Olympiad problems

2005 Pan African, 1

Tags: inequalities

For any positive real numbers $a,b$ and $c$, prove: \[ \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} \geq \dfrac{2}{a+b} + \dfrac{2}{b+c} + \dfrac{2}{c+a} \geq \dfrac{9}{a+b+c} \]

2018 239 Open Mathematical Olympiad, 8-9.6

Tags: number theory , inequalities

Petya wrote down 100 positive integers $n, n+1, \ldots, n+99$, and Vasya wrote down 99 positive integers $m, m-1, \ldots, m-98$. It turned out that for each of Petya's numbers, there is a number from Vasya that divides it. Prove that $m>n^3/10, 000, 000$. [i]Proposed by Ilya Bogdanov[/i]

2012 China Northern MO, 2

Tags: inequalities , algebra

Positive integers $x_1,x_2,...,x_n$ ($n \in N_+$) satisfy $x_1^2 +x_2^2+...+x_n^2=111$, find the maximum possible value of $S =\frac{x_1 +x_2+...+x_n}{n}$.

2003 Alexandru Myller, 1

Tags: complex number , inequalities , algebra

Let be a natural number $ n, $ a positive real number $ \lambda , $ and a complex number $ z. $ Prove the following inequalities. $$ 0\le -\lambda +\frac{1}{n}\sum_{\stackrel{w\in\mathbb{C}}{w^n=1 }} \left| z-\lambda w \right|\le |z| $$ [i]Gheorghe Iurea[/i]

2017 Hanoi Open Mathematics Competitions, 13

Tags: inequalities , distance , geometric inequalities , geometry

Let $ABC$ be a triangle. For some $d>0$ let $P$ stand for a point inside the triangle such that $|AB| - |P B| \ge d$, and $|AC | - |P C | \ge d$. Is the following inequality true $|AM | - |P M | \ge d$, for any position of $M \in BC $?

2016 Taiwan TST Round 1, 2

Tags: inequalities

Let $a,b,c$ be nonnegative real numbers such that $(a+b)(b+c)(c+a) \neq0$. Find the minimum of $(a+b+c)^{2016}(\frac{1}{a^{2016}+b^{2016}}+\frac{1}{b^{2016}+c^{2016}}+\frac{1}{c^{2016}+a^{2016}})$.

1989 Spain Mathematical Olympiad, 6

Tags: combinatorics , inequalities

Prove that among any seven real numbers there exist two,$ a$ and $b$, such that $\sqrt3|a-b|\le |1+ab|$. Give an example of six real numbers not having this property.

2022 Dutch IMO TST, 3

Tags: inequalities , algebra

For real numbers $x$ and $y$ we define $M(x, y)$ to be the maximum of the three numbers $xy$, $(x- 1)(y - 1)$, and $x + y - 2xy$. Determine the smallest possible value of $M(x, y)$ where $x$ and $y$ range over all real numbers satisfying $0 \le x, y \le 1$.

2014 Kyiv Mathematical Festival, 2

Tags: inequalities

Let $x,y,z$ be real numbers such that $(x-z)(y-z)=x+y+z-3.$ Prove that $x^2+y^2+z^2\ge3.$

1986 IMO Longlists, 78

Tags: inequalities , trigonometry , function , geometry

If $T$ and $T_1$ are two triangles with angles $x, y, z$ and $x_1, y_1, z_1$, respectively, prove the inequality \[\frac{\cos x_1}{\sin x}+\frac{\cos y_1}{\sin y}+\frac{\cos z_1}{\sin z} \leq \cot x+\cot y+\cot z.\]

2014 Contests, 1a

Tags: algebra , inequalities

Assume that $x, y \ge 0$. Show that $x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$.

2013 Estonia Team Selection Test, 3

Tags: inequalities , sum , algebra

Let $x_1,..., x_n$ be non-negative real numbers, not all of which are zeros. (i) Prove that $$1 \le \frac{\left(x_1+\frac{x_2}{2}+\frac{x_3}{3}+...+\frac{x_n}{n}\right)(x_1+2x_2+3x_3+...+nx_n)}{(x_1+x_2+x_3+...+x_n)^2} \le \frac{(n+1)^2}{4n}$$ (ii) Show that, for each $n > 1$, both inequalities can hold as equalities.

2013 Abels Math Contest (Norwegian MO) Final, 1b

Tags: sequence , algebra , inequalities

The sequence $a_1, a_2, a_3,...$ is defined so that $a_1 = 1$ and $a_{n+1} =\frac{a_1 + a_2 + ...+ a_n}{n}+1$ for $n \ge 1$. Show that for every positive real number $b$ we can find $a_k$ so that $a_k < bk$.

2012 Abels Math Contest (Norwegian MO) Final, 4b

Tags: algebra , inequalities

Positive numbers $b_1, b_2,..., b_n$ are given so that $b_1 + b_2 + ...+ b_n \le 10$. Further, $a_1 = b_1$ and $a_m = sa_{m-1} + b_m$ for $m > 1$, where $0 \le s < 1$. Show that $a^2_1 + a^2_2 + ... + a^2_n \le \frac{100}{1 - s^2} $

2009 Brazil Team Selection Test, 3

Tags: polynomial , inequalities , monic polynomial , algebra

Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be a monic polynomial of degree $4$. It is known that all the roots of $P$ are real, distinct and belong to the interval $[-1, 1]$. (a) Prove that $P(x) > -4$ for all real $x$. (b) Find the highest value of the real constant $k$ such that $P(x) > k$ for every real $x$ and for every polynomial $P(x)$ satisfying the given conditions.

