Olympiad problems

2017 Harvard-MIT Mathematics Tournament, 7

Tags: inequalities

Determine the largest real number $c$ such that for any $2017$ real numbers $x_1, x_2, \dots, x_{2017}$, the inequality $$\sum_{i=1}^{2016}x_i(x_i+x_{i+1})\ge c\cdot x^2_{2017}$$ holds.

2008 China Team Selection Test, 2

Tags: polynomial , inequalities , algebra

Let $ x,y,z$ be positive real numbers, show that $ \frac {xy}{z} \plus{} \frac {yz}{x} \plus{} \frac {zx}{y} > 2\sqrt [3]{x^3 \plus{} y^3 \plus{} z^3}.$

1987 IMO Shortlist, 15

Tags: inequalities , pigeonhole principle , linear algebra , inequality system

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$. [i](IMO Problem 3)[/i] [i]Proposed by Germany, FR[/i]

2005 All-Russian Olympiad, 3

Tags: inequalities , algebra

Given three reals $a_1,\,a_2,\,a_3>1,\,S=a_1+a_2+a_3$. Provided ${a_i^2\over a_i-1}>S$ for every $i=1,\,2,\,3$ prove that \[\frac{1}{a_1+a_2}+\frac{1}{a_2+a_3}+\frac{1}{a_3+a_1}>1.\]

2014 IFYM, Sozopol, 6

Tags: inequalities

$x_1,...,x_n$ are non-negative reals and $n \geq 3$. Prove that at least one of the following inequalities is true: \[ \sum_{i=1} ^n \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}, \] \[ \sum_{i=1} ^n \frac{x_i}{x_{i-1}+x_{i-2}} \geq \frac{n}{2} . \]

2017 German National Olympiad, 5

Tags: symmetric inequality , inequalities

Prove that for all non-negative numbers $x,y,z$ satisfying $x+y+z=1$, one has \[1 \le \frac{x}{1-yz}+\frac{y}{1-zx}+\frac{z}{1-xy} \le \frac{9}{8}.\]

2008 Grigore Moisil Intercounty, 2

Tags: calculus , integration , inequalities , function , real analysis

Let $ n\in \mathbb{N^*}$ and $ f: [0,1]\rightarrow \mathbb{R}$ a continuos function with the prop. $ \int_{0}^{1}(1\minus{}x^n)f(x)dx\equal{}0$. Prove that $ \int_{0}^{1}f^2(x)dx \geq 2(n\plus{}1)\left(\int_{0}^{1}f(x)dx\right)^2$

2009 Ukraine National Mathematical Olympiad, 4

Tags: inequalities

Let $x \leq y \leq z \leq t$ be real numbers such that $xy + xz + xt + yz + yt + zt = 1.$ [b]a)[/b] Prove that $xt < \frac 13,$ b) Find the least constant $C$ for which inequality $xt < C$ holds for all possible values $x$ and $t.$

2006 Tournament of Towns, 3

Tags: inequalities

Let $a$ be some positive number. Find the number of integer solutions $x$ of inequality $100 < xa < 1000$ given that inequality $10 < xa < 100$ has exactly $5$ integer solutions. Consider all possible cases. [i](4 points)[/i]

2023 Francophone Mathematical Olympiad, 1

Tags: polynomial , algebra , inequalities

Let $P(X) = a_n X^n + a_{n-1} X^{n-1} + \cdots + a_1 X + a_0$ be a polynomial with real coefficients such that $0 \leqslant a_i \leqslant a_0$ for $i = 1, 2, \ldots, n$. Prove that, if $P(X)^2 = b_{2n} X^{2n} + b_{2n-1} X^{2n-1} + \cdots + b_{n+1} X^{n+1} + \cdots + b_1 X + b_0$, then $4 b_{n+1} \leqslant P(1)^2$.

