Olympiad problems

2017 China Team Selection Test, 5

Tags: inequalities

Given integer $m\geq2$,$x_1,...,x_m$ are non-negative real numbers,prove that:$$(m-1)^{m-1}(x_1^m+...+x_m^m)\geq(x_1+...+x_m)^m-m^mx_1...x_m$$and please find out when the equality holds.

1988 Irish Math Olympiad, 10

Tags: inequalities

Let $0\le x\le 1$. Show that if $n$ is any positive integer, then $$(1+x)^n\ge (1-x)^n+2nx(1-x^2)^{\frac{n-1}{2}}$$.

2024 ITAMO, 6

Tags: inequalities , integer , extermal , fraction , maximal

For each integer $n$, determine the smallest real number $M_n$ such that \[\frac{1}{a_1}+\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots+\frac{a_{n-1}}{a_n} \le M_n\] for any $n$-tuple $(a_1,a_2,\dots,a_n)$ of integers such that $1<a_1<a_2<\dots<a_n$.

2002 Federal Math Competition of S&M, Problem 3

Tags: rhombus , geometry , geometric transformation , inequalities , circumcircle , trigonometry

Let $ ABCD$ be a rhombus with $ \angle BAD \equal{} 60^{\circ}$. Points $ S$ and $ R$ are chosen inside the triangles $ ABD$ and $ DBC$, respectively, such that $ \angle SBR \equal{} \angle RDS \equal{} 60^{\circ}$. Prove that $ SR^2\geq AS\cdot CR$.

2000 Brazil Team Selection Test, Problem 4

Tags: number theory , inequalities , relatively prime

[b]Problem:[/b]For a positive integer $ n$,let $ V(n; b)$ be the number of decompositions of $ n$ into a product of one or more positive integers greater than $ b$. For example,$ 36 \equal{} 6.6 \equal{}4.9 \equal{} 3.12 \equal{} 3 .3. 4$, so that $ V(36; 2) \equal{} 5$.Prove that for all positive integers $ n$; b it holds that $ V(n;b)<\frac{n}{b}$. :)

2024 Czech-Polish-Slovak Junior Match, 4

Tags: inequalities , integer

Let $a,b,c$ be integers satisfying $a+b+c=1$ and $ab+bc+ca<abc$. Show that $ab+bc+ca<2abc$.

1996 Estonia Team Selection Test, 2

Tags: inequalities , geometry , geometric inequality , inradius

Let $a,b,c$ be the sides of a triangle, $\alpha ,\beta ,\gamma$ the corresponding angles and $r$ the inradius. Prove that $$a\cdot sin\alpha+b\cdot sin\beta+c\cdot sin\gamma\geq 9r$$

2008 Moldova National Olympiad, 9.8

Tags: inequalities , algebra , four variables , 4-variable inequality , function

Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.

2019 Regional Olympiad of Mexico Center Zone, 2

Tags: algebra , functional , functional inequality , inequalities

Find all functions $ f: \mathbb {R} \rightarrow \mathbb {R} $ such that $ f (x + y) \le f (xy) $ for every pair of real $ x $, $ y$.

2005 Brazil Undergrad MO, 2

Tags: function , integration , real analysis , induction , inequalities

Let $f$ and $g$ be two continuous, distinct functions from $[0,1] \rightarrow (0,+\infty)$ such that $\int_{0}^{1}f(x)dx = \int_{0}^{1}g(x)dx$ Let $y_n=\int_{0}^{1}{\frac{f^{n+1}(x)}{g^{n}(x)}dx}$, for $n\geq 0$, natural. Prove that $(y_n)$ is an increasing and divergent sequence.

2007 Today's Calculation Of Integral, 184

Tags: function , calculus , real analysis , integration , calculus computations , logarithm , inequalities

(1) For real numbers $x,\ a$ such that $0<x<a,$ prove the following inequality. \[\frac{2x}{a}<\int_{a-x}^{a+x}\frac{1}{t}\ dt<x\left(\frac{1}{a+x}+\frac{1}{a-x}\right). \] (2) Use the result of $(1)$ to prove that $0.68<\ln 2<0.71.$

1961 IMO, 5

Tags: geometry , perpendicular bisector , trigonometry , inequalities

Construct a triangle $ABC$ if $AC=b$, $AB=c$ and $\angle AMB=w$, where $M$ is the midpoint of the segment $BC$ and $w<90$. Prove that a solution exists if and only if \[ b \tan{\dfrac{w}{2}} \leq c <b \] In what case does the equality hold?

