Olympiad problems

1945 Moscow Mathematical Olympiad, 092

Tags: inequalities , algebra

Prove that for any positive integer $n\ge 2$ the following inequality holds: $$\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}>\frac{1}{2}$$

1957 Polish MO Finals, 4

Tags: algebra , inequalities

Prove that if $ a \geq 0 $ and $ b \geq 0 $, then $$ \sqrt{a^2 + b^2} \geq a + b - (2 - \sqrt{2}) \sqrt{ab}.$$

1990 IMO Longlists, 77

Tags: inequalities

Let $a, b, c \in \mathbb R$. Prove that \[(a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2) \geq (ab + bc + ca)^3.\] When does the equality hold?

1982 AMC 12/AHSME, 29

Tags: calculus , algebra , symmetry , inequalities , function , domain

Let $ x$,$ y$, and $ z$ be three positive real numbers whose sum is $ 1$. If no one of these numbers is more than twice any other, then the minimum possible value of the product $ xyz$ is $ \textbf{(A)}\ \frac{1}{32}\qquad \textbf{(B)}\ \frac{1}{36}\qquad \textbf{(C)}\ \frac{4}{125}\qquad \textbf{(D)}\ \frac{1}{127}\qquad \textbf{(E)}\ \text{none of these}$

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 $$\frac{1}{a_1+ a_2}+\frac{1}{a_2+ a_3}+\frac{1}{a_3+ a_1}>1$$

2007 China Girls Math Olympiad, 3

Tags: inequalities , induction

Let $ n$ be an integer greater than $ 3$, and let $ a_1, a_2, \cdots, a_n$ be non-negative real numbers with $ a_1 \plus{} a_2 \plus{} \cdots \plus{} a_n \equal{} 2$. Determine the minimum value of \[ \frac{a_1}{a_2^2 \plus{} 1}\plus{} \frac{a_2}{a^2_3 \plus{} 1}\plus{} \cdots \plus{} \frac{a_n}{a^2_1 \plus{} 1}.\]

2024 Regional Olympiad of Mexico West, 6

Tags: inequalities , geometric inequality , geometry

We say that a triangle of sides $a,b,c$ is [i] virtual[/i] if such measures satisfy $$\begin{cases} a^{2024}+b^{2024}> c^{2024},\\ b^{2024}+c^{2024}> a^{2024},\\ c^{2024}+a^{2024}> b^{2024} \end{cases}$$ Find the number of ordered triples $(a,b,c)$ such that $a,b,c$ are integers between $1$ and $2024$ (inclusive) and $a,b,c$ are the sides of a [i]virtual [/i] triangle.

2011 AMC 12/AHSME, 23

Tags: inequalities , linear algebra , algebra , polynomial , function , matrix , induction

Let $f(z)=\frac{z+a}{z+b}$ and $g(z)=f(f(z))$, where $a$ and $b$ are complex numbers. Suppose that $|a|=1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $|b|$? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \sqrt{2}-1 \qquad \textbf{(C)}\ \sqrt{3}-1 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

2019 New Zealand MO, 8

Tags: algebra , inequalities

Suppose that $x_1, x_2, x_3, . . . x_n$ are real numbers between $0$ and $ 1$ with sum $s$. Prove that $$\prod_{i=1}^{n} \frac{x_i}{s + 1 - x_i} + \prod_{i=1}^{n} (1 - x_i) \le 1.$$

2012 JBMO TST - Turkey, 3

Tags: function , inequalities

Show that for all real numbers $x, y$ satisfying $x+y \geq 0$ \[ (x^2+y^2)^3 \geq 32(x^3+y^3)(xy-x-y) \]

2011 AIME Problems, 13

Tags: trig identities , symmetry , inequalities , circumcircle , trigonometry , law of sines , geometry

Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP>CP$. Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB=12$ and $\angle O_1 P O_2 = 120^\circ$, then $AP=\sqrt{a}+\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$.

2000 Czech and Slovak Match, 1

Tags: geometry , algebra , inequalities , geometric inequalities

$a,b,c$ are positive real numbers which satisfy $5abc>a^3+b^3+c^3$. Prove that $a,b,c$ can form a triangle.

