Found problems: 6530
2023 Quang Nam Province Math Contest (Grade 11), Problem 5
a) Given an acute triangle $ABC(AB>AC).$ The circle $(O)$ with diameter $BC$ intersects $AB,AC$ at $F,E$, respectively. Let $H$ be the intersection point of $BE,CF,$ the line $AH$ intersects the line $BC$ at $D,$ the line $EF$ intersects the line $BC$ at $K.$ The line passing through $D$ and parallel to $EF$ intersects $AB,AC$ at $M,N,$ respectively.
Prove that: $M,O,N,K$ are on the same circle.
b) Given $\triangle ABC, \angle BAC=\angle BCA=30^{\circ}.$ $D,E,F$ are moving points on the side $AB,BC,CA$ such that: $\angle BFD=\angle BFE=60^{\circ}.$ Let $p,p_1$ be the perimeter of $\triangle ABC,\triangle DEF,$ respectively. Prove that: $p\le 2p_1.$
2012 APMO, 5
Let $ n $ be an integer greater than or equal to $ 2 $. Prove that if the real numbers $ a_1 , a_2 , \cdots , a_n $ satisfy $ a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n $, then
\[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \]
must hold.
2008 Saint Petersburg Mathematical Olympiad, 4
The numbers $x_1,...x_{100}$ are written on a board so that $ x_1=\frac{1}{2}$ and for every $n$ from $1$ to $99$, $x_{n+1}=1-x_1x_2x_3*...*x_{100}$. Prove that $x_{100}>0.99$.
2003 VJIMC, Problem 4
Let $f,g:[0,1]\to(0,+\infty)$ be two continuous functions such that $f$ and $\frac gf$ are increasing. Prove that
$$\int^1_0\frac{\int^x_0f(t)\text dt}{\int^x_0g(t)\text dt}\text dx\le2\int^1_0\frac{f(t)}{g(t)}\text dt.$$
2016 Hong Kong TST, 2
Find the largest possible positive integer $n$ , so that there exist$n$ distinct positive real numbers $x_1,x_2,...,x_n$ satisfying the following inequality : for any $1\le i,j \le n,$ $(3x_i-x_j) (x_i-3x_j)\geq (1-x_ix_j)^2$
2007 Italy TST, 3
Let $p \geq 5$ be a prime.
(a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$
(b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that:
\[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]
2007 Estonia Math Open Junior Contests, 5
In a school tennis tournament with $ m \ge 2$ participants, each match consists of 4 sets. A player who wins more than half of all sets during a match gets 2 points for this match. A player who wins exactly half of all sets during the match gets 1 point, and a player who wins less than half of all sets gets 0 points. During the tournament, each participant plays exactly one match against each remaining player. Find the least number of participants m for which it is possible that some participant wins more sets than any other participant but obtains less points than any other participant.
2000 Romania Team Selection Test, 2
Let $n\ge 1$ be a positive integer and $x_1,x_2\ldots ,x_n$ be real numbers such that $|x_{k+1}-x_k|\le 1$ for $k=1,2,\ldots ,n-1$. Prove that
\[\sum_{k=1}^n|x_k|-\left|\sum_{k=1}^nx_k\right|\le\frac{n^2-1}{4}\]
[i]Gh. Eckstein[/i]
2017 Balkan MO Shortlist, C3
In the plane, there are $n$ points ($n\ge 4$) where no 3 of them are collinear. Let $A(n)$ be the number of parallelograms whose vertices are those points with area $1$. Prove the following inequality:
$A(n)\leq \frac{n^2-3n}{4}$ for all $n\ge 4$
2022 Germany Team Selection Test, 1
Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$
2002 India IMO Training Camp, 20
Let $a,b,c$ be positive real numbers. Prove that
\[\frac{a}b+\frac{b}c+\frac{c}a \geq \frac{c+a}{c+b}+\frac{a+b}{a+c}+\frac{b+c}{b+a}\]
2022 China Second Round A1, 1
$a,b,c,d$ are real numbers so that $a\geq b,c\geq d$,\[|a|+2|b|+3|c|+4|d|=1.\]
Let $P=(a-b)(b-c)(c-d)$,find the maximum and minimum value of $P$.
2012 AMC 8, 20
What is the correct ordering of the three numbers $\frac5{19}$, $\frac7{21}$, and $\frac9{23}$, in increasing order?
