Olympiad problems

1897 Eotvos Mathematical Competition, 2

Tags: inequalities

Show that, if $\alpha$, $\beta$ and $\gamma$ are angles of an arbitrary triangle, $$\text{sin } \frac{\alpha}{2} \text{ sin } \frac{\beta}{2} \text{ sin } \frac{\gamma}{2} < \frac14.$$.

2004 Croatia National Olympiad, Problem 2

Tags: inequalities , geometric inequality

If $a,b,c$ are the sides and $\alpha,\beta,\gamma$ the corresponding angles of a triangle, prove the inequality $$\frac{\cos\alpha}{a^3}+\frac{\cos\beta}{b^3}+\frac{\cos\gamma}{c^3}\ge\frac3{2abc}.$$

2014 Mediterranean Mathematics Olympiad, 1

Tags: induction , inequalities , triangle inequality , algebra , n-variable inequality

Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that $ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$ (Proposed by Gerhard Woeginger, Austria)

1998 Belarus Team Selection Test, 4

Tags: inequalities , trigonometry , algebra

Prove the inequality $$\sum_{k=1}^{n}\frac{\sin (k+1)x}{\sin kx}< 2\frac{\cos x}{\sin^2x}$$ where $0 < nx < \pi/2$, $n \in N$.

2015 Thailand TSTST, 1

Tags: inequalities

Let $x, y, z$ be positive real numbers satisfying $x + y + z =\frac{2}{x+y}+\frac{2}{y+z}+\frac{2}{z+x}$. Prove that $$\frac{3}{2}\leq\frac{3}{\sqrt[3]{xyz}(\sqrt[3]{xyz}+1)}\leq\frac{1}{x(y+1)}+\frac{1}{y(z+1)}+\frac{1}{z(x+1)}.$$

2018 China Western Mathematical Olympiad, 7

Tags: number theory , least common multiple , inequalities

Let $p$ and $c$ be an prime and a composite, respectively. Prove that there exist two integers $m,n,$ such that $$0<m-n<\frac{\textup{lcm}(n+1,n+2,\cdots,m)}{\textup{lcm}(n,n+1,\cdots,m-1)}=p^c.$$

2024 Kyiv City MO Round 2, Problem 1

Tags: algebra , inequalities

Prove that for any real numbers $x, y, z$ at least one of numbers $x^2 + y + \frac{1}{4}, y^2 + z + \frac{1}{4}, z^2 + x + \frac{1}{4}$ is nonnegative. [i]Proposed by Oleksii Masalitin[/i]

2011 ISI B.Math Entrance Exam, 5

Tags: inequalities , number theory

Consider a sequence denoted by $F_n$ of non-square numbers . $F_1=2$,$F_2=3$,$F_3=5$ and so on . Now , if $m^2\leq F_n<(m+1)^2$ . Then prove that $m$ is the integer closest to $\sqrt{n}$.

2003 Putnam, 2

Tags: inequalities , putnam inequalities

Let $a_1, a_2, \cdots , a_n$ and $b_1, b_2,\cdots, b_n$ be nonnegative real numbers. Show that \[(a_1a_2 \cdots a_n)^{1/n}+ (b_1b_2 \cdots b_n)^{1/n} \le ((a_1 + b_1)(a_2 + b_2) \cdots (a_n + b_n))^{1/n}\]

2012 Irish Math Olympiad, 5

Tags: inequalities

(a) Show that if $x$ and $y$ are positive real numbers, then $$(x+y)^5\ge 12xy(x^3+y^3)$$ (b) Prove that the constant $12$ is the best possible. In other words, prove that for any $K>12$ there exist positive real numbers $x$ and $y$ such that $$(x+y)^5<Kxy(x^3+y^3)$$

2012 Uzbekistan National Olympiad, 4

Tags: inequalities , linear algebra , matrix

Given $a,b$ and $c$ positive real numbers with $ab+bc+ca=1$. Then prove that $\frac{a^3}{1+9b^2ac}+\frac{b^3}{1+9c^2ab}+\frac{c^3}{1+9a^2bc} \geq \frac{(a+b+c)^3}{18}$

1996 Estonia National Olympiad, 1

Tags: inequalities , algebra

Prove that for any positive numbers $x,y$ it holds that $x^xy^y \ge x^yy^x$.

2008 All-Russian Olympiad, 8

Tags: induction , algorithm , linear algebra , matrix , combinatorics , inequalities

We are given $ 3^{2k}$ apparently identical coins,one of which is fake,being lighter than the others. We also dispose of three apparently identical balances without weights, one of which is broken (and yields outcomes unrelated to the actual situations). How can we find the fake coin in $ 3k\plus{}1$ weighings?

