Olympiad problems

2010 Romanian Masters In Mathematics, 4

Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions: (i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers); (ii) $|a_1-b_1|+|a_2-b_2|=2010$; (iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$; (iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$. [i]Massimo Gobbino, Italy[/i]

2002 Putnam, 3

Tags: inequalities , logarithm

Show that for all integers $n>1$, \[ \dfrac {1}{2ne} < \dfrac {1}{e} - \left( 1 - \dfrac {1}{n} \right)^n < \dfrac {1}{ne}. \]

2006 Princeton University Math Competition, 5

Tags: calculus , integration , function , inequalities , floor function

Find the greatest integer less than the number $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{1000000}}$

Oliforum Contest III 2012, 4

Tags: algebra , inequalities

Show that if $a \ge b \ge c \ge 0$ then $$a^2b(a - b) + b^2c(b - c) + c^2a(c - a) \ge 0.$$

1997 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: algebra , inequalities

Let $x_1, x_2, ... , x_n$ be non-negative numbers $n\ge3$ such that $x_1 + x_2 + ... + x_n = 1$. Determine the maximum possible value of the expression $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n$.

2010 Contests, 3

Tags: trigonometry , calculus , function , complex number , algebra , inequalities

Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.

1999 Hungary-Israel Binational, 1

Tags: inequalities , polynomial , algebra

$ c$ is a positive integer. Consider the following recursive sequence: $ a_1\equal{}c, a_{n\plus{}1}\equal{}ca_{n}\plus{}\sqrt{(c^2\minus{}1)(a_n^2\minus{}1)}$, for all $ n \in N$. Prove that all the terms of the sequence are positive integers.

1988 IMO Longlists, 92

Tags: inequalities , algebra

Let $p \geq 2$ be a natural number. Prove that there exist an integer $n_0$ such that \[ \sum^{n_0}_{i=1} \frac{1}{i \cdot \sqrt[p]{i + 1}} > p. \]

1997 Korea National Olympiad, 5

Tags: geometry , inequalities

Let $a,b,c$ be the side lengths of any triangle $\triangle ABC$ opposite to $A,B$ and $C,$ respectively. Let $x,y,z$ be the length of medians from $A,B$ and $C,$ respectively. If $T$ is the area of $\triangle ABC$, prove that $\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\ge\sqrt{\sqrt{3}T}$

1968 Miklós Schweitzer, 5

Tags: inequalities , real analysis

Let $ k$ be a positive integer, $ z$ a complex number, and $ \varepsilon <\frac12$ a positive number. Prove that the following inequality holds for infinitely many positive integers $ n$: \[ \mid \sum_{0\leq l \leq \frac{n}{k+1}} \binom{n-kl}{l}z^l \mid \geq (\frac 12-\varepsilon)^n.\] [i]P. Turan[/i]

1994 India Regional Mathematical Olympiad, 1

Tags: integration , inequalities

A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is $15000$. What are the page numbers on the torn leaf?

2023 Taiwan TST Round 3, 4

Tags: inequalities , am-gm , number theory , diophantine equation

Find all positive integers $a$, $b$ and $c$ such that $ab$ is a square, and \[a+b+c-3\sqrt[3]{abc}=1.\] [i]Proposed by usjl[/i]

1997 China National Olympiad, 3

Tags: induction , inequalities

Let $(a_n)$ be a sequence of non-negative real numbers satisfying $a_{n+m}\le a_n+a_m$ for all non-negative integers $m,n$. Prove that if $n\ge m$ then $a_n\le ma_1+\left(\dfrac{n}{m}-1\right)a_m$ holds.