$\textbf{(A)}\hspace{.05in}\dfrac9{23} < \dfrac7{21} < \dfrac5{19}$
$\textbf{(B)}\hspace{.05in}\dfrac5{19} < \dfrac7{21} < \dfrac9{23}$
$\textbf{(C)}\hspace{.05in}\dfrac9{23} < \dfrac5{19} < \dfrac7{21}$
$\textbf{(D)}\hspace{.05in}\dfrac5{19} < \dfrac9{23} < \dfrac7{21}$
$\textbf{(E)}\hspace{.05in}\dfrac7{21} < \dfrac5{19} < \dfrac9{23}$
2013 Bogdan Stan, 3
Let $ a,b,c $ be three real numbers such that $ \cos a+\cos b+\cos c=\sin a+\sin b+\sin c=0. $ Prove that
[b]i)[/b] $ \cos 6a+\cos 6b+\cos 6c=3\cos (2a+2b+2c) $
[b]ii)[/b] $ \sin 6a+\sin 6b+\sin 6c=3\sin (2a+2b+2c) $
[i]Vasile Pop[/i]
2022 Latvia Baltic Way TST, P2
Prove that for positive real numbers $a,b,c$ satisfying $abc=1$ the following inequality holds:
$$ \frac{a}{b}+\frac{b}{c}+\frac{c}{a} \ge \frac{a^2+1}{2a}+\frac{b^2+1}{2b}+\frac{c^2+1}{2c}.$$
2014 Contests, 1
Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that
$ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$
(Proposed by Gerhard Woeginger, Austria)
2008 National Olympiad First Round, 31
If the inequality
\[
((x+y)^2+4)((x+y)^2-2)\geq A\cdot (x-y)^2
\]
is hold for every real numbers $x,y$ such that $xy=1$, what is the largest value of $A$?
$
\textbf{(A)}\ 12
\qquad\textbf{(B)}\ 14
\qquad\textbf{(C)}\ 16
\qquad\textbf{(D)}\ 18
\qquad\textbf{(E)}\ 20
$
1986 Kurschak Competition, 2
Let $n>2$ be a positive integer. Find the largest value $h$ and the smallest value $H$ for which
\[h<{a_1\over a_1+a_2}+{a_2\over a_2+a_3}+\cdots+{a_n\over a_n+a_1}<H\]
holds for any positive reals $a_1,\dots,a_n$.
2024 Brazil EGMO TST, 2
Let \( n, k \geq 1 \). In a school, there are \( n \) students and \( k \) clubs. Each student participates in at least one of the clubs. One day, a school uniform was established, which could be either blue or red. Each student chose only one of these colors. Every day, the principal visited one of the clubs, forcing all the students in it to switch the colors of the uniforms they wore.
Assuming that the students are distributed in clubs in such a way that any initial choice of uniforms they make, after a certain number of days, it is possible to have at most one student with one of the colors. Show that
\[
n \geq 2^{n-k-1} - 1.
\]
2020 China Team Selection Test, 5
Let $a_1,a_2,\cdots,a_n$ be a permutation of $1,2,\cdots,n$. Among all possible permutations, find the minimum of $$\sum_{i=1}^n \min \{ a_i,2i-1 \}.$$
2012 Bosnia Herzegovina Team Selection Test, 2
Prove for all positive real numbers $a,b,c$, such that $a^2+b^2+c^2=1$:
\[\frac{a^3}{b^2+c}+\frac{b^3}{c^2+a}+\frac{c^3}{a^2+b}\ge \frac{\sqrt{3}}{1+\sqrt{3}}.\]
2003 Irish Math Olympiad, 4
Given real positive a,b , find the larget real c such that $c\leq max(ax+\frac{1}{ax},bx+\frac{1}{bx})$ for all positive ral x.
There is a solution here,,,,
http://www.kalva.demon.co.uk/irish/soln/sol039.html
but im wondering if there is a better one .
Thank you.
1993 Austrian-Polish Competition, 6
If $a,b \ge 0$ are real numbers, prove the inequality
$$\left(\frac{\sqrt{a}+\sqrt{b}}{2}\right)^2\leq\frac{a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b}{4}\leq\frac{a+\sqrt{ab}+b}{3} \leq \sqrt{\left(\frac{a^{2/3}+b^{2/3}}{2}\right)^{3}}$$
For each of the inequalities, find the cases of equality.
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:
$$(1-(a_1^2+\cdots+a_n^2))(1-(b_1^2+\cdots+b_n^2)) \le (1-(a_1b_1+\cdots+a_nb_n))^2$$
1983 IMO Longlists, 16
Suppose that ${x_1, x_2, \dots , x_n}$ are positive integers for which $x_1 + x_2 + \cdots+ x_n = 2(n + 1)$. Show that there exists an integer $r$ with $0 \leq r \leq n - 1$ for which the following $n - 1$ inequalities hold:
\[x_{r+1} + \cdots + x_{r+i} \leq 2i+ 1, \qquad \qquad \forall i, 1 \leq i \leq n - r; \]
\[x_{r+1} + \cdots + x_n + x_1 + \cdots+ x_i \leq 2(n - r + i) + 1, \qquad \qquad \forall i, 1 \leq i \leq r - 1.\]
Prove that if all the inequalities are strict, then $r$ is unique and that otherwise there are exactly two such $r.$