2017 ELMO Problems, 6

Tags: inequalities , algebra

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$: (i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$ (ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$ [i]Proposed by Ashwin Sah[/i]

1978 All Soviet Union Mathematical Olympiad, 264

Tags: algebra , inequalities

Given $0 < a \le x_1\le x_2\le ... \le x_n \le b$. Prove that $$(x_1+x_2+...+x_n)\left ( \frac{1}{x_1}+ \frac{1}{x_2}+...+ \frac{1}{x_n}\right)\le \frac{(a+b)^2}{4ab}n^2$$

2001 National High School Mathematics League, 10

Tags: inequalities

The solution to inequality $\left|\frac{1}{\log_{\frac{1}{2}}x}+2\right|>\frac{3}{2}$ is________(express answer with a set).

2012 Mediterranean Mathematics Olympiad, 3

Tags: inequalities , linear algebra , matrix , induction , combinatorics

Consider a binary matrix $M$(all entries are $0$ or $1$) on $r$ rows and $c$ columns, where every row and every column contain at least one entry equal to $1$. Prove that there exists an entry $M(i,j) = 1$, such that the corresponding row-sum $R(i)$ and column-sum $C(j)$ satisfy $r R(i)\ge c C(j)$. (Proposed by Gerhard Woeginger, Austria)

2009 China Northern MO, 5

Tags: algebra , inequalities

Assume : $x,y,z>0$ , $ x^2+y^2+z^2 = 3 $ . Prove the following inequality : $${\frac{x^{2009}-2008(x-1)}{y+z}+\frac{y^{2009}-2008(y-1)}{x+z}+\frac{z^{2009}-2008(z-1)}{x+y}\ge\frac{1}{2}(x+y+z)}$$

1955 Polish MO Finals, 4

Tags: trigonometry , inequalities , algebra

Prove that $$ \sin^2 \alpha + \sin^2 \beta \geq \sin \alpha \sin \beta + \sin \alpha + \sin \beta - 1.$$

2009 Indonesia TST, 2

Tags: inequalities

Let $ x_1,x_2,\ldots,x_n$ be real numbers greater than 1. Show that \[ \frac{x_1x_2}{x_3}\plus{}\frac{x_2x_3}{x_4}\plus{}\cdots\plus{}\frac{x_nx_1}{x_2}\ge4n\] and determine when the equality holds.

2004 Bulgaria Team Selection Test, 2

Tags: inequalities , algebra , inequality

Prove that if $a,b,c \ge 1$ and $a+b+c=9$, then $\sqrt{ab+bc+ca} \le \sqrt{a} +\sqrt{b} + \sqrt{c}$

1990 IMO Longlists, 52

Tags: inequalities

Let real numbers $a_1, a_2, \ldots, a_n$ satisfy $0 < a_i \leq a, \ i = 1, 2, \ldots, n$. Prove that (i) If $n = 4$, then \[\frac 1a \sum_{i=1}^4 a_i - \frac{a_1a_2 + a_2a_3 + a_3 a_4 + a_4 a_1}{a^2} \leq 2.\] (ii) If $n = 6$, then \[\frac 1a \sum_{i=1}^6 a_i - \frac{a_1a_2 + a_2a_3 + \cdots + a_5 a_6 + a_6 a_1}{a^2} \leq 3.\]

2005 China Team Selection Test, 1

Tags: rearrangement inequality , combinatorics , inequalities

Let $a_{1}$, $a_{2}$, …, $a_{6}$; $b_{1}$, $b_{2}$, …, $b_{6}$ and $c_{1}$, $c_{2}$, …, $c_{6}$ are all permutations of $1$, $2$, …, $6$, respectively. Find the minimum value of $\sum_{i=1}^{6}a_{i}b_{i}c_{i}$.

2022 CHMMC Winter (2022-23), 3

Tags: algebra , complex number , inequalities

Suppose that $a,b,c$ are complex numbers with $a+b+c = 0$, $|abc| = 1$, $|b| = |c|$, and $$\frac{9-\sqrt{33}}{48} \le \cos^2 \left( arg \left( \frac{b}{a} \right) \right)\le \frac{9+\sqrt{33}}{48} .$$ Find the maximum possible value of $|-a^6+b^6+c^6|$.

Ukrainian TYM Qualifying - geometry, XI.15

Tags: geometry , incenter , inequalities , circumcircle

Let $I$ be the point of intersection of the angle bisectors of the $\vartriangle ABC$, $W_1,W_2,W_3$ be point of intersection of lines $AI, BI, CI$ with the circle circumscribed around the triangle, $r$ and $R$ be the radii of inscribed and circumscribed circles respectively. Prove the inequality $$IW_1+ IW_2 + IW_3\ge 2R + \sqrt{2Rr.}$$