2013 Macedonian Team Selection Test, Problem 4

Tags: inequalities

Let $a>0,b>0,c>0$ and $a+b+c=1$. Show the inequality $$\frac{a^4+b^4}{a^2+b^2}+\frac{b^3+c^3}{b+c} + \frac{2a^2+b^2+2c^2}{2} \geq \frac{1}{2}$$

2011 China Team Selection Test, 1

Tags: inequalities

Let $n\geq 3$ be an integer. Find the largest real number $M$ such that for any positive real numbers $x_1,x_2,\cdots,x_n$, there exists an arrangement $y_1,y_2,\cdots,y_n$ of real numbers satisfying \[\sum_{i=1}^n \frac{y_i^2}{y_{i+1}^2-y_{i+1}y_{i+2}+y_{i+2}^2}\geq M,\] where $y_{n+1}=y_1,y_{n+2}=y_2$.

2017 Czech And Slovak Olympiad III A, 2

Tags: inequalities , geometric inequality

Find all pairs of real numbers $k, l$ such that inequality $ka^2 + lb^2> c^2$ applies to the lengths of sides $a, b, c$ of any triangle.

2009 India IMO Training Camp, 7

Tags: geometry , inequalities

Let $ P$ be any point in the interior of a $ \triangle ABC$.Prove That $ \frac{PA}{a}\plus{}\frac{PB}{b}\plus{}\frac{PC}{c}\ge \sqrt{3}$.

1987 Greece National Olympiad, 3

Tags: algebra , inequalities

Prova that for any real $a$, expresssion $A=(a-1)(a-3)(a-4)(a-6)+10$ is always positive. What is the minimum value that expression $A$ can take and for which values of $a$?

1995 All-Russian Olympiad Regional Round, 10.5

Tags: inequalities , algebra , trinomial

Consider all quadratic functions $f(x) = ax^2 +bx+c$ with $a < b$ and $f(x) \ge 0$ for all $x$. What is the smallest possible value of the expression $\frac{a+b+c}{b-a}$?

2007 ITest, 38

Tags: logarithm , floor function , 3d geometry , inequalities , geometry

Find the largest positive integer that is equal to the cube of the sum of its digits.

2022 Taiwan TST Round 1, A

Tags: inequalities

Let $a_1, a_2, a_3, \ldots$ be a sequence of reals such that there exists $N\in\mathbb{N}$ so that $a_n=1$ for all $n\geq N$, and for all $n\geq 2$ we have \[a_{n}\leq a_{n-1}+2^{-n}a_{2n}.\] Show that $a_k>1-2^{-k}$ for all $k\in\mathbb{N}$. [i] Proposed by usjl[/i]

2007 Junior Balkan Team Selection Tests - Romania, 2

Tags: calculus , inequalities

Let $x, y, z \ge 0$ be real numbers. Prove that: \[\frac{x^{3}+y^{3}+z^{3}}{3}\ge xyz+\frac{3}{4}|(x-y)(y-z)(z-x)| .\] [hide="Additional task"]Find the maximal real constant $\alpha$ that can replace $\frac{3}{4}$ such that the inequality is still true for any non-negative $x,y,z$.[/hide]

2002 Irish Math Olympiad, 5

Tags: inequalities

Let $ 0<a,b,c<1$. Prove the inequality: $ \frac{a}{1\minus{}a}\plus{}\frac{b}{1\minus{}b}\plus{}\frac{c}{1\minus{}c} \ge \frac {3 \sqrt[3]{abc}}{1\minus{} \sqrt[3]{abc}}.$ Determine the cases of equality.

1997 China Team Selection Test, 1

Tags: induction , inequalities , vieta , algebra , polynomial

Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions: [b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0$; [b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}$; [b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.

2014 South africa National Olympiad, 5

Tags: inequalities , combinatorics

Let $n > 1$ be an integer. An $n \times n$-square is divided into $n^2$ unit squares. Of these unit squares, $n$ are coloured green and $n$ are coloured blue, and all remaining ones are coloured white. Are there more such colourings for which there is exactly one green square in each row and exactly one blue square in each column; or colourings for which there is exactly one green square and exactly one blue square in each row?

2019 LIMIT Category A, Problem 3

Tags: inequalities , geometry , geometric inequalities

In $\triangle ABC$, $\left|\overline{AB}\right|=\left|\overline{AC}\right|$, $D$ is the foot of the perpendicular from $C$ to $AB$ and $E$ the foot of the perpendicular from $B$ to $AC$, then $\textbf{(A)}~\left|\overline{BC}\right|^3>\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(B)}~\left|\overline{BC}\right|^3<\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(C)}~\left|\overline{BC}\right|^3=\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(D)}~\text{None of the above}$