Olympiad problems

2020 Jozsef Wildt International Math Competition, W57

In all triangles $ABC$ does it hold that: $$\sum\sin^2\frac A2\cos^2A\ge\frac{3\left(s^2-(2R+r)^2\right)}{8R^2}$$ [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

2006 Taiwan TST Round 1, 1

Tags: trigonometry , inequalities

Let the three sides of $\triangle ABC$ be $a,b,c$. Prove that $\displaystyle \frac{\sin^2A}{a}+\frac{\sin^2B}{b}+\frac{\sin^2C}{c} \le \frac{S^2}{abc}$ where $\displaystyle S=\frac{a+b+c}{2}$. Find the case where equality holds.

2002 India IMO Training Camp, 5

Tags: inequalities

Let $a,b,c$ be positive reals such that $a^2+b^2+c^2=3abc$. Prove that \[\frac{a}{b^2c^2}+\frac{b}{c^2a^2}+\frac{c}{a^2b^2} \geq \frac{9}{a+b+c}\]

2007 Junior Balkan Team Selection Tests - Moldova, 2

Tags: algebra , inequalities

The real numbers $a_1, a_2, a_3$ are greater than $1$ and have the sum equal to $S$. If for any $i = 1, 2, 3$, holds the inequality $\frac{a_i^2}{a_i-1}>S$ , prove the inequality $$\frac{1}{a_1+ a_2}+\frac{1}{a_2+ a_3}+\frac{1}{a_3+ a_1}>1$$

2019 BAMO, A

Tags: algebra , inequalities , minimum

Let $a$ and $b$ be positive whole numbers such that $\frac{4.5}{11}<\frac{a}{b}<\frac{5}{11}$. Find the fraction $\frac{a}{b}$ for which the sum $a+b$ is as small as possible. Justify your answer

2022 Bulgaria JBMO TST, 2

Tags: inequalities , am-gm , titu's lemma , cauchy schwarz

Let $a$, $b$ and $c$ be positive real numbers with $abc = 1$. Determine the minimum possible value of $$ \left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right) \cdot \left(\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}\right) $$ as well as all triples $(a,b,c)$ which attain the minimum.

1961 AMC 12/AHSME, 38

Tags: inequalities , pythagorean theorem , geometry

Triangle $ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$: $ \textbf{(A)}\ s^2\le8r^2$ $\qquad\textbf{(B)}\ s^2=8r^2$ $\qquad\textbf{(C)}\ s^2 \ge 8r^2$ ${\qquad\textbf{(D)}\ s^2\le4r^2 }$ ${\qquad\textbf{(E)}\ x^2=4r^2 } $

2010 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: number theory , inequalities

Suppose $a$, $b$, $c$, and $d$ are distinct positive integers such that $a^b$ divides $b^c$, $b^c$ divides $c^d$, and $c^d$ divides $d^a$. [list](a) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the smallest? (b) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the largest?[/list]

1982 Spain Mathematical Olympiad, 6

Tags: algebra , inequalities

Prove that if $u, v$ are any nonnegative real numbers, and $a,b$ positive real numbers such that $a + b = 1$, then $$u^a v^b \le au + bv.$$

I Soros Olympiad 1994-95 (Rus + Ukr), 10.5

Tags: algebra , inequalities

Let $a_1,a_2,...,a_{1994}$ be real numbers in the interval $[-1,1]$, $$S=\frac{a_1+a_2+...+a_{1994}}{1994}.$$ Prove that for an arbitrary natural , $1\le n \le 1994$, holds the inequality $$| a_1+a_2+...+a_n - nS | \le 997.$$

2013 AMC 12/AHSME, 24

Tags: probability , geometry , inequalities , trigonometry , triangle inequality , trig identities , law of sines

Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? $ \textbf{(A)} \ \frac{553}{715} \qquad \textbf{(B)} \ \frac{443}{572} \qquad \textbf{(C)} \ \frac{111}{143} \qquad \textbf{(D)} \ \frac{81}{104} \qquad \textbf{(E)} \ \frac{223}{286}$

1979 Chisinau City MO, 175

Tags: algebra , inequalities

Prove that if the sum of positive numbers $a, b, c$ is equal to $1$, then $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 9.$

2017 India IMO Training Camp, 1

Tags: inequalities

Let $a,b,c$ be distinct positive real numbers with $abc=1$. Prove that $$\sum_{\text{cyc}} \frac{a^6}{(a-b)(a-c)}>15.$$

1974 All Soviet Union Mathematical Olympiad, 203

Tags: function , algebra , inequalities

Given a function $f(x)$ on the segment $0\le x\le 1$. For all $x, f(x)\ge 0, f(1)=1$. For all the couples of $(x_1,x_2)$ such, that all the arguments are in the segment $$f(x_1+x_2)\ge f(x_1)+f(x_2).$$ a) Prove that for all $x$ holds $f(x) \le 2x$. b) Is the inequality $f(x) \le 1.9x$ valid?

