Olympiad problems

2004 Gheorghe Vranceanu, 1

Tags: sequence , inequalities

Find all infinite sequences of real numbers $ \left( a_n \right)_{n\ge 1} $ that verify, for any natural number $ n, $ the inequalities $$ \frac{1}{2\sqrt{a_{n+1}}} <\sqrt{n+1} -\sqrt{n} <\frac{1}{ 2\sqrt{a_n}} . $$

2012 Israel National Olympiad, 3

Tags: inequality , symmetric inequality , algebra , inequalities

Let $a,b,c$ be real numbers such that $a^3(b+c)+b^3(a+c)+c^3(a+b)=0$. Prove that $ab+bc+ca\leq0$.

2008 Hong Kong TST, 2

Tags: limit , inequalities

Let $ a$, $ b$, $ c$ be the three sides of a triangle. Determine all possible values of \[ \frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}\]

2001 IMC, 2

Tags: convergence , inequalities , real analysis

Let $a_{0}=\sqrt{2}, b_{0}=2,a_{n+1}=\sqrt{2-\sqrt{4-a_{n}^{2}}},b_{n+1}=\frac{2b_{n}}{2+\sqrt{4+b_{n}^{2}}}$. a) Prove that the sequences $(a_{n})$ and $(b_{n})$ are decreasing and converge to $0$. b) Prove that the sequence $(2^{n}a_{n})$ is increasing, the sequence $(2^{n}b_{n})$ is decreasing and both converge to the same limit. c) Prove that there exists a positive constant $C$ such that for all $n$ the following inequality holds: $0 <b_{n}-a_{n} <\frac{C}{8^{n}}$.

1992 APMO, 5

Tags: inequalities , algebra

Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.

2023 239 Open Mathematical Olympiad, 8

Tags: derivative , inequalities , algebra

Let $r\geqslant 0$ be a real number and define $f(x)=1/(1+x^2)^r$. Prove that \[|f^{(k)}(x)|\leqslant\frac{2r\cdot(2r+1)\cdots(2r+k-1)}{(1+x^2)^{r+k/2}},\]for every natural number $k{}$. Here, $f^{(k)}(x)$ denotes the $k^{\text{th}}$ derivative of $f$.

2012 Bogdan Stan, 1

Tags: inequalities , algebra

Let be three real numbers $ a,b,c\in [0,1] $ satisfying the condition $ ab+bc+ca=1. $ Prove that $$ a^2+b^2+c^2\le 2, $$ and determine the cases in which equality is attained.

2003 Irish Math Olympiad, 5

Tags: algebra , inequalities , function

show that thee is no function f definedonthe positive real numbes such that : $f(y) > (y-x)f(x)^2$

2016 Germany Team Selection Test, 2

Tags: inequalities , integer , n-variable inequality , number theory

The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]

2014 Peru IMO TST, 8

Tags: inequalities

Let $x, y, z$ be real numbers such that $$\displaystyle{\begin{cases} x^2+y^2+z^2+(x+y+z)^2=9 \\ xyz \leq \frac{15}{32} \end{cases}} $$ Find the maximum possible value of $x.$

2011 National Olympiad First Round, 35

Tags: calculus , derivative , function , inequalities

Which of these has the smallest maxima on positive real numbers? $\textbf{(A)}\ \frac{x^2}{1+x^{12}} \qquad\textbf{(B)}\ \frac{x^3}{1+x^{11}} \qquad\textbf{(C)}\ \frac{x^4}{1+x^{10}} \qquad\textbf{(D)}\ \frac{x^5}{1+x^{9}} \qquad\textbf{(E)}\ \frac{x^6}{1+x^{8}}$

2003 Turkey MO (2nd round), 3

Tags: inequalities , function

Let $ f: \mathbb R \rightarrow \mathbb R$ be a function such that $ f(tx_1\plus{}(1\minus{}t)x_2)\leq tf(x_1)\plus{}(1\minus{}t)f(x_2)$ for all $ x_1 , x_2 \in \mathbb R$ and $ t\in (0,1)$. Show that $ \sum_{k\equal{}1}^{2003}f(a_{k\plus{}1})a_k \geq \sum_{k\equal{}1}^{2003}f(a_k)a_{k\plus{}1}$ for all real numbers $ a_1,a_2,...,a_{2004}$ such that $ a_1\geq a_2\geq ... \geq a_{2003}$ and $ a_{2004}\equal{}a_1$

