Olympiad problems

Today's calculation of integrals, 864

Let $m,\ n$ be positive integer such that $2\leq m<n$. (1) Prove the inequality as follows. \[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\] (2) Prove the inequality as follows. \[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\] (3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows. \[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]

2011 USAMTS Problems, 3

Tags: floor function , induction , function , inequalities , algebra

Find all integers $b$ such that there exists a positive real number $x$ with \[ \dfrac {1}{b} = \dfrac {1}{\lfloor 2x \rfloor} + \dfrac {1}{\lfloor 5x \rfloor} \] Here, $\lfloor y \rfloor$ denotes the greatest integer that is less than or equal to $y$.

2001 IMO Shortlist, 2

Tags: inequalities , calculus

Let $a_0, a_1, a_2, \ldots$ be an arbitrary infinite sequence of positive numbers. Show that the inequality $1 + a_n > a_{n-1} \sqrt[n]{2}$ holds for infinitely many positive integers $n$.

2002 China Girls Math Olympiad, 5

Tags: floor function , inequalities

There are $ n \geq 2$ permutations $ P_1, P_2, \ldots, P_n$ each being an arbitrary permutation of $ \{1,\ldots,n\}.$ Prove that \[ \sum^{n\minus{}1}_{i\equal{}1} \frac{1}{P_i \plus{} P_{i\plus{}1}} > \frac{n\minus{}1}{n\plus{}2}.\]

2001 Irish Math Olympiad, 4

Tags: inequalities

Prove that for all positive integers $ n$: $ \frac{2n}{3n\plus{}1} \le \displaystyle\sum_{k\equal{}n\plus{}1}^{2n}\frac{1}{k} \le \frac{3n\plus{}1}{4(n\plus{}1)}$.

2008 South africa National Olympiad, 3

Tags: inequalities

Let $a,b,c$ be positive real numbers. Prove that \[(a+b)(b+c)(c+a)\ge 8(a+b-c)(b+c-a)(c+a-b)\] and determine when equality occurs.

2011 Czech-Polish-Slovak Match, 1

Tags: inequalities

Let $a$, $b$, $c$ be positive real numbers satisfying $a^2<bc$. Prove that $b^3+ac^2>ab(a+c)$.

2007 Bulgaria Team Selection Test, 4

Tags: inequalities , combinatorics

Let $G$ is a graph and $x$ is a vertex of $G$. Define the transformation $\varphi_{x}$ over $G$ as deleting all incident edges with respect of $x$ and drawing the edges $xy$ such that $y\in G$ and $y$ is not connected with $x$ with edge in the beginning of the transformation. A graph $H$ is called $G-$[i]attainable[/i] if there exists a sequece of such transformations which transforms $G$ in $H.$ Let $n\in\mathbb{N}$ and $4|n.$ Prove that for each graph $G$ with $4n$ vertices and $n$ edges there exists $G-$[i]attainable[/i] graph with at least $9n^{2}/4$ triangles.

1896 Eotvos Mathematical Competition, 1

Tags: algebra , inequalities

If $k$ is the number of distinct prime divisors of a natural number $n$, prove that log $n \geq k$ log $2$.

2011 IFYM, Sozopol, 2

Tags: inequalities

prove that $(\frac{1}{a+c}+\frac{1}{b+d})(\frac{1}{\frac{1}{a}+\frac{1}{c}}+\frac{1}{\frac{1}{b}+\frac{1}{d}}) \leq 1$ for $0 < a < b \leq c < d$ and when $(\frac{1}{a+c}+\frac{1}{b+d})(\frac{1}{\frac{1}{a}+\frac{1}{c}}+\frac{1}{\frac{1}{b}+\frac{1}{d}}) = 1 $

2006 Belarusian National Olympiad, 1

Tags: vector , inequalities , algebra

Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be unit vectors in $R^3$. Prove that $$\sqrt{1-\overrightarrow{a}\cdot\overrightarrow{b}}\le \sqrt{1-\overrightarrow{a}\cdot\overrightarrow{c}}+\sqrt{1-\overrightarrow{c}\cdot\overrightarrow{b}}$$ (A.Mirotin)

2022-IMOC, A1

Tags: inequalities

If positive real numbers $x,y,z$ satisfies $x+y+z=3,$ prove that $$\sum_{\text{cyc}} y^2z^2<3+\sum_{\text{cyc}} yz.$$ [i]Proposed by Li4 and Untro368.[/i]

2007 Today's Calculation Of Integral, 187

Tags: calculus , integration , trigonometry , inequalities , calculus computations

For a constant $a,$ let $f(x)=ax\sin x+x+\frac{\pi}{2}.$ Find the range of $a$ such that $\int_{0}^{\pi}\{f'(x)\}^{2}\ dx \geq f\left(\frac{\pi}{2}\right).$

2011 All-Russian Olympiad, 1

Tags: inequalities , relatively prime , number theory

Two natural numbers $d$ and $d'$, where $d'>d$, are both divisors of $n$. Prove that $d'>d+\frac{d^2}{n}$.

