Found problems: 6530
2007 Peru MO (ONEM), 1
Find all values of $A$ such that $0^o < A < 360^o$ and also
$\frac{\sin A}{\cos A - 1} \ge 1$ and $\frac{3\cos A - 1}{\sin A} \ge 1.$
2011 Indonesia MO, 3
Given an acute triangle $ABC$, let $l_a$ be the line passing $A$ and perpendicular to $AB$, $l_b$ be the line passing $B$ and perpendicular to $BC$, and $l_c$ be the line passing $C$ and perpendicular to $CA$. Let $D$ be the intersection of $l_b$ and $l_c$, $E$ be the intersection of $l_c$ and $l_a$, and $F$ be the intersection of $l_a$ and $l_b$. Prove that the area of the triangle $DEF$ is at least three times of the area of $ABC$.
2020 Jozsef Wildt International Math Competition, W59
If $a_k>0~(k=1,2,\ldots,n)$ then prove that
$$\sum_{\text{cyc}}\left(\frac{(a_1+a_2+\ldots+a_{n-1})^2}{a_n}+\frac{a_n^2}{a_1}\right)\ge\frac{n^2}2\sum_{k=1}^na_k$$
[i]Proposed by Mihály Bencze[/i]
1981 AMC 12/AHSME, 12
If $p$, $q$ and $M$ are positive numbers and $q<100$, then the number obtained by increasing $M$ by $p\%$ and decreasing the result by $q\%$ exceeds $M$ if and only if
$\text{(A)}\ p>q ~~ \text{(B)}\ p>\frac{q}{100-q} ~~ \text{(C)}\ p>\frac{q}{1-q} ~~ \text{(D)}\ p>\frac{100q}{100+q} ~~ \text{(E)}\ p>\frac{100q}{100-q}$
2019 Centers of Excellency of Suceava, 1
For $ a,b,c,d $ positive, prove:
$$ \frac{2a}{a^2+bc} +\frac{2b}{b^2+cd} +\frac{2c}{c^2+da} +\frac{2d}{d^2+ab}\le \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d} $$
[i]Dan Popescu[/i]
2019 China Second Round Olympiad, 2
Let $a_1,a_2,\cdots,a_n$ be integers such that $1=a_1\le a_2\le \cdots\le a_{2019}=99$. Find the minimum $f_0$ of the expression $$f=(a_1^2+a_2^2+\cdots+a_{2019}^2)-(a_1a_3+a_2a_4+\cdots+a_{2017}a_{2019}),$$
and determine the number of sequences $(a_1,a_2,\cdots,a_n)$ such that $f=f_0$.
2010 Indonesia TST, 2
Consider a polynomial with coefficients of real numbers $ \phi(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ with three positive real roots. Assume that $ \phi(0)<0$, prove that \[ 2b^3\plus{}9a^2d\minus{}7abc \le 0.\]
[i]Hery Susanto, Malang[/i]
2016 Baltic Way, 7
Find all positive integers $n$ for which $$3x^n + n(x + 2) - 3 \geq nx^2$$ holds for all real numbers $x.$
2020 Jozsef Wildt International Math Competition, W24
Let $M=\{3,4,5,6,7,8,9,10,11,12,13,15,17,19,21,23\}$. Prove that for any $a_i>0,i=\overline{1,n},n\in M$ the inequality holds:
$$\frac{a_1^2}{(a_2+a_3)^4}+\frac{a_2^2}{(a_3+a_4)^4}+\ldots+\frac{a_{n-1}^2}{(a_n+a_1)^4}+\frac{a_n^2}{(a_1+a_2)^4}\ge\frac{n^3}{16s^2},$$
where $s=\sum_{i=1}^na_i$.
[i]Proposed by Marius Olteanu[/i]
1968 Czech and Slovak Olympiad III A, 1
Let $a_1,\ldots,a_n\ (n>2)$ be real numbers with at most one zero. Solve the system
\begin{align*}
x_1x_2 &= a_1, \\
x_2x_3 &= a_2, \\
&\ \vdots \\
x_{n-1}x_n &= a_{n-1}, \\
x_nx_1 &\ge a_n.
\end{align*}
2018 Slovenia Team Selection Test, 3
Let $a$, $b$ and $c$ be positive real numbers satisfying $abc=1$. Prove that the following inequality holds:
$$\frac{a+b+c}{3}\geq\frac{a}{a^2b+2}+\frac{b}{b^2c+2}+\frac{c}{c^2a+2}.$$
2013 District Olympiad, 2
Let $a,b\in \mathbb{C}$. Prove that $\left| az+b\bar{z} \right|\le 1$, for every $z\in \mathbb{C}$, with $\left| z \right|=1$, if and only if $\left| a \right|+\left| b \right|\le 1$.
