Olympiad problems

1961 IMO, 4

Tags: ratio , inequalities , geometry , triangle , intersection

Consider triangle $P_1P_2P_3$ and a point $p$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers \[ \dfrac{P_1P}{PQ_1}, \dfrac{P_2P}{PQ_2}, \dfrac{P_3P}{PQ_3} \] at least one is $\leq 2$ and at least one is $\geq 2$

2000 Romania National Olympiad, 2b

Tags: geometry , inequalities , geometric inequality

If $a, b, c$ represent the lengths of the sides of a triangle, prove that: $$\frac{a}{b-a+c}+ \frac{b}{b-a+c}+ \frac{c}{b-a+c} \ge 3$$

2016 Belarus Team Selection Test, 1

Tags: inequality , inequalities , algebra

Prove for positive $a,b,c$ that $$ (a^2+\frac{b^2}{c^2})(b^2+\frac{c^2}{a^2})(c^2+\frac{a^2}{b^2}) \geq abc (a+\frac{1}{a})(b+\frac{1}{b})(c+\frac{1}{c})$$

2008 China National Olympiad, 3

Tags: inequalities , algebra

Given a positive integer $n$ and $x_1 \leq x_2 \leq \ldots \leq x_n, y_1 \geq y_2 \geq \ldots \geq y_n$, satisfying \[\displaystyle\sum_{i = 1}^{n} ix_i = \displaystyle\sum_{i = 1}^{n} iy_i\] Show that for any real number $\alpha$, we have \[\displaystyle\sum_{i =1}^{n} x_i[i\alpha] \geq \displaystyle\sum_{i =1}^{n} y_i[i\alpha]\] Here $[\beta]$ denotes the greastest integer not larger than $\beta$.

2018 Bulgaria JBMO TST, 1

Tags: inequalities

For real numbers $a$ and $b$, define $$f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}.$$ Find the smallest possible value of the expression $$f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b).$$

2014 IFYM, Sozopol, 8

Tags: combinatorics , inequalities

Some number of coins is firstly separated into 200 groups and then to 300 groups. One coin is [i]special[/i], if on the second grouping it is in a group that has less coins than the previous one, in the first grouping, that it was in. Find the least amount of [i]special[/i] coins we can have.

2006 IMO, 3

Tags: inequalities , function , algebra

Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.

2018 Moldova EGMO TST, 5

Tags: inequalities

Let $a$ and $b$ be real numbers such that $a + b = 1$. Prove the inequality $$\sqrt{1+5a^2} + 5\sqrt{2+b^2} \geq 9.$$ [i]Proposed by Baasanjav Battsengel[/i]

2010 JBMO Shortlist, 2

Tags: rectangle , rotation , inequalities , combinatorics , geometry

A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.

2008 Romania National Olympiad, 4

Tags: linear algebra , matrix , polynomial , inequalities , algebra

Let $ A\equal{}(a_{ij})_{1\leq i,j\leq n}$ be a real $ n\times n$ matrix, such that $ a_{ij} \plus{} a_{ji} \equal{} 0$, for all $ i,j$. Prove that for all non-negative real numbers $ x,y$ we have \[ \det(A\plus{}xI_n)\cdot \det(A\plus{}yI_n) \geq \det (A\plus{}\sqrt{xy}I_n)^2.\]

2009 Iran MO (3rd Round), 4

Tags: inequalities , function , algebra

Does there exists two functions $f,g :\mathbb{R}\rightarrow \mathbb{R}$ such that: $\forall x\not =y : |f(x)-f(y)|+|g(x)-g(y)|>1$ Time allowed for this problem was 75 minutes.

2006 China Northern MO, 8

Tags: inequalities

Given a sequence $\{ a_{n}\}$ such that $a_{n+1}=a_{n}+\frac{1}{2006}a_{n}^{2}$ , $n \in N$, $a_{0}=\frac{1}{2}$. Prove that $1-\frac{1}{2008}< a_{2006}< 1$.

2021 BMT, Tie 3

Tags: algebra , inequalities

For integers $a$ and $b$, $a + b$ is a root of $x^2 + ax + b = 0$. Compute the smallest possible value of $ab$.

