1,\alpha_2,\alpha_3$ are distinct and noncollinear on the plane, and satisfy $\alpha_1+\alpha_2+\alpha_3=0$, then there holds
\[\sum_{i=1}^{3}\left(\frac{|\alpha_{i+1}-\alpha_{i+2}|}{\sqrt{|\alpha_i|}}\left(\frac{1}{\sqrt{|\alpha_{i+1}|}}+\frac{1}{\sqrt{|\alpha_{i+2}|}}-\frac{2}{\sqrt{|\alpha_{i}|}}\right)\right)\leq 0......(*)\]
where $\alpha_4=\alpha_1, \alpha_5=\alpha_2$. Verify further the sufficient and necessary condition for the equality holding in $(*)$.
2009 Kyrgyzstan National Olympiad, 9
For any positive $ a_1 ,a_2 ,...,a_n$ prove that $ \frac {{a_1 }} {{a_2 \plus{} a_3 }} \plus{} \frac {{a_2 }} {{a_3 \plus{} a_4 }} \plus{} ... \plus{} \frac {{a_n }} {{a_1 \plus{} a_2 }} > \frac {n} {4}$ holds.
2012 JBMO ShortLists, 2
Let $a$ , $b$ , $c$ be positive real numbers such that $abc=1$ . Show that :
\[\frac{1}{a^3+bc}+\frac{1}{b^3+ca}+\frac{1}{c^3+ab} \leq \frac{ \left (ab+bc+ca \right )^2 }{6}\]
2000 Belarus Team Selection Test, 5.1
Let $AM$ and $AL$ be the median and bisector of a triangle $ABC$ ($M,L \in BC$).
If $BC = a, AM = m_a, AL = l_a$, prove the inequalities:
(a) $a\tan \frac{a}{2} \le 2m_a \le a \cot \frac{a}{2} $ if $a < \frac{\pi}{2}$ and $a\tan \frac{a}{2} \ge 2m_a \ge a \cot \frac{a}{2} $ if $a > \frac{\pi}{2}$
(b) $2l_a \le a\cot \frac{a}{2} $.
2004 Poland - First Round, 3
3. In acute-angled triangle ABC point D is the perpendicular projection of C on the side AB. Point E is the perpendicular projection of D on the side BC. Point F lies on the side DE and:
$\frac{EF}{FD}=\frac{AD}{DB}$
Prove that $CF \bot AE$
2014 HMNT, 3
The side lengths of a triangle are distinct positive integers. One of the side lengths is a multiple of $42,$ and another is a multiple of $72$. What is the minimum possible length of the third side?
2016 CCA Math Bonanza, L2.3
Let $ABC$ be a right triangle with $\angle{ACB}=90^{\circ}$. $D$ is a point on $AB$ such that $CD\perp AB$. If the area of triangle $ABC$ is $84$, what is the smallest possible value of $$AC^2+\left(3\cdot CD\right)^2+BC^2?$$
[i]2016 CCA Math Bonanza Lightning #2.3[/i]