Olympiad problems

2013 Tournament of Towns, 3

Denote by $[a, b]$ the least common multiple of $a$ and $b$. Let $n$ be a positive integer such that $[n, n + 1] > [n, n + 2] >...> [n, n + 35]$. Prove that $[n, n + 35] > [n,n + 36]$.

2011 India IMO Training Camp, 1

Tags: trigonometry , inequalities

Let $ABC$ be an acute-angled triangle. Let $AD,BE,CF$ be internal bisectors with $D, E, F$ on $BC, CA, AB$ respectively. Prove that \[\frac{EF}{BC}+\frac{FD}{CA}+\frac{DE}{AB}\geq 1+\frac{r}{R}\]

1990 Nordic, 2

Tags: inequalities , power

Let $a_1, a_2, . . . , a_n$ be real numbers. Prove $\sqrt[3]{a_1^3+ a_2^3+ . . . + a_n^3} \le \sqrt{a_1^2+ a_2^2+ . . . + a_n^2} $ (1) When does equality hold in (1)?

1966 IMO Longlists, 50

Tags: trigonometry , inequalities , geometry

For any quadrilateral with the side lengths $a,$ $b,$ $c,$ $d$ and the area $S,$ prove the inequality$S\leq \frac{a+c}{2}\cdot \frac{b+d}{2}.$

2017 Turkey Junior National Olympiad, 4

Tags: inequalities , factorization

If real numbers $a>b>1$ satisfy the inequality$$(ab+1)^2+(a+b)^2\leq 2(a+b)(a^2-ab+b^2+1)$$what is the minimum possible value of $\dfrac{\sqrt{a-b}}{b-1}$

1998 AIME Problems, 9

Tags: geometry , probability , analytic geometry , inequalities

Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ mintues. The probability that either one arrives while the other is in the cafeteria is $40 \%,$ and $m=a-b\sqrt{c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$

1974 IMO Longlists, 5

Tags: geometry , inequalities , 3d geometry

A straight cone is given inside a rectangular parallelepiped $B$, with the apex at one of the vertices, say $T$ , of the parallelepiped, and the base touching the three faces opposite to $T .$ Its axis lies at the long diagonal through $T.$ If $V_1$ and $V_2$ are the volumes of the cone and the parallelepiped respectively, prove that \[V_1 \leq \frac{\sqrt 3 \pi V_2}{27}.\]

1970 IMO, 2

Tags: inequalities , decimal representation , digit , algebra

We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.

1996 Romania National Olympiad, 1

Tags: algebra , inequalities

Let $a, b, c \in R,$ $a \ne 0$, such that $a$ and $4a+3b+2c$ have the same sign. Show that the equation $ax^2+bx+c=0$ cannot have both roots in the interval $(1,2)$.

2010 Balkan MO Shortlist, A3

Tags: algebra , inequalities

Let $a,b,c,d$ be positive real numbers. Prove that \[(\frac{a}{a+b})^{5}+(\frac{b}{b+c})^{5}+(\frac{c}{c+d})^{5}+(\frac{d}{d+a})^{5}\ge \frac{1}{8}\]

1946 Moscow Mathematical Olympiad, 115

Tags: trigonometry , inequalities , acute

Prove that if $\alpha$ and $\beta$ are acute angles and $\alpha$ < $\beta$ , then $\frac{tan \alpha}{\alpha} < \frac{tan \beta}{\beta} $

1953 Czech and Slovak Olympiad III A, 3

Tags: inequalities

Prove that the inequality $$\left(a_1+\cdots+a_n\right)\left(\frac{1}{a_1}+\cdots+\frac{1}{a_n}\right)\ge n^2$$ holds for any positive numbers $a_1,\ldots,a_n$ and determine when equality occurs.

2004 Romania Team Selection Test, 2

Tags: analytic geometry , inequalities , rectangle , geometry

Let $\{R_i\}_{1\leq i\leq n}$ be a family of disjoint closed rectangular surfaces with total area 4 such that their projections of the $Ox$ axis is an interval. Prove that there exist a triangle with vertices in $\displaystyle \bigcup_{i=1}^n R_i$ which has an area of at least 1. [Thanks Grobber for the correction]

2018 Belarusian National Olympiad, 11.7

Tags: inequalities , number theory

Consider the expression $M(n, m)=|n\sqrt{n^2+a}-bm|$, where $n$ and $m$ are arbitrary positive integers and the numbers $a$ and $b$ are fixed, moreover $a$ is an odd positive integer and $b$ is a rational number with an odd denominator of its representation as an irreducible fraction. Prove that there is [b]a)[/b] no more than a finite number of pairs $(n, m)$ for which $M(n, m)=0$; [b]b)[/b] a positive constant $C$ such that the inequality $M(n, m)\geqslant0$ holds for all pairs $(n, m)$ with $M(n, m)\ne 0$.