2001 National Olympiad First Round, 18

Tags: inequalities , triangle inequality , geometry , perpendicular bisector

A convex polygon has at least one side with length $1$. If all diagonals of the polygon have integer lengths, at most how many sides does the polygon have? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ \text{None of the preceding} $

1989 National High School Mathematics League, 15

Tags: inequalities

For any positive integer $n$, $a_n>0$, and $\sum_{j=1}^{n}a_j^3=\left(\sum_{j=1}^{n}a_j\right)^2$. Prove that $a_n=n$

2020 Saint Petersburg Mathematical Olympiad, 5.

Tags: geometric inequality , inequalities

The altitudes $BB_1$ and $CC_1$ of the acute triangle $\triangle ABC$ intersect at $H$. The circle centered at $O_b$ passes through points $A,C_1$, and the midpoint of $BH$. The circle centered at $O_c$ passes through $A,B_1$ and the midpoint of $CH$. Prove that $B_1 O_b +C_1O_c > \frac{BC}{4}$

1949-56 Chisinau City MO, 52

Tags: combinatorics , algebra , inequalities

Prove that for any natural number $n$ the following inequality holds $$4^n < (2n+1)C_{2n}^n$$

1993 China National Olympiad, 2

Tags: inequalities

Given a natural number $k$ and a real number $a (a>0)$, find the maximal value of $a^{k_1}+a^{k_2}+\cdots +a^{k_r}$, where $k_1+k_2+\cdots +k_r=k$ ($k_i\in \mathbb{N} ,1\le r \le k$).

2018 District Olympiad, 3

Tags: inequalities , logarithm

Let $a, b, c$ be strictly positive real numbers such that $1 < b \le c^2 \le a^{10}$, and \[\log_ab + 2\log_bc + 5\log_ca = 12.\] Prove that \[2\log_ac + 5\log_cb + 10\log_ba \ge 21.\]

2017 China Northern MO, 1

Tags: sequence , inequalities

A sequence $\{a_n\}$ is defined as follows: $a_1 = 1$, $a_2 = \frac{1}{3}$, and for all $n \geq 1,$ $\frac{(1+a_n)(1+a_{n+2})}{(1+a_n+1)^2} = \frac{a_na_{n+2}}{a_{n+1}^2}$. Prove that, for all $n \geq 1$, $a_1 + a_2 + ... + a_n < \frac{34}{21}$.

2019 Korea USCM, 6

Tags: real analysis , inequalities , integral inequality , function

A function $f:[0,\infty)\to[0,\infty)$ is integrable and $$\int_0^\infty f(x)^2 dx<\infty,\quad \int_0^\infty xf(x) dx <\infty$$ Prove the following inequality. $$\left(\int_0^\infty f(x) dx \right)^3 \leq 8\left(\int_0^\infty f(x)^2 dx \right) \left(\int_0^\infty xf(x) dx \right)$$

2006 Kyiv Mathematical Festival, 3

Tags: inequalities

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $x,y>0$ and $xy\ge1.$ Prove that $x^3+y^3+4xy\ge x^2+y^2+x+y+2.$ Let $x,y>0$ and $xy\ge1.$ Prove that $2(x^3+y^3+xy+x+y)\ge5(x^2+y^2).$

2009 Argentina Team Selection Test, 2

Tags: inequalities , clevermath

Let $ a_1, a_2, ..., a_{300}$ be nonnegative real numbers, with $ \sum_{i\equal{}1}^{300} a_i \equal{} 1$. Find the maximum possible value of $ \sum_{i \neq j, i|j} a_ia_j$.

2014 Serbia JBMO TST, 1

Tags: inequalities

For $a, b, c, d, e$ in the interval $[0,1]$, prove that $(1+a+b+c+d+e)^2=>4(a^2+b^2+c^2+d^2+e^2)$