Olympiad problems

2006 USAMO, 2

For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k + 1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $\tfrac{N}{2}.$

2016 Sharygin Geometry Olympiad, P8

Tags: inequalities , geometry

Let $ABCDE$ be an inscribed pentagon such that $\angle B +\angle E = \angle C +\angle D$.Prove that $\angle CAD < \pi/3 < \angle A$. [i](Proposed by B.Frenkin)[/i]

2011 Austria Beginners' Competition, 3

Tags: algebra , inequalities

Let $x, y$ be positive real numbers with $x + y + xy= 3$. Prove that$$x + y\ge 2.$$ When does equality holds? (K. Czakler, GRG 21, Vienna)

1992 IMO Longlists, 21

Tags: inequalities , trigonometry , three variable inequality

Prove that if $x,y,z >1$ and $\frac 1x +\frac 1y +\frac 1z = 2$, then \[\sqrt{x+y+z} \geq \sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]

2014 Junior Balkan Team Selection Tests - Moldova, 1

Tags: sum , inequalities , algebra

Prove that $$\frac{2 }{2013 +1} +\frac{2^{2}}{2013^{2^{1}}+1} +\frac{2^{3}}{2013^{2^{2}}+1} + ...+ \frac{2^{2014}}{2013^{2^{2013}}+1} < \frac{1}{1006}$$

2013 Middle European Mathematical Olympiad, 1

Tags: inequalities

Let $ a, b, c$ be positive real numbers such that \[ a+b+c=\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} . \] Prove that \[ 2(a+b+c) \ge \sqrt[3]{7 a^2 b +1 } + \sqrt[3]{7 b^2 c +1 } + \sqrt[3]{7 c^2 a +1 } . \] Find all triples $ (a,b,c) $ for which equality holds.

2010 Cuba MO, 9

Tags: number theory , algebra , inequalities

Let $A$ be the subset of the natural numbers such that the sum of Its digits are multiples of$ 2009$. Find $x, y \in A$ such that $y - x > 0$ is minimum and $x$ is also minimum.

2015 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: inequalities , algebra

Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality: $$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \leq \frac{a^2+b^2+c^2}{2}$$

2011 SEEMOUS, Problem 1

Tags: function , real analysis , integral inequality , inequalities

Let $f:[0,1]\rightarrow R$ be a continuous function and n be an integer number,n>0.Prove that $\int_0^1f(x)dx \le (n+1)*\int_0^1 x^n*f(x)dx $

2010 239 Open Mathematical Olympiad, 6

Tags: algebra , inequalities

We have six positive numbers $a_1, a_2, \ldots , a_6$ such that $a_1a_2\ldots a_6 =1$. Prove that: $$ \frac{1}{a_1(a_2 + 1)} + \frac{1}{a_2(a_3 + 1)} + \ldots + \frac{1}{a_6(a_1 + 1)} \geq 3.$$

2010 Moldova Team Selection Test, 2

Tags: trigonometry , inequalities

Let $ x_1, x_2, \ldots, x_n$ be positive real numbers with sum $ 1$. Find the integer part of: $ E\equal{}x_1\plus{}\dfrac{x_2}{\sqrt{1\minus{}x_1^2}}\plus{}\dfrac{x_3}{\sqrt{1\minus{}(x_1\plus{}x_2)^2}}\plus{}\cdots\plus{}\dfrac{x_n}{\sqrt{1\minus{}(x_1\plus{}x_2\plus{}\cdots\plus{}x_{n\minus{}1})^2}}$

2018 Bulgaria JBMO TST, 4

Tags: inequalities

The real numbers $a_1 \leq a_2 \leq \cdots \leq a_{672}$ are given such that $$a_1 + a_2 + \cdots + a_{672} = 2018.$$ For any $n \leq 672$, there are $n$ of these numbers with an integer sum. What is the smallest possible value of $a_{672}$?

2021 Thailand TST, 2

Tags: algebra , inequality , 4-variable inequality , inequalities

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

1998 Baltic Way, 9

Tags: trigonometry , inequalities

Let the numbers $\alpha ,\beta $ satisfy $0<\alpha <\beta <\frac{\pi}{2}$ and let $\gamma $ and $\delta $ be the numbers defined by the conditions: $(\text{i})\ 0<\gamma<\frac{\pi}{2}$, and $\tan\gamma$ is the arithmetic mean of $\tan\alpha$ and $\tan\beta$; $(\text{ii})\ 0<\delta<\frac{\pi}{2}$, and $\frac{1}{\cos\delta}$ is the arithmetic mean of $\frac{1}{\cos\alpha}$ and $\frac{1}{\cos\beta}$. Prove that $\gamma <\delta $.