1952 AMC 12/AHSME, 45

Tags: inequalities

If $ a$ and $ b$ are two unequal positive numbers, then: $ \textbf{(A)}\ \frac {2ab}{a \plus{} b} > \sqrt {ab} > \frac {a \plus{} b}{2} \qquad\textbf{(B)}\ \sqrt {ab} > \frac {2ab}{a \plus{} b} > \frac {a \plus{} b}{2}$ $ \textbf{(C)}\ \frac {2ab}{a \plus{} b} > \frac {a \plus{} b}{2} > \sqrt {ab} \qquad\textbf{(D)}\ \frac {a \plus{} b}{2} > \frac {2ab}{a \plus{} b} > \sqrt {ab}$ $ \textbf{(E)}\ \frac {a \plus{} b}{2} > \sqrt {ab} > \frac {2ab}{a \plus{} b}$

2005 Germany Team Selection Test, 2

Tags: inequalities , function , 3-variable inequality , nesbitt

If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that \[\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).\]

IV Soros Olympiad 1997 - 98 (Russia), 11.2

Tags: number theory , algebra , inequalities

Find all values of the parameter $a$ for which there are exactly $1998$ integers $x$ satisfying the inequality $$x^2 -\pi x +a < 0.$$

1997 Hungary-Israel Binational, 2

Tags: inequalities

Find all the real numbers $ \alpha$ satisfy the following property: for any positive integer $ n$ there exists an integer $ m$ such that $ \left |\alpha\minus{}\frac{m}{n}\right|<\frac{1}{3n}$.

2017 India PRMO, 7

Tags: inequalities , radical , algebra

Find the number of positive integers $n$, such that $\sqrt{n} + \sqrt{n + 1} < 11$.

1993 China Team Selection Test, 2

Tags: quadratic , function , algebra , inequalities

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 $n.$

2022 Kosovo National Mathematical Olympiad, 4

Tags: inequalities , algebra

Let $a,b$ and $c$ be positive real numbers such that $a+b+c+3abc\geq (ab)^2+(bc)^2+(ca)^2+3$. Show that the following inequality hold, $$\frac{a^3+b^3+c^3}{3}\geq\frac{abc+2021}{2022}.$$

1986 China Team Selection Test, 2

Tags: 3d geometry , tetrahedron , perimeter , inequalities , function , geometry

Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that: i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}. ii) The same as above replacing "area" for "perimeter".

2014 IMO, 1

Tags: algebra , inequalities , sequence , unique

Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that \[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\] [i]Proposed by Gerhard Wöginger, Austria.[/i]

2002 Kazakhstan National Olympiad, 2

Tags: inequalities , trigonometry , algebra

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]

2023 USA TSTST, 2

Tags: algebra , inequalities

Let $n\ge m\ge 1$ be integers. Prove that \[\sum_{k=m}^n \left (\frac 1{k^2}+\frac 1{k^3}\right) \ge m\cdot \left(\sum_{k=m}^n \frac 1{k^2}\right)^2.\] [i]Raymond Feng and Luke Robitaille[/i]

2007 Nicolae Coculescu, 3

Tags: inequalities

Show that for any three numbers $ a,b,c\in (1,\infty ) , $ the following inequality is true: $$ \log_{ab} c +\log_{bc} a +\log_{ca} b\ge log_{a^2bc} bc +log_{b^2ca} ca +log_{c^2ab} ab$$ [i]Costel Anghel[/i]

2013 Bosnia Herzegovina Team Selection Test, 5

Tags: inequalities

Let $x_1,x_2,\ldots,x_n$ be nonnegative real numbers of sum equal to $1$. Let $F_n=x_1^{2}+x_2^{2}+\cdots +x_n^{2}-2(x_1x_2+x_2x_3+\cdots +x_nx_1)$. Find: a) $\min F_3$; b) $\min F_4$; c) $\min F_5$.

1992 Romania Team Selection Test, 2

Tags: inequalities

Let $ a_1, a_2, ..., a_k $ be distinct positive integers such that the $2^k$ sums $\displaystyle\sum\limits_{i=1}^{k}{\epsilon_i a_i}$, $\epsilon_i\in\left\{0,1\right\}$ are distinct. a) Show that $ \dfrac{1}{a_1}+\dfrac{1}{a_2}+...+\dfrac{1}{a_k}\le2(1-2^{-k}) $; b) Find the sequences $(a_1,a_2,...,a_k)$ for which the equality holds. [i]Șerban Buzețeanu[/i]

2022 Indonesia TST, A

Tags: function , algebra , fe , inequalities

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying \[ f(a^2) - f(b^2) \leq (f(a)+b)(a-f(b)) \] for all $a,b \in \mathbb{R}$.