1995 Belarus Team Selection Test, 3

Tags: inequalities , algebra

If $0<a,b<1$ and $p,q\geq 0 ,\ p+q=1$ are real numbers , then prove that: \[a^pb^q+(1-a)^p(1-b)^q\le 1\]

2007 China Northern MO, 1

Tags: trigonometry , inequalities

Let $ \alpha$, $ \beta$ be acute angles. Find the maximum value of \[ \frac{\left(1-\sqrt{\tan\alpha\tan\beta}\right)^{2}}{\cot\alpha+\cot\beta}\]

2012 Polish MO Finals, 6

Tags: inequality , algebra , inequalities

Show that for any positive real numbers $a, b, c$ true is inequality: $\left(\frac{a - b}{c}\right)^2 + \left(\frac{b - c}{a}\right)^2 + \left(\frac{c - a}{b}\right)^2 \ge 2\sqrt{2}\left(\frac{a - b}{c} + \frac{b - c}{a} + \frac{c - a}{b} \right)$.

2003 National High School Mathematics League, 7

Tags: inequalities

The solution set for inequality $|x|^3-2x^2-4|x|+3<0$ is________.

2012 Romania National Olympiad, 3

Tags: calculus , function , inequalities

[color=darkred]Let $a,b\in\mathbb{R}$ with $0<a<b$ . Prove that: [list] [b]a)[/b] $2\sqrt {ab}\le\frac {x+y+z}3+\frac {ab}{\sqrt[3]{xyz}}\le a+b$ , for $x,y,z\in [a,b]\, .$ [b]b)[/b] $\left\{\frac {x+y+z}3+\frac {ab}{\sqrt[3]{xyz}}\, |\, x,y,z\in [a,b]\right\}=[2\sqrt {ab},a+b]\, .$ [/list][/color]

2021-IMOC qualification, A3

Tags: inequalities , function , functional inequality , functional , algebra

Find all injective function $f: N \to N$ satisfying that for all positive integers $m,n$, we have: $f(n(f(m)) \le nm$

2016 Korea USCM, 7

Tags: holder inequality , integral , inequalities , real analysis , function

$M$ is a postive real and $f:[0,\infty)\to[0,M]$ is a continuous function such that $$\int_0^\infty (1+x)f(x) dx<\infty$$ Then, prove the following inequality. $$\left(\int_0^\infty f(x) dx \right)^2 \leq 4M \int_0^\infty x f(x) dx$$ (@below, Thank you. I fixed.)

2006 Lithuania National Olympiad, 3

Tags: algebra , inequalities

Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.

2004 Korea Junior Math Olympiad, 1

Tags: geometry , inequalities

For positive reals $a_1, a_2, ..., a_5$ such that $a^2_1+a^2_2+...+a^2_5=2$, consider five squares with sides $a_1, a_2, ..., a_5$ respectively. Show that these squares can be placed inside (including boundaries) a square with side length of $2$ so that the square themselves do not overlap each other.

1978 All Soviet Union Mathematical Olympiad, 257

Tags: sequence , algebra , inequalities

Prove that there exists such an infinite sequence $\{x_i\}$, that for all $m$ and all $k$ ($m\ne k$) holds the inequality $$|x_m-x_k|>1/|m-k|$$

2010 Germany Team Selection Test, 2

Tags: reflection , geometry , incenter , inequalities

Let $ABC$ be a triangle with incenter $I$ and let $X$, $Y$ and $Z$ be the incenters of the triangles $BIC$, $CIA$ and $AIB$, respectively. Let the triangle $XYZ$ be equilateral. Prove that $ABC$ is equilateral too. [i]Proposed by Mirsaleh Bahavarnia, Iran[/i]

2020 Jozsef Wildt International Math Competition, W35

Tags: geometric inequalities , inequalities , geometry

In all triangles $ABC$ does it hold: $$(b^n+c^p)\tan^{n+p}\frac A2+(c^n+a^p)\tan^{n+p}\frac B2+(a^n+b^p)\tan^{n+p}\frac C2\ge6\sqrt{\left(\frac{4r^2}{R\sqrt3}\right)^{n+p}}$$ where $n,p\in(0,\infty)$. [i]Proposed by Nicolae Papacu[/i]

2001 Moldova National Olympiad, Problem 8

Tags: algebra , inequalities

If $a_1,a_2,\ldots,a_n$ are positive real numbers, prove the inequality $$\dfrac1{\dfrac1{1+a_1}+\dfrac1{1+a_2}+\ldots+\dfrac1{1+a_n}}-\dfrac1{\dfrac1{a_1}+\dfrac1{a_2}+\ldots+\dfrac1{a_n}}\ge\frac1n.$$