Olympiad problems

2011 Croatia Team Selection Test, 1

Tags: function , inequalities

Let $a,b,c$ be positive reals such that $a+b+c=3$. Prove the inequality \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geq \frac{3}{2}.\]

2007 China Team Selection Test, 3

Tags: inequalities , combinatorial geometry , analytic geometry , geometry

Assume there are $ n\ge3$ points in the plane, Prove that there exist three points $ A,B,C$ satisfying $ 1\le\frac{AB}{AC}\le\frac{n\plus{}1}{n\minus{}1}.$

2010 All-Russian Olympiad, 2

Tags: inequalities , combinatorics

There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.

2007 India National Olympiad, 1

Tags: inequalities , trigonometry , function , geometry

In a triangle $ ABC$ right-angled at $ C$ , the median through $ B$ bisects the angle between $ BA$ and the bisector of $ \angle B$. Prove that \[ \frac{5}{2} < \frac{AB}{BC} < 3\]

2004 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities

The real numbers $a_1,a_2,\ldots,a_{100}$ satisfy the relationship \[ a_1^2+ a_2^2 + \cdots +a_{100}^2 + ( a_1+a_2 + \cdots + a_{100})^2 = 101. \] Prove that $|a_k|\leq 10$, for all $k=1,2,\ldots,100$.

2005 Baltic Way, 2

Tags: inequalities , trigonometry , algebra

Let $\alpha$, $\beta$ and $\gamma$ be three acute angles such that $\sin \alpha+\sin \beta+\sin \gamma = 1$. Show that \[\tan^{2}\alpha+\tan^{2}\beta+\tan^{2}\gamma \geq \frac{3}{8}. \]

2010 India IMO Training Camp, 10

Tags: geometry , parallelogram , inequalities , analytic geometry , trigonometry , complex number

Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$

2009 South East Mathematical Olympiad, 3

Tags: inequalities

Let $x,y,z $ be positive reals such that $\sqrt{a}=x(y-z)^2$, $\sqrt{b}=y(z-x)^2$ and $\sqrt{c}=z(x-y)^2$. Prove that \[a^2+b^2+c^2 \geq 2(ab+bc+ca)\]

2004 Bulgaria Team Selection Test, 2

Tags: inequalities , algebra , inequality

Prove that if $a,b,c \ge 1$ and $a+b+c=9$, then $\sqrt{ab+bc+ca} \le \sqrt{a} +\sqrt{b} + \sqrt{c}$

2019 India IMO Training Camp, P1

Tags: number theory , inequalities

Let $a_1,a_2,\ldots, a_m$ be a set of $m$ distinct positive even numbers and $b_1,b_2,\ldots,b_n$ be a set of $n$ distinct positive odd numbers such that \[a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019\] Prove that \[5m+12n\le 581.\]

2007 Grigore Moisil Intercounty, 4

Tags: inequalities , solve inequality , algebra

Solve in the set of real numbers the fractional part inequality $ \{ x \}\le\{ nx \} , $ where $ n $ is a fixed natural number.

2016 Czech-Polish-Slovak Junior Match, 2

Tags: inequalities , algebra

Let $x$ and $y$ be real numbers such that $x^2 + y^2 - 1 < xy$. Prove that $x + y - |x - y| < 2$. Slovakia

1991 Baltic Way, 5

Tags: inequalities , search

For any positive numbers $a, b, c$ prove the inequalities \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge \frac{2}{a+b}+\frac{2}{b+c}+\frac{2}{c+a}\ge \frac{9}{a+b+c}.\]

IV Soros Olympiad 1997 - 98 (Russia), 11.9

Tags: algebra , inequalities

The numbers $a$, $b$ and $c$ satisfy the conditions $$0 < a \le b \le c\,\,\,,\,\,\, a+b+ c = 7\,\,\,, \,\,\,abc = 9.$$ Within what limits can each of the numbers $a$, $b$ and $c$ vary?

2022 Princeton University Math Competition, B1

Tags: algebra , inequalities

Let $a, b, c, d$ be real numbers for which $a^2 + b^2 + c^2 + d^2 = 1$. Show the following inequality: $$a^2 + b^2 - c^2 - d^2 \le \sqrt{2 + 4(ac + bd)}.$$

1974 Putnam, B5

Tags: factorial , inequalities

Show that $$1+\frac{n}{1!} + \frac{n^{2}}{2!} +\ldots+ \frac{n^{n}}{n!} > \frac{e^{n}}{2}$$ for every integer $n\geq 0.$

1981 Romania Team Selection Tests, 5.

Tags: inequalities , geometry , geometric inequality

Let $ABC$ be an equilateral triangle, $M$ be a point inside it, and $A',B',C'$ be the intersections of $AM,\; BM,\; CM$ with the sides of $ABC$. If $A'',\; B'',\; C''$ are the midpoints of $BC$, $CA$, $AB$, show that there is a triangle with sides $A'A''$, $B'B''$ and $C'C''$. [i]Laurențiu Panaitopol[/i]

1999 Swedish Mathematical Competition, 5

Tags: algebra , inequalities

$x_i$ are non-negative reals. $x_1 + x_2 + ...+ x_n = s$. Show that $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n \le \frac{s^2}{4}$.

2017 District Olympiad, 4

Tags: algebra , inequality , inequalities

If $ a,b,c>0 $ and $ ab+bc+ca+abc=4, $ then $ \sqrt{ab} +\sqrt{bc} +\sqrt{ca} \le 3\le a+b+c. $

OMMC POTM, 2023 9

Tags: inequalities , algebra

Show that for any $8$ distinct positive real numbers, one can choose a quadraple of them $(a,b,c,d)$ , all distinct such that $$(ac+bd)^2 \ge \frac{2+\sqrt3}{4}\left(a^2+b^2 \right)\left(c^2+d^2 \right)$$ [i]Proposed by Evan Chang (squareman), USA[/i]

2006 IMO Shortlist, 8

Tags: inequalities , geometry , circumcircle , triangle inequality

Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that \[\angle{PAB}+\angle{PDC}\leq 90^\circ\qquad\text{and}\qquad\angle{PBA}+\angle{PCD}\leq 90^\circ.\] Prove that $AB+CD \geq BC+AD$. [i]Proposed by Waldemar Pompe, Poland[/i]

2003 Switzerland Team Selection Test, 3

Tags: inequalities

Find the largest real number $ C_1 $ and the smallest real number $ C_2 $, such that, for all reals $ a,b,c,d,e $, we have \[ C_1 < \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a} < C_2 \]

2011 District Olympiad, 1

Tags: algebra , inequalities

Find the real numbers $x$ and $y$ such that $$(x^2 -x +1)(3y^2-2y + 3) -2=0.$$

2009 Ukraine Team Selection Test, 2

Tags: inequalities

Let $ a$, $ b$, $ c$ are sides of a triangle. Find the least possible value $ k$ such that the following inequality always holds: $ \left|\frac{a\minus{}b}{a\plus{}b}\plus{}\frac{b\minus{}c}{b\plus{}c}\plus{}\frac{c\minus{}a}{c\plus{}a}\right|<k$ [i](Vitaly Lishunov)[/i]

2022 Czech-Polish-Slovak Junior Match, 1

Tags: inequalities , algebra

Determine the largest possible value of the expression $ab+bc+ 2ac$ for non-negative real numbers $a, b, c$ whose sum is $1$.