Olympiad problems

2008 Harvard-MIT Mathematics Tournament, 3

Tags: function , inequalities , analytic geometry , quadratic , conic , parabola , absolute value

Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.

2012 Today's Calculation Of Integral, 813

Tags: integration , calculus , geometry , quadratic , inequalities , calculus computations

Let $a$ be a real number. Find the minimum value of $\int_0^1 |ax-x^3|dx$. How many solutions (including University Mathematics )are there for the problem? Any advice would be appreciated.

2015 Stars Of Mathematics, 4

Tags: geometric inequality , inequalities

Let $S$ be a finite set of points in the plane,situated in general position(any three points in $S$ are not collinear),and let $$D(S,r)=\{\{x,y\}:x,y\in S,\text{dist}(x,y)=r\},$$ where $R$ is a positive real number,and $\text{dist}(x,y)$ is the euclidean distance between points $x$ and $y$.Prove that $$\sum_{r>0}|D(S,r)|^2\le\frac{3|S|^2(|S|-1)}{4}.$$

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.

2024 Stars of Mathematics, P3

Tags: inequalities

Fix postive integer $n\geq 2$. Let $a_1,a_2,\dots ,a_n$ be real numbers in the interval $[1,2024]$. Prove that $$\sum_{i=1}^n\frac{1}{a_i}(a_1+a_2+\dots +a_i)>\frac{1}{44}n(n+33).$$ [i]Proposed by Radu-Andrei Lecoiu[/i]

1990 Turkey Team Selection Test, 2

Tags: inequalities , calculus , derivative , polynomial , algebra

For real numbers $x_i$, the statement \[ x_1 + x_2 + x_3 = 0 \Rightarrow x_1x_2 + x_2x_3 + x_3x_1 \leq 0\] is always true. (Prove!) For which $n\geq 4$ integers, the statement \[x_1 + x_2 + \dots + x_n = 0 \Rightarrow x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1 \leq 0\] is always true. Justify your answer.

2022 Sharygin Geometry Olympiad, 8.8

Tags: trapezoid , isosceles , geometry , geometric inequality , inequalities

An isosceles trapezoid $ABCD$ ($AB = CD$) is given. A point $P$ on its circumcircle is such that segments $CP$ and $AD$ meet at point $Q$. Let $L$ be tha midpoint of$ QD$. Prove that the diagonal of the trapezoid is not greater than the sum of distances from the midpoints of the lateral sides to ana arbitrary point of line $PL$.

2020 Jozsef Wildt International Math Competition, W8

Tags: inequalities

If $a,b>0$ then prove: $$\left(\frac{a+b}2-\frac{2ab}{a+b}\right)\operatorname{arctan}\left(\frac{\sqrt{2ab}-\sqrt{a^2+b^2}}{\sqrt2+\sqrt{ab}\left(a^2+b^2\right)}\right)+\left(\sqrt{\frac{a^2+b^2}2}-\sqrt{ab}\right)\arctan\left(\frac{(a-b)^2}{2+2ab}\right)\ge0$$ [i]Proposed by Daniel Sitaru[/i]

2010 Indonesia TST, 1

Tags: inequalities

Let $ a$, $ b$, and $ c$ be non-negative real numbers and let $ x$, $ y$, and $ z$ be positive real numbers such that $ a\plus{}b\plus{}c\equal{}x\plus{}y\plus{}z$. Prove that \[ \dfrac{a^3}{x^2}\plus{}\dfrac{b^3}{y^2}\plus{}\dfrac{c^3}{z^2} \ge a\plus{}b\plus{}c.\] [i]Hery Susanto, Malang[/i]

2009 Argentina Team Selection Test, 2

Tags: inequalities , clevermath

Let $ a_1, a_2, ..., a_{300}$ be nonnegative real numbers, with $ \sum_{i\equal{}1}^{300} a_i \equal{} 1$. Find the maximum possible value of $ \sum_{i \neq j, i|j} a_ia_j$.

2004 Regional Olympiad - Republic of Srpska, 2

Tags: trigonometry , inequalities

Let $0<x<\pi/2$. Prove the inequality \[\sin x>\frac{4x}{x^2+4} .\]

2015 Canada National Olympiad, 2

Tags: geometry , inequalities , geometric inequality

Let $ABC$ be an acute-angled triangle with altitudes $AD,BE,$ and $CF$. Let $H$ be the orthocentre, that is, the point where the altitudes meet. Prove that \[\frac{AB\cdot AC+BC\cdot CA+CA\cdot CB}{AH\cdot AD+BH\cdot BE+CH\cdot CF}\leq 2.\]

2011 Singapore Junior Math Olympiad, 1

Tags: inequalities , algebra

Suppose $a,b,c,d> 0$ and $x = \sqrt{a^2+b^2}, y = \sqrt{c^2+d^2}$. Prove that $xy \ge ac + bd$.