1975 AMC 12/AHSME, 3

Tags: inequalities

Which of the following inequalities are satisfied for all real numbers $ a$, $ b$, $ c$, $ x$, $ y$, $ z$ which satisfy the conditions $ x < a$, $ y < b$, and $ z < c$? $ \text{I}. \ xy \plus{} yz \plus{} zx < ab \plus{} bc \plus{} ca$ $ \text{II}. \ x^2 \plus{} y^2 \plus{} z^2 < a^2 \plus{} b^2 \plus{} c^2$ $ \text{III}. \ xyz < abc$ $ \textbf{(A)}\ \text{None are satisfied.} \qquad \textbf{(B)}\ \text{I only} \qquad \textbf{(C)}\ \text{II only} \qquad$ $ \textbf{(D)}\ \text{III only} \qquad \textbf{(E)}\ \text{All are satisfied.}$

1997 Junior Balkan MO, 3

Tags: inequalities

Let $ABC$ be a triangle and let $I$ be the incenter. Let $N$, $M$ be the midpoints of the sides $AB$ and $CA$ respectively. The lines $BI$ and $CI$ meet $MN$ at $K$ and $L$ respectively. Prove that $AI+BI+CI>BC+KL$. [i]Greece[/i]

2020 Baltic Way, 3

Tags: sequence , inequalities

A real sequence $(a_n)_{n=0}^\infty$ is defined recursively by $a_0 = 2$ and the recursion formula $$ a_{n} = \begin{dcases} a_{n-1}^2 & \text{if $a_{n-1}<\sqrt3$} \\ \frac{a_{n-1}^2}{3} & \text{if $a_{n-1}\geq\sqrt 3$.} \end{dcases} $$ Another real sequence $(b_n)_{n=1}^\infty$ is defined in terms of the first by the formula $$ b_{n} = \begin{dcases} 0 & \text{if $a_{n-1}<\sqrt3$} \\ \frac{1}{2^{n}} & \text{if $a_{n-1}\geq\sqrt 3$,} \end{dcases} $$ valid for each $n\geq 1$. Prove that $$ b_1 + b_2 + \cdots + b_{2020} < \frac23. $$

2000 Macedonia National Olympiad, 3

Tags: inequalities , circumcircle , power of a point , geometry

In a triangle with sides $a,b,c,t_a,t_b,t_c$ are the corresponding medians and $D$ the diameter of the circumcircle. Prove that \[\frac{a^2+b^2}{t_c}+\frac{b^2+c^2}{t_a}+\frac{c^2+a^2}{t_b}\le 6D\]

2010 CHMMC Fall, 8

Tags: inequalities , algebra

Rachel writes down a simple inequality: one $2$-digit number is greater than another. Matt is sitting across from Rachel and peeking at her paper. If Matt, reading upside down, sees a valid inequality between two $2$-digit numbers, compute the number of different inequalities that Rachel could have written. Assume that each digit is either a $1, 6, 8$, or $9$.

2023 Czech and Slovak Olympiad III A., 2

Tags: combinatorics , inequalities

Let $n$ be a positive integer, where $n \geq 3$ and let $a_1, a_2, ..., a_n$ be the lengths of sides of some $n$-gon. Prove that $$a_1 + a_2 + ... + a_n \geq \sqrt{2 \cdot (a_1^2 + a_2^2 + ... + a_n^2)} $$

2004 Romania Team Selection Test, 16

Tags: inequalities , geometry , circumcircle , geometric transformation , homothety , parallelogram , complex number

Three circles $\mathcal{K}_1$, $\mathcal{K}_2$, $\mathcal{K}_3$ of radii $R_1,R_2,R_3$ respectively, pass through the point $O$ and intersect two by two in $A,B,C$. The point $O$ lies inside the triangle $ABC$. Let $A_1,B_1,C_1$ be the intersection points of the lines $AO,BO,CO$ with the sides $BC,CA,AB$ of the triangle $ABC$. Let $ \alpha = \frac {OA_1}{AA_1} $, $ \beta= \frac {OB_1}{BB_1} $ and $ \gamma = \frac {OC_1}{CC_1} $ and let $R$ be the circumradius of the triangle $ABC$. Prove that \[ \alpha R_1 + \beta R_2 + \gamma R_3 \geq R. \]

2009 China Girls Math Olympiad, 3

Tags: analytic geometry , inequalities

Let $ n$ be a given positive integer. In the coordinate set, consider the set of points $ \{P_{1},P_{2},...,P_{4n\plus{}1}\}\equal{}\{(x,y)|x,y\in \mathbb{Z}, xy\equal{}0, |x|\le n, |y|\le n\}.$ Determine the minimum of $ (P_{1}P_{2})^{2} \plus{} (P_{2}P_{3})^{2} \plus{}...\plus{} (P_{4n}P_{4n\plus{}1})^{2} \plus{} (P_{4n\plus{}1}P_{1})^{2}.$

2008 Tournament Of Towns, 3

Tags: ratio , geometry , inequalities

In his triangle $ABC$ Serge made some measurements and informed Ilias about the lengths of median $AD$ and side $AC$. Based on these data Ilias proved the assertion: angle $CAB$ is obtuse, while angle $DAB$ is acute. Determine a ratio $AD/AC$ and prove Ilias' assertion (for any triangle with such a ratio).

1974 Kurschak Competition, 3

Tags: algebra , polynomial , inequalities

Let $$p_k(x) = 1 -x + \frac{x^2}{2! } - \frac{x^3}{3!}+ ... + \frac{(-x)^{2k}}{(2k)!}$$ Show that it is non-negative for all real $x$ and all positive integers $k$.