Olympiad problems

2012 China Team Selection Test, 2

Tags: pigeonhole principle , floor function , ceiling function , function , algebra , difference of squares , inequalities

Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that \[0<|xy-zw|<C\alpha ^{-4}\] where $\alpha =\frac{|X|}{n}$.

2011 Indonesia MO, 3

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

Given an acute triangle $ABC$, let $l_a$ be the line passing $A$ and perpendicular to $AB$, $l_b$ be the line passing $B$ and perpendicular to $BC$, and $l_c$ be the line passing $C$ and perpendicular to $CA$. Let $D$ be the intersection of $l_b$ and $l_c$, $E$ be the intersection of $l_c$ and $l_a$, and $F$ be the intersection of $l_a$ and $l_b$. Prove that the area of the triangle $DEF$ is at least three times of the area of $ABC$.

2023 Bangladesh Mathematical Olympiad, P10

Tags: algebra , polynomial , inequalities

Let all possible $2023$-degree real polynomials: $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$, where $P(0)+P(1)=0$, and the polynomial has 2023 real roots $r_1, r_2,\cdots r_{2023}$ [not necessarily distinct] so that $0\leq r_1,r_2,\cdots r_{2023}\leq1$. What is the maximum value of $r_1 \cdot r_2 \cdots r_{2023}?$

2017 China Team Selection Test, 2

Tags: inequalities , algebra

Let $x>1$ ,$n$ be positive integer. Prove that$$\sum_{k=1}^{n}\frac{\{kx \}}{[kx]}<\sum_{k=1}^{n}\frac{1}{2k-1}$$ Where $[kx ]$ be the integer part of $kx$ ,$\{kx \}$ be the decimal part of $kx$.

1979 All Soviet Union Mathematical Olympiad, 273

Tags: sequence , algebra , inequalities

For every $n$, the decreasing sequence $\{x_k\}$ satisfies a condition $$x_1+x_4/2+x_9/3+...+x_n^2/n \le 1$$ Prove that for every $n$, it also satisfies $$x_1+x_2/2+x_3/3+...+x_n/n\le 3$$

PEN A Problems, 49

Tags: inequalities , divisibility theory

Prove that there is no positive integer $n$ such that, for $k=1, 2, \cdots, 9,$ the leftmost digit of $(n+k)!$ equals $k$.

1997 Croatia National Olympiad, Problem 2

Tags: inequalities

Let $a,b,c$ be positive reals. Prove that $$a^ab^bc^c \geq a^bb^cc^a$$

2006 Baltic Way, 4

Tags: inequalities

Let $a,b,c,d,e,f$ be non-negative real numbers satisfying $a+b+c+d+e+f=6$. Find the maximal possible value of $\color{white}\ .\quad \ \color{black}\ \quad abc+bcd+cde+def+efa+fab$ and determine all $6$-tuples $(a,b,c,d,e,f)$ for which this maximal value is achieved.

1982 Tournament Of Towns, (020) 1

Tags: inequalities , algebra

(a) Prove that for any positive numbers $x_1,x_2,...,x_k$ ($k > 3$), $$\frac{x_1}{x_k+x_2}+ \frac{x_2}{x_1+x_3}+...+\frac{x_k}{x_{k-1}+x_1}\ge 2$$ (b) Prove that for every $k$ this inequality cannot be sharpened, i.e. prove that for every given $k$ it is not possible to change the number $2$ in the right hand side to a greater number in such a way that the inequality remains true for every choice of positive numbers $x_1,x_2,...,x_k$. (A Prokopiev)

1992 Denmark MO - Mohr Contest, 3

Tags: algebra , inequalities

Let $x$ and $y$ be positive numbers with $x +y=1$. Show that $$\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right) \ge 9.$$

2007 Moldova National Olympiad, 12.4

Tags: calculus , integration , inequalities , function , calculus computations

If the function $f\colon [1,2]\to R$ is such that $\int_{1}^{2}f(x) dx=\frac{73}{24}$, then show that there exists a $x_{0}\in (1;2)$ such that \[x_{0}^{2}<f(x_{0})<x_{0}^{3}\] [Edit: $f$ is continuous]

2012 India Regional Mathematical Olympiad, 2

Tags: algebra , polynomial , vieta , inequalities , binomial theorem , sum of roots

Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ be a polynomial of degree $n\geq 3.$ Knowing that $a_{n-1}=-\binom{n}{1}$ and $a_{n-2}=\binom{n}{2},$ and that all the roots of $P$ are real, find the remaining coefficients. Note that $\binom{n}{r}=\frac{n!}{(n-r)!r!}.$

2006 Austria Beginners' Competition, 2

Tags: algebra , inequalities

For which real numbers $a$ is the set of all solutions of the inequality $$(x^2 + ax + 4)(x^2 - 5x + 6) < 0$$ an interval?