2011 Mongolia Team Selection Test, 1

Tags: inequalities

Let $t,k,m$ be positive integers and $t>\sqrt{km}$. Prove that $\dbinom{2m}{0}+\dbinom{2m}{1}+\cdots+\dbinom{2m}{m-t-1}<\dfrac{2^{2m}}{2k}$ (proposed by B. Amarsanaa, folklore)

1974 IMO Longlists, 32

Tags: inequalities

Let $a_1,a_2,\ldots ,a_n$ be $n$ real numbers such that $0<a\le a_k\le b$ for $k=1,2,\ldots ,n$. If $m_1=\frac{1}{n}(a_1+a_2+\cdots+a_n)$ and $m_2=\frac{1}{n}(a_1^2+a_2^2+\cdots + a_n^2)$, prove that $m_2\le\frac{(a+b)^2}{4ab}m_1^2$ and find a necessary and sufficient condition for equality.

2017 Balkan MO Shortlist, A4

Tags: algebra , three variable inequality , inequalities

Let $M = \{(a,b,c)\in R^3 :0 <a,b,c<\frac12$ with $a+b+c=1 \}$ and $f: M\to R$ given as $$f(a,b,c)=4\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-\frac{1}{abc}$$ Find the best (real) bounds $\alpha$ and $\beta$ such that $f(M) = \{f(a,b,c): (a,b,c)\in M\}\subseteq [\alpha,\beta]$ and determine whether any of them is achievable.

2018 Ukraine Team Selection Test, 5

Tags: combinatorial geometry , geometry , geometric inequality , inequalities

Find the smallest positive number $\lambda$ such that for an arbitrary $12$ points on the plane $P_1,P_2,...P_{12}$ (points may coincide), with distance between arbitrary two of them does not exceeds $1$, holds the inequality $\sum_{1\le i\le j\le 12} P_iP_j^2 \le \lambda$

1986 IMO Longlists, 24

Tags: geometry , inequalities

Two families of parallel lines are given in the plane, consisting of $15$ and $11$ lines, respectively. In each family, any two neighboring lines are at a unit distance from one another; the lines of the first family are perpendicular to the lines of the second family. Let $V$ be the set of $165$ intersection points of the lines under consideration. Show that there exist not fewer than $1986$ distinct squares with vertices in the set $V .$

2013 Saudi Arabia IMO TST, 2

Tags: function , algebra , inequalities , increasing

Let $S = f\{0.1. 2.3,...\}$ be the set of the non-negative integers. Find all strictly increasing functions $f : S \to S$ such that $n + f(f(n)) \le 2f(n)$ for every $n$ in $S$

2000 Nordic, 4

Tags: calculus , functional inequality , inequalities

The real-valued function $f$ is defined for $0 \le x \le 1, f(0) = 0, f(1) = 1$, and $\frac{1}{2} \le \frac{ f(z) - f(y)}{f(y) - f(x)} \le 2$ for all $0 \le x < y < z \le 1$ with $z - y = y -x$. Prove that $\frac{1}{7} \le f (\frac{1}{3} ) \le \frac{4}{7}$.

2003 Gheorghe Vranceanu, 2

Tags: inequalities

Let be a natural number $ n $ and $ 2n $ positive real numbers $ v_1,v_2,\ldots ,v_{2n} $ such that the last $ n $ of them are greater than $ 1. $ Prove that: $$ \sum_{i=1}^n v_iv_{n+i}\le \max_{1\le k\le n}\left( \left( -1+\prod_{l=n}^{2n} v_l \right) v_k +\sum_{m=1}^n v_m \right) $$

1981 Czech and Slovak Olympiad III A, 1

Tags: parameterization , algebra , inequalities

Determine all $a\in\mathbb R$ such that the inequality \[x^4+x^3-2(a+1)x^2-ax+a^2<0\] has at least one real solution $x.$

2009 Belarus Team Selection Test, 1

Tags: algebra , inequalities

Prove that any positive real numbers a,b,c satisfy the inequlaity $$\frac{1}{(a+b)b}+\frac{1}{(b+c)c}+\frac{1}{(c+a)a}\ge \frac{9}{2(ab+bc+ca)}$$ I.Voronovich

2014 IFYM, Sozopol, 6

Tags: inequalities , algebra

The positive real numbers $a,b,c$ are such that $21ab+2bc+8ca\leq 12$. Find the smallest value of $\frac{1}{a}+\frac{2}{b}+\frac{3}{c}$.

2017 Princeton University Math Competition, 4

Tags: inequalities

Ayase chooses three numbers $a, b, c$ independently and uniformly from the interval $[-1, 1]$. The probability that $0 < a + b < a < a + b + c$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p + q$?

2007 India National Olympiad, 3

Tags: inequalities , quadratic , number theory

Let $ m$ and $ n$ be positive integers such that $ x^2 \minus{} mx \plus{}n \equal{} 0$ has real roots $ \alpha$ and $ \beta$. Prove that $ \alpha$ and $ \beta$ are integers [b]if and only if[/b] $ [m\alpha] \plus{} [m\beta]$ is the square of an integer. (Here $ [x]$ denotes the largest integer not exceeding $ x$)