2005 India IMO Training Camp, 3

Tags: inequalities

If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

2003 Alexandru Myller, 1

Tags: inequalities , algebra , complex number

Let be a natural number $ n, $ a positive real number $ \lambda , $ and a complex number $ z. $ Prove the following inequalities. $$ 0\le -\lambda +\frac{1}{n}\sum_{\stackrel{w\in\mathbb{C}}{w^n=1 }} \left| z-\lambda w \right|\le |z| $$ [i]Gheorghe Iurea[/i]

2012 Thailand Mathematical Olympiad, 10

Tags: number theory , inequalities , algebra , irrational

Let $x$ be an irrational number. Show that there are integers $m$ and $n$ such that $\frac{1}{2555}< mx + n <\frac{1}{2012}$

2016 Saudi Arabia Pre-TST, 1.2

Tags: inequalities , algebra

Let $a, b, c$ be positive numbers such that $a^2+b^2+c^2+abc = 4$. Prove that $$\frac{a + b}{c} +\frac{b + c}{a} +\frac{c + a}{b} \ge a + b + c + \frac{1}{a} + \frac{1}{b} +\frac{1}{c}$$

1995 IMO Shortlist, 4

Tags: function , inequalities , optimization , 133 , system of equations

Find all of the positive real numbers like $ x,y,z,$ such that : 1.) $ x \plus{} y \plus{} z \equal{} a \plus{} b \plus{} c$ 2.) $ 4xyz \equal{} a^2x \plus{} b^2y \plus{} c^2z \plus{} abc$ Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.

1970 Poland - Second Round, 1

Tags: trigonometry , algebra , inequalities

Prove that $$ |\cos n\beta - \cos n\alpha| \leq n^2 |\cos \beta - \cos\alpha|,$$ where $n$ is a natural number . Check for what values of $ n $, $ \alpha $, $ \beta $ equality holds.

1980 Putnam, B1

Tags: exponential , inequalities

For which real numbers $c$ is $$\frac{e^x +e^{-x} }{2} \leq e^{c x^2 }$$ for all real $x?$

2008 Moldova MO 11-12, 1

Tags: inequalities , algebra , polynomial , calculus

Consider the equation $ x^4 \minus{} 4x^3 \plus{} 4x^2 \plus{} ax \plus{} b \equal{} 0$, where $ a,b\in\mathbb{R}$. Determine the largest value $ a \plus{} b$ can take, so that the given equation has two distinct positive roots $ x_1,x_2$ so that $ x_1 \plus{} x_2 \equal{} 2x_1x_2$.

1968 Polish MO Finals, 2

Tags: algebra , inequalities

Prove that for every natural $n$ $$\frac{1}{3} + \frac{2}{3\cdot 5} + \frac{3}{3 \cdot 5 \cdot 7} + ...+ \frac{n}{3 \cdot 5 \cdot 7 \cdot ...\cdot (2n+1)} < \frac{1}{2}.$$

2008 Polish MO Finals, 1

Tags: linear algebra , inequalities , matrix

In each cell of a matrix $ n\times n$ a number from a set $ \{1,2,\ldots,n^2\}$ is written --- in the first row numbers $ 1,2,\ldots,n$, in the second $ n\plus{}1,n\plus{}2,\ldots,2n$ and so on. Exactly $ n$ of them have been chosen, no two from the same row or the same column. Let us denote by $ a_i$ a number chosen from row number $ i$. Show that: \[ \frac{1^2}{a_1}\plus{}\frac{2^2}{a_2}\plus{}\ldots \plus{}\frac{n^2}{a_n}\geq \frac{n\plus{}2}{2}\minus{}\frac{1}{n^2\plus{}1}\]

1999 IMC, 4

Tags: function , limit , real analysis , inequalities

Prove that there's no function $f: \mathbb{R}^+\rightarrow\mathbb{R}^+$ such that $f(x)^2\ge f(x+y)\left(f(x)+y\right)$ for all $x,y>0$.

2011 Ukraine Team Selection Test, 11

Tags: integer polynomial , algebra , inequalities , polynomial

Let $ P (x) $ and $ Q (x) $ be polynomials with real coefficients such that $ P (0)> 0 $ and all coefficients of the polynomial $ S (x) = P (x) \cdot Q (x) $ are integers. Prove that for any positive $ x $ the inequality holds: $$S ({{x} ^ {2}}) - {{S} ^ {2}} (x) \le \frac {1} {4} ({{P} ^ {2}} ({{ x} ^ {3}}) + Q ({{x} ^ {3}})). $$