2017 Harvard-MIT Mathematics Tournament, 32

Tags: inequalities

Let $a$, $b$, $c$ be non-negative real numbers such that $ab + bc + ca = 3$. Suppose that \[a^3 b + b^3 c + c^3 a + 2abc(a + b + c) = \frac{9}{2}.\] What is the maximum possible value of $ab^3 + bc^3 + ca^3$?

2003 Iran MO (2nd round), 1

Tags: inequalities

Let $x,y,z\in\mathbb{R}$ and $xyz=-1$. Prove that: \[ x^4+y^4+z^4+3(x+y+z)\geq\frac{x^2}{y}+\frac{x^2}{z}+\frac{y^2}{x}+\frac{y^2}{z}+\frac{z^2}{x}+\frac{z^2}{y}. \]

2005 IMAR Test, 2

Tags: function , inequalities

Let $n \geq 3$ be an integer and let $a,b\in\mathbb{R}$ such that $nb\geq a^2$. We consider the set \[ X = \left\{ (x_1,x_2,\ldots,x_n)\in\mathbb{R}^n \mid \sum_{k=1}^n x_k = a, \ \sum_{k=1}^n x_k^2 = b \right\} . \] Find the image of the function $M: X\to \mathbb{R}$ given by \[ M(x_1,x_2,\ldots,x_n) = \max_{1\leq k\leq n} x_k . \] [i]Dan Schwarz[/i]

2013 Hong kong National Olympiad, 1

Tags: function , inequalities

Let $a,b,c$ be positive real numbers such that $ab+bc+ca=1$. Prove that \[\sqrt[4]{\frac{\sqrt{3}}{a}+6\sqrt{3}b}+\sqrt[4]{\frac{\sqrt{3}}{b}+6\sqrt{3}c}+\sqrt[4]{\frac{\sqrt{3}}{c}+6\sqrt{3}a}\le\frac{1}{abc}\] When does inequality hold?

1964 Kurschak Competition, 3

Tags: inequalities , algebra

Show that for any positive reals $w, x, y, z$ we have $$\sqrt{\frac{w^2 + x^2 + y^2 + z^2}{4}}\ge \sqrt[3]{ \frac{wxy + wxz + wyz + xyz}{4}}$$

2016 Sharygin Geometry Olympiad, P8

Tags: inequalities , geometry

Let $ABCDE$ be an inscribed pentagon such that $\angle B +\angle E = \angle C +\angle D$.Prove that $\angle CAD < \pi/3 < \angle A$. [i](Proposed by B.Frenkin)[/i]

1993 China Team Selection Test, 2

Tags: inequalities , algebra , function , quadratic

Let $n \geq 2, n \in \mathbb{N}$, $a,b,c,d \in \mathbb{N}$, $\frac{a}{b} + \frac{c}{d} < 1$ and $a + c \leq n,$ find the maximum value of $\frac{a}{b} + \frac{c}{d}$ for fixed $n.$

1992 India Regional Mathematical Olympiad, 1

Tags: polynomial , algebra , inequalities

Determine the set of integers $n$ for which $n^2+19n+92$ is a square.

2012 Online Math Open Problems, 7

Tags: probability , function , trigonometry , inequalities , relatively prime , number theory

Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m+n$. [i]Ray Li.[/i]

2017 Greece Junior Math Olympiad, 2

Tags: function , algebra , inequalities

Let $x,y,z$ is positive. Solve: $\begin{cases}{x\left( {6 - y} \right) = 9}\\ {y\left( {6 - z} \right) = 9}\\ {z\left( {6 - x} \right) = 9}\end{cases}$

2009 Jozsef Wildt International Math Competition, w. 24

Tags: geometry , triangle , inequalities

If $K$, $L$, $M$ denote the midpoints of the sides $AB$, $BC$, $CA$ in triangle $\triangle ABC$, then for all $P$ in the plane of triangle $\triangle ABC$, we have $$\frac{AB}{PK}+\frac{BC}{PL}+\frac{CA}{PM} \geq \frac{AB\cdot BC \cdot CA}{4\cdot PK\cdot PL\cdot PM}$$