2016 Kyiv Mathematical Festival, P3

Tags: inequalities

1) Let $a,b,c\ge0$ and $ab+bc+ca=2.$ Prove that \[\frac{ab}{c+1}+\frac{bc}{a+1}+\frac{ca}{b+1}+2(a+b+c)\ge6.\] 2) Let $a,b,c\ge0$ and $ab+bc+ca=3.$ Prove that \[\frac{ab}{c+1}+\frac{bc}{a+1}+\frac{ca}{b+1}\ge\frac{3}{2}.\]

2004 India IMO Training Camp, 3

Tags: polynomial , inequalities , algebra

Suppose the polynomial $P(x) \equiv x^3 + ax^2 + bx +c$ has only real zeroes and let $Q(x) \equiv 5x^2 - 16x + 2004$. Assume that $P(Q(x)) = 0$ has no real roots. Prove that $P(2004) > 2004$

2019 LIMIT Category C, Problem 2

Tags: inequalities , absolute value

Let $x,y\in[0,\infty)$. Which of the following is true? $\textbf{(A)}~\left|\log\left(1+x^2\right)-\log\left(1+y^2\right)\right|\le|x-y|$ $\textbf{(B)}~\left|\sin^2x-\sin^2y\right|\le|x-y|$ $\textbf{(C)}~\left|\tan^{-1}x-\tan^{-1}y\right|\le|x-y|$ $\textbf{(D)}~\text{None of the above}$

2009 Abels Math Contest (Norwegian MO) Final, 4a

Tags: algebra , inequalities

Show that $\left(\frac{2010}{2009}\right)^{2009}> 2$.

1985 Tournament Of Towns, (092) T3

Tags: algebra , inequalities

Three real numbers $a, b$ and $c$ are given . It is known that $a + b + c >0 , bc+ ca + ab > 0$ and $abc > 0$ . Prove that $a > 0 , b > 0$ and $c > 0$ .

2005 Taiwan TST Round 2, 2

Tags: inequalities

Find all positive integers $n \ge 3$ such that there exists a positive constant $M_n$ satisfying the following inequality for any $n$ positive reals $a_1, a_2,\dots\>,a_n$: \[\displaystyle \frac{a_1+a_2+\cdots\>+a_n}{\sqrt[n]{a_1a_2\cdots\>a_n}} \le M_n \biggl( \frac{a_2}{a_1} + \frac{a_3}{a_2} +\cdots\>+ \frac{a_n}{a_{n-1}} + \frac {a_1}{a_n} \biggr).\] Moreover, find the minimum value of $M_n$ for such $n$. The difficulty is finding $M_n$...

2005 Moldova National Olympiad, 10.4

Tags: inequalities

Real numbers $ x_{1},x_{2},..,x_{n}$ are positive. Prove the inequality: $ \frac{x_{1}}{x_{2}\plus{}x_{3}}\plus{}\frac{x_{2}}{x_{3}\plus{}x_{4}}\plus{}...\plus{} \frac{x_{n\minus{}1}}{x_{n}\plus{}x_{1}}\plus{}\frac{x_{n}}{x_{1}\plus{}x_{2}}>(\sqrt{2}\minus{}1)n$

1996 USAMO, 3

Tags: geometry , geometric transformation , reflection , trapezoid , inequalities , ratio , angle bisector

Let $ABC$ be a triangle. Prove that there is a line $\ell$ (in the plane of triangle $ABC$) such that the intersection of the interior of triangle $ABC$ and the interior of its reflection $A'B'C'$ in $\ell$ has area more than $\frac23$ the area of triangle $ABC$.

1990 All Soviet Union Mathematical Olympiad, 520

Tags: algebra , inequalities , minimum

Let $x_1, x_2, ..., x_n$ be positive reals with sum $1$. Show that $$\frac{x_1^2}{x_1 + x_2}+ \frac{x_2^2}{x_2 + x_3} +... + \frac{x_{n-1}^2}{x_{n-1} + x_n} + \frac{x_n^2}{x_n + x_1} \ge \frac12$$

2003 China Team Selection Test, 3

Tags: trigonometry , ratio , geometry , inequalities

(1) $D$ is an arbitary point in $\triangle{ABC}$. Prove that: \[ \frac{BC}{\min{AD,BD,CD}} \geq \{ \begin{array}{c} \displaystyle 2\sin{A}, \ \angle{A}< 90^o \\ \\ 2, \ \angle{A} \geq 90^o \end{array} \] (2)$E$ is an arbitary point in convex quadrilateral $ABCD$. Denote $k$ the ratio of the largest and least distances of any two points among $A$, $B$, $C$, $D$, $E$. Prove that $k \geq 2\sin{70^o}$. Can equality be achieved?

2012 South East Mathematical Olympiad, 4

Tags: trigonometry , inequalities

Let $a, b, c, d$ be real numbers satisfying inequality $a\cos x+b\cos 2x+c\cos 3x+d\cos 4x\le 1$ holds for arbitrary real number $x$. Find the maximal value of $a+b-c+d$ and determine the values of $a,b,c,d$ when that maximum is attained.

2002 IMO Shortlist, 2

Tags: inequalities , algebra , sequence , bounded

Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that \[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \] Prove that $c\geq1$.