Olympiad problems

2008 Postal Coaching, 3

Let $a$ and $b$ be two complex numbers. Prove the inequality $$|1 + ab| + |a + b| \ge \sqrt{|a^2 - 1| \cdot |b^2 - 1|}$$

2012 Korea Junior Math Olympiad, 7

Tags: inequalities , algebra , maximum , positive real , sum

If all $x_k$ ($k = 1, 2, 3, 4, 5)$ are positive reals, and $\{a_1,a_2, a_3, a_4, a_5\} = \{1, 2,3 , 4, 5\}$, find the maximum of $$\frac{(\sqrt{s_1x_1} +\sqrt{s_2x_2}+\sqrt{s_3x_3}+\sqrt{s_4x_4}+\sqrt{s_5x_5})^2}{a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 + a_5x_5}$$ ($s_k = a_1 + a_2 +... + a_k$)

2007 Hungary-Israel Binational, 2

Tags: algebra , inequalities

Let $ a,b,c,d$ be real numbers, such that $ a^2\le 1, a^2 \plus{} b^2\le 5, a^2 \plus{} b^2 \plus{} c^2\le 14, a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2\le 30$. Prove that $ a \plus{} b \plus{} c \plus{} d\le 10$.

2021 Bosnia and Herzegovina Team Selection Test, 1

Tags: inequalities , max , algebra

Let $x,y,z$ be real numbers from the interval $[0,1]$. Determine the maximum value of expression $$W=y\cdot \sqrt{1-x}+z\cdot\sqrt{1-y}+x\cdot\sqrt{1-z}$$

1969 Bulgaria National Olympiad, Problem 2

Tags: inequalities , sequence

Prove that $$S_n=\frac1{1^2}+\frac1{2^2}+\ldots+\frac1{n^2}<2$$for every $n\in\mathbb N$.

2018 Brazil Team Selection Test, 6

Tags: inequalities , algebra

An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have $$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$ Find the largest constant $K = K(n)$ such that $$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$ holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.

2004 Romania Team Selection Test, 13

Tags: function , inequalities , modular arithmetic , euler , relatively prime , number theory

Let $m\geq 2$ be an integer. A positive integer $n$ has the property that for any positive integer $a$ coprime with $n$, we have $a^m - 1\equiv 0 \pmod n$. Prove that $n \leq 4m(2^m-1)$. Created by Harazi, modified by Marian Andronache.

2009 Jozsef Wildt International Math Competition, W. 14

Tags: integration , inequalities , calculus

If the function $f:[0,1]\to (0.+\infty)$ is increasing and continuous, then for every $a\geq 0$ the following inequality holds: $$\int \limits_0^1 \frac{x^{a+1}}{f(x)}dx \leq \frac{a+1}{a+2} \int \limits_0^1 \frac{x^{a}}{f(x)}dx$$

1990 Romania Team Selection Test, 5

Tags: circles , geometry , circumcircle , geometric inequalities , area , inequalities

Let $O$ be the circumcenter of an acute triangle $ABC$ and $R$ be its circumcenter. Consider the disks having $OA,OB,OC$ as diameters, and let $\Delta$ be the set of points in the plane belonging to at least two of the disks. Prove that the area of $\Delta$ is greater than $R^2/8$.

2002 Regional Competition For Advanced Students, 3

Tags: inequalities , geometric inequality , geometry , convex , hexagon , perimeter

In the convex $ABCDEF$ (has all interior angles less than $180^o$) with the perimeter $s$ the triangles $ACE$ and $BDF$ have perimeters $u$ and $v$ respectively. a) Show the inequalities $\frac{1}{2} \le \frac{s}{u+v}\le 1$ b) Check whether $1$ is replaced by a smaller number or $1/2$ by a larger number can the inequality remains valid for all convex hexagons.

2003 Iran MO (2nd round), 2

Tags: geometry , inequalities , trigonometry , circumcircle , trapezoid

$\angle{A}$ is the least angle in $\Delta{ABC}$. Point $D$ is on the arc $BC$ from the circumcircle of $\Delta{ABC}$. The perpendicular bisectors of the segments $AB,AC$ intersect the line $AD$ at $M,N$, respectively. Point $T$ is the meet point of $BM,CN$. Suppose that $R$ is the radius of the circumcircle of $\Delta{ABC}$. Prove that: \[ BT+CT\leq{2R}. \]

2012 USA TSTST, 6

Tags: inequalities , polynomial , algebra , vieta , quadratic

Positive real numbers $x, y, z$ satisfy $xyz+xy+yz+zx = x+y+z+1$. Prove that \[ \frac{1}{3} \left( \sqrt{\frac{1+x^2}{1+x}} + \sqrt{\frac{1+y^2}{1+y}} + \sqrt{\frac{1+z^2}{1+z}} \right) \le \left( \frac{x+y+z}{3} \right)^{5/8} . \]

2013 Iran MO (3rd Round), 2

Tags: inequalities

Real numbers $a_1 , a_2 , \dots, a_n$ add up to zero. Find the maximum of $a_1 x_1 + a_2 x_2 + \dots + a_n x_n$ in term of $a_i$'s, when $x_i$'s vary in real numbers such that $(x_1 - x_2)^2 + (x_2 - x_3)^2 + \dots + (x_{n-1} - x_n)^2 \leq 1$. (15 points)

2016 India Regional Mathematical Olympiad, 2

Tags: algebra , inequalities

Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$ Prove that $abc \le \frac{1}{8}$.

2021 Romania National Olympiad, 3

Tags: inequalities , algebra

If $a,b,c>0,a+b+c=1$,then: $\frac{1}{abc}+\frac{4}{a^{2}+b^{2}+c^{2}}\geq\frac{13}{ab+bc+ca}$

2016 India National Olympiad, P2

Tags: algebra , inequalities

For positive real numbers $a,b,c$ which of the following statements necessarily implies $a=b=c$: (I) $a(b^3+c^3)=b(c^3+a^3)=c(a^3+b^3)$, (II) $a(a^3+b^3)=b(b^3+c^3)=c(c^3+a^3)$ ? Justify your answer.

2008 Switzerland - Final Round, 10

Tags: algebra , inequalities

Find all pairs$ (a, b)$ of positive real numbers with the following properties: (i) For all positive real numbers $x, y, z,w$ holds $x + y^2 + z^3 + w^6 \ge a (xyzw)^{b}$ . (ii) There is a quadruple $(x, y, z,w)$ of positive real numbers such that in equality (i) applies.

2025 Philippine MO, P5

Tags: algebra , inequalities , four variables

Find the largest real constant $k$ for which the inequality \[(a^2 + 