2014 Chile TST IMO, 1
Given positive real numbers \(a\), \(b\), and \(c\) such that \(a+b+c \leq \frac{3}{2}\), find the minimum of
\[
a+b+c + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.
\]
2003 AIME Problems, 14
The decimal representation of $m/n$, where $m$ and $n$ are relatively prime positive integers and $m < n$, contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of $n$ for which this is possible.
2003 AIME Problems, 7
Point $B$ is on $\overline{AC}$ with $AB = 9$ and $BC = 21$. Point $D$ is not on $\overline{AC}$ so that $AD = CD$, and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\triangle ACD$. Find $s$.
2001 District Olympiad, 4
Prove that:
a) the sequence $a_n=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+n},\ n\ge 1$ is monotonic.
b) there is a sequence $(a_n)_{n\ge 1}\in \{0,1\}$ such that:
\[\lim_{n\to \infty} \left(\frac{a_1}{n+1}+\frac{a_2}{n+2}+\ldots +\frac{a_n}{n+n}\right)=\frac{1}{2}\]
[i]Radu Gologan[/i]
1974 Bulgaria National Olympiad, Problem 6
In triangle pyramid $MABC$ at least two of the plane angles next to the edge $M$ are not equal to each other. Prove that if the bisectors of these angles form the same angle with the angle bisector of the third plane angle, the following inequality is true
$$8a_1b_1c_1\le a^2a_1+b^2b_1+c^2c_1$$
where $a,b,c$ are sides of triangle $ABC$ and $a_1,b_1,c_1$ are edges crossed respectively with $a,b,c$.
[i]V. Petkov[/i]
1999 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 5
Find the smallest positive integer $ u$ such that there exists only one positive integer $ a$ and satisfies the inequality
\[ 20u < 19a < 21u \ \text{?}
\]
1984 IMO Longlists, 29
Let $S_n = \{1, \cdots, n\}$ and let $f$ be a function that maps every subset of $S_n$ into a positive real number and satisfies the following condition: For all $A \subseteq S_n$ and $x, y \in S_n, x \neq y, f(A \cup \{x\})f(A \cup \{y\}) \le f(A \cup \{x, y\})f(A)$. Prove that for all $A,B \subseteq S_n$ the following inequality holds:
\[f(A) \cdot f(B) \le f(A \cup B) \cdot f(A \cap B)\]
2011 Belarus Team Selection Test, 2
Positive real $a,b,c$ satisfy the condition $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=1+\frac{1}{6}\left( \frac{a}{c}+\frac{b}{a}+\frac{c}{b} \right)$$ Prove that $$\frac{a^3bc}{b+c}+\frac{b^3ca}{a+c}+\frac{c^3ab}{a+b}\ge \frac{1}{6}(ab+bc+ca)^2$$
I.Voronovich
2023 Saint Petersburg Mathematical Olympiad, 1
Let $a, b>1$ be reals such that $a+\frac{1}{a^2} \geq 5b-\frac{3}{b^2}$. Show that $a>5b-\frac{4}{b^2}$.
1987 India National Olympiad, 4
If $ x$, $ y$, $ z$, and $ n$ are natural numbers, and $ n\geq z$ then prove that the relation $ x^n \plus{} y^n \equal{} z^n$ does not hold.
2001 Brazil National Olympiad, 1
Show that for any $a,b,c$ positive reals,
\[ (a+b)(a+c) \geq 2 \sqrt{abc(a+b+c)} \]
2010 District Olympiad, 4
Let $ f: [0,1]\rightarrow \mathbb{R}$ a derivable function such that $ f(0)\equal{}f(1)$, $ \int_0^1f(x)dx\equal{}0$ and $ f^{\prime}(x) \neq 1\ ,\ (\forall)x\in [0,1]$.
i)Prove that the function $ g: [0,1]\rightarrow \mathbb{R}\ ,\ g(x)\equal{}f(x)\minus{}x$ is strictly decreasing.
ii)Prove that for each integer number $ n\ge 1$, we have:
$ \left|\sum_{k\equal{}0}^{n\minus{}1}f\left(\frac{k}{n}\right)\right|<\frac{1}{2}$
2016 Estonia Team Selection Test, 4
Prove that for any positive integer $n\ge $, $2 \cdot \sqrt3 \cdot \sqrt[3]{4} ...\sqrt[n-1]{n} > n$