2013 Baltic Way, 16

Tags: inequalities , number theory

We call a positive integer $n$ [i]delightful[/i] if there exists an integer $k$, $1 < k < n$, such that \[1+2+\cdots+(k-1)=(k+1)+(k+2)+\cdots+n\] Does there exist a delightful number $N$ satisfying the inequalities \[2013^{2013}<\dfrac{N}{2013^{2013}}<2013^{2013}+4 ?\]

2014 Putnam, 4

Tags: probability , inequalities , polynomial , expected value , algebra

Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$) Determine the smallest possible value of the probability of the event $X=0.$

2017 China Northern MO, 8

Tags: inequalities

Let $n>1$ be an integer, and let $x_1, x_2, ..., x_n$ be real numbers satisfying $x_1, x_2, ..., x_n \in [0,n]$ with $x_1x_2...x_n = (n-x_1)(n-x_2)...(n-x_n)$. Find the maximum value of $y = x_1 + x_2 + ... + x_n$.

2008 Germany Team Selection Test, 3

Tags: polynomial , inequalities , algebra

Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that \[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0. \]

1986 Vietnam National Olympiad, 1

Tags: function , inequalities

Let $ \frac{1}{2}\le a_1, a_2, \ldots, a_n \le 5$ be given real numbers and let $ x_1, x_2, \ldots, x_n$ be real numbers satisfying $ 4x_i^2\minus{} 4a_ix_i \plus{} \left(a_i \minus{} 1\right)^2 \le 0$. Prove that \[ \sqrt{\sum_{i\equal{}1}^n\frac{x_i^2}{n}}\le\sum_{i\equal{}1}^n\frac{x_i}{n}\plus{}1\]

2008 National Olympiad First Round, 16

Tags: inequalities

A class of $50$ students took an exam with $4$ questions. At least $1$ of any $40$ students gave exactly $3$, at least $2$ of any $40$ gave exactly $2$, and at least $3$ of any $40$ gave exactly $1$ correct answers. At least $4$ of any $40$ students gave exactly $4$ wrong answers. What is the least number of students who gave an odd number of correct answers? $ \textbf{(A)}\ 18 \qquad\textbf{(B)}\ 24 \qquad\textbf{(C)}\ 26 \qquad\textbf{(D)}\ 28 \qquad\textbf{(E)}\ \text{None of the above} $

2011 ISI B.Math Entrance Exam, 3

Tags: induction , inequalities

For $n\in\mathbb{N}$ prove that \[\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n}\leq\frac{1}{\sqrt{2n+1}}.\]

2014 VJIMC, Problem 4

Tags: inequalities , integration , calculus

Let $0<a<b$ and let $f:[a,b]\to\mathbb R$ be a continuous function with $\int^b_af(t)dt=0$. Show that $$\int^b_a\int^b_af(x)f(y)\ln(x+y)dxdy\le0.$$

2011 JBMO Shortlist, 3

Tags: inequalities , algebra

$\boxed{\text{A3}}$If $a,b$ be positive real numbers, show that:$$ \displaystyle{\sqrt{\dfrac{a^2+ab+b^2}{3}}+\sqrt{ab}\leq a+b}$$

2014 BMT Spring, P1

Tags: inequalities

Suppose that $a,b,c,d$ are non-negative real numbers such that $a^2+b^2+c^2+d^2=2$ and $ab+bc+cd+da=1$. Find the maximum value of $a+b+c+d$ and determine all equality cases.

2023 Abelkonkurransen Finale, 4a

Tags: inequalities , geometry

Assuming $a,b,c$ are the side-lengths of a triangle, show that \begin{align*} \frac{a^2+b^2-c^2}{ab} + \frac{b^2+c^2-a^2}{bc} + \frac{c^2+a^2-b^2}{ca} > 2. \end{align*} Also show that the inequality does not necessarily hold if you replace $2$ (on the right-hand side) by a bigger by a bigger number.

2015 Swedish Mathematical Competition, 3

Tags: min , inequalities , algebra

Let $a$, $b$, $c$ be positive real numbers. Determine the minimum value of the following expression $$ \frac{a^2+2b^2+4c^2}{b(a+2c)}$$