2014 Contests, 1

Tags: inequalities

Let $a,b,c$ be real numbers such that $a+b+c=1$ and $abc>0$ . Prove that\[bc+ca+ab<\frac{\sqrt{abc}}{2}+\frac{1}{4}.\]

1994 Abels Math Contest (Norwegian MO), 3a

Tags: inequalities , algebra , product

Let $x_1,x_2,...,x_{1994}$ be positive real numbers. Prove that $$\left(\frac{x_1}{x_2}\right)^{\frac{x_1}{x_2}}\left(\frac{x_2}{x_3}\right)^{\frac{x_2}{x_3}}...\left(\frac{x_{1993}}{x_{1994}}\right)^{\frac{x_{1993}}{x_{1994}}} \ge \left(\frac{x_1}{x_2}\right)^{\frac{x_2}{x_1}}\left(\frac{x_2}{x_3}\right)^{\frac{x_3}{x_2}}...\left(\frac{x_{1993}}{x_{1994}}\right)^{\frac{x_{1994}}{x_{1993}}}$$

2023 Francophone Mathematical Olympiad, 1

Tags: inequalities , number theory , sequence

Let $u_0, u_1, u_2, \ldots$ be integers such that $u_0 = 100$; $u_{k+2} \geqslant 2 + u_k$ for all $k \geqslant 0$; and $u_{\ell+5} \leqslant 5 + u_\ell$ for all $\ell \geqslant 0$. Find all possible values for the integer $u_{2023}$.

2006 Romania National Olympiad, 2

Tags: trigonometry , inequalities

Prove that for all $\displaystyle a,b \in \left( 0 ,\frac{\pi}{4} \right)$ and $\displaystyle n \in \mathbb N^\ast$ we have \[ \frac{\sin^n a + \sin^n b}{\left( \sin a + \sin b \right)^n} \geq \frac{\sin^n 2a + \sin^n 2b}{\left( \sin 2a + \sin 2b \right)^n} . \]

2009 China Team Selection Test, 3

Tags: induction , inequalities

Let $ x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n}$ be positive real numbers. Denote by $ X \equal{} \sum_{i \equal{} 1}^{m}x,Y \equal{} \sum_{j \equal{} 1}^{n}y.$ Prove that $ 2XY\sum_{i \equal{} 1}^{m}\sum_{j \equal{} 1}^{n}|x_{i} \minus{} y_{j}|\ge X^2\sum_{j \equal{} 1}^{n}\sum_{l \equal{} 1}^{n}|y_{i} \minus{} y_{l}| \plus{} Y^2\sum_{i \equal{} 1}^{m}\sum_{k \equal{} 1}^{m}|x_{i} \minus{} x_{k}|$

1999 APMO, 2

Tags: induction , logarithm , inequalities

Let $a_1, a_2, \dots$ be a sequence of real numbers satisfying $a_{i+j} \leq a_i+a_j$ for all $i,j=1,2,\dots$. Prove that \[ a_1 + \frac{a_2}{2} + \frac{a_3}{3} + \cdots + \frac{a_n}{n} \geq a_n \] for each positive integer $n$.

2006 Bulgaria Team Selection Test, 2

Tags: inequalities , algebra

Prove that if $a,b,c>0,$ then \[ \frac{ab}{3a+4b+5c}+\frac{bc}{3b+4c+5a}+\frac{ca}{3c+4a+5b}\le \frac{a+b+c}{12}. \] [i] Nikolai Nikolov[/i]

2006 China National Olympiad, 1

Tags: induction , inequalities

Let $a_1,a_2,\ldots,a_k$ be real numbers and $a_1+a_2+\ldots+a_k=0$. Prove that \[ \max_{1\leq i \leq k} a_i^2 \leq \frac{k}{3} \left( (a_1-a_2)^2+(a_2-a_3)^2+\cdots +(a_{k-1}-a_k)^2\right). \]

1985 IMO Longlists, 63

Tags: algebra , sequence , inequality , calculus , recurrence relation , inequalities

Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that \[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]

1996 Romania National Olympiad, 1

Tags: algebra , inequalities

Let $a$ and $b$ be real numbers such that $a + b = 2$. Show that: $$\min \{|a|,|b|\} < 1 < \max \{|a|,|b|\} \Leftrightarrow a, b \in (-3,1)$$

2018 Cyprus IMO TST, 3

Tags: inequalities

Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$ obtains its minimum value.

1991 Arnold's Trivium, 20

Tags: calculus , derivative , parameterization , function , inequalities

Find the derivative of the solution of the equation $\ddot{x} =x + A\dot{x}^2$, with initial conditions $x(0) = 1$, $\dot{x}(0) = 0$, with respect to the parameter $A$ for $A = 0$.