2010 Contests, 2

Tags: induction , inequalities , prime factorization , number theory

Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

2002 Korea - Final Round, 1

Tags: polynomial , inequalities , algebra

For $n \ge 3$, let $S=a_1+a_2+\cdots+a_n$ and $T=b_1b_2\cdots b_n$ for positive real numbers $a_1,a_2,\ldots,a_n, b_1,b_2 ,\ldots,b_n$, where the numbers $b_i$ are pairwise distinct. (a) Find the number of distinct real zeroes of the polynomial \[f(x)=(x-b_1)(x-b_2)\cdots(x-b_n)\sum_{j=1}^n \frac{a_j}{x-b_j}\] (b) Prove the inequality \[\frac1{n-1}\sum_{j=1}^n \left(1-\frac{a_j}{S}\right)b_j > \left(\frac{T}{S}\sum_{j=1}^{n} \frac{a_j}{b_j}\right)^{\frac1{n-1}}\]

2023 Auckland Mathematical Olympiad, 10

Tags: algebra , inequalities

Find the maximum of the expression $$||...||x_1 - x_2|- x_3| -... | - x_{2023}|,$$ where $x_1,x_2,..., x_{2023}$ are distinct natural numbers between $1$ and $2023$.

2020 Taiwan APMO Preliminary, P6

Tags: algebra , inequalities , inequality

Let $a,b,c$ be positive reals. Find the minimum value of $$\dfrac{13a+13b+2c}{2a+2b}+\dfrac{24a-b+13c}{2b+2c}+\dfrac{(-a+24b+13c)}{2c+2a}$$. (1)What is the minimum value? (2)If the minimum value occurs when $(a,b,c)=(a_0,b_0,c_0)$,then find $\frac{b_0}{a_0}+\frac{c_0}{b_0}$.

2009 Jozsef Wildt International Math Competition, W. 17

Tags: function , inequalities

If $a$, $b$, $c>0$ and $abc=1$, $\alpha = max\{a,b,c\}$; $f,g : (0, +\infty )\to \mathbb{R}$, where $f(x)=\frac{2(x+1)^2}{x}$ and $g(x)= (x+1)\left (\frac{1}{\sqrt{x}}+1\right )^2$, then $$(a+1)(b+1)(c+1)\geq min\{ \{f(x),g(x) \}\ |\ x\in\{a,b,c\} \backslash \{ \alpha \}\} $$

2005 Miklós Schweitzer, 9

Tags: complex number , inequalities , rational function , complex analysis

prove that if $r_n$ is a rational function whose numerator and denominator have at most degrees $n$, then $$||r_n||_{1/2}+\left\|\frac{1}{r_n}\right\|_2\geq\frac{1}{2^{n-1}}$$ where $||\cdot||_a$ denotes the supremum over a circle of radius $a$ around the origin.

2004 Poland - First Round, 4

Tags: inequalities

4.Given is $n \in \mathbb Z$ and positive reals a,b. Find possible maximal value of the sum: $x_1y_1 + x_2y_2 + ... + x_ny_n$ when $x_1,x_2,...,x_n$ and $y_1,y_2,...,y_n$ are in $<0;1>$ and satisfies: $x_1 + x_2 + ... + x_n \leq a$ and $y_1 + y_2 + ... + y_n \leq b$

2005 France Team Selection Test, 6

Tags: triangle inequality , algebra , inequalities

Let $P$ be a polynom of degree $n \geq 5$ with integer coefficients given by $P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_0 \quad$ with $a_i \in \mathbb{Z}$, $a_n \neq 0$. Suppose that $P$ has $n$ different integer roots (elements of $\mathbb{Z}$) : $0,\alpha_2,\ldots,\alpha_n$. Find all integers $k \in \mathbb{Z}$ such that $P(P(k))=0$.

2004 IMO, 4

Tags: inequalities , triangle inequality , n-variable inequality

Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that \[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\] Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.

1991 Baltic Way, 7

Tags: inequalities , trigonometry

If $\alpha,\beta,\gamma$ are the angles of an acute-angled triangle, prove that \[\sin \alpha + \sin \beta > \cos \alpha + \cos\beta + \cos\gamma.\]

2013 Switzerland - Final Round, 8

Tags: algebra , inequalities

Let $a, b, c > 0$ be real numbers. Show the following inequality: $$a^2 \cdot \frac{a - b}{a + b}+ b^2\cdot \frac{b - c}{b + c}+ c^2\cdot \frac{c - a}{c + a} \ge 0 .$$ When does equality holds?