1993 China Team Selection Test, 2

Tags: quadratic , function , inequalities , algebra

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 $n.$

2013 Canada National Olympiad, 4

Tags: inequalities , ceiling function , algebra

Let $n$ be a positive integer. For any positive integer $j$ and positive real number $r$, define $f_j(r)$ and $g_j(r)$ by \[f_j(r) = \min (jr, n) + \min\left(\frac{j}{r}, n\right), \text{ and } g_j(r) = \min (\lceil jr\rceil, n) + \min \left(\left\lceil\frac{j}{r}\right\rceil, n\right),\] where $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Prove that \[\sum_{j=1}^n f_j(r)\leq n^2+n\leq \sum_{j=1}^n g_j(r)\] for all positive real numbers $r$.

1999 China National Olympiad, 2

Tags: algebra , polynomial , inequalities

Determine the maximum value of $\lambda$ such that if $f(x) = x^3 +ax^2 +bx+c$ is a cubic polynomial with all its roots nonnegative, then \[f(x)\geq\lambda(x -a)^3\] for all $x\geq0$. Find the equality condition.

2002 AMC 12/AHSME, 25

Tags: parabola , inequalities , analytic geometry , geometry , special factorizations , conic

Let $ f(x)\equal{}x^2\plus{}6x\plus{}1$, and let $ R$ denote the set of points $ (x,y)$ in the coordinate plane such that \[ f(x)\plus{}f(y)\le0\text{ and }f(x)\minus{}f(y)\le0 \]The area of $ R$ is closest to $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 22 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 25$

2013 Switzerland - Final Round, 2

Tags: inequalities , number theory , prime

Let $n$ be a natural number and $p_1, ..., p_n$ distinct prime numbers. Show that $$p_1^2 + p_2^2 + ... + p_n^2 > n^3$$

2015 Korea National Olympiad, 3

Tags: inequalities

Reals $a,b,c,x,y$ satisfies $a^2+b^2+c^2+x^2+y^2=1$. Find the maximum value of $$(ax+by)^2+(bx+cy)^2$$

2010 Turkey Team Selection Test, 2

Tags: inequalities

Show that \[ \sum_{cyc} \sqrt[4]{\frac{(a^2+b^2)(a^2-ab+b^2)}{2}} \leq \frac{2}{3}(a^2+b^2+c^2)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right) \] for all positive real numbers $a, \: b, \: c.$

2002 VJIMC, Problem 3

Tags: inequalities

Positive numbers $x_1,\ldots,x_n$ satisfy $$\frac1{1+x_1}+\frac1{1+x_2}+\ldots+\frac1{1+x_n}=1.$$Prove that $$\sqrt{x_1}+\sqrt{x_2}+\ldots+\sqrt{x_n}\ge(n-1)\left(\frac1{\sqrt{x_1}}+\frac1{\sqrt{x_2}}+\ldots+\frac1{\sqrt{x_n}}\right).$$

2023 Tuymaada Olympiad, 6

Tags: inequalities , geometry , plane , geometry inequality

In the plane $n$ segments with lengths $a_1, a_2, \dots , a_n$ are drawn. Every ray beginning at the point $O$ meets at least one of the segments. Let $h_i$ be the distance from $O$ to the $i$-th segment (not the line!) Prove the inequality \[\frac{a_1}{h_1}+\frac{a_2}{h_2} + \ldots + \frac{a_i}{h_i} \geqslant 2 \pi.\]

2008 Indonesia TST, 2

Tags: algebra , inequalities , sequence , sum

Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$. Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$ for all positive integers $n$.

2003 Regional Competition For Advanced Students, 1

Tags: function , inequalities

Find the minimum value of the expression $ \frac{a\plus{}1}{a(a\plus{}2)}\plus{}\frac{b\plus{}1}{b(b\plus{}2)}\plus{}\frac{c\plus{}1}{c(c\plus{}2)}$, where $ a,b,c$ are positive real numbers with $ a\plus{}b\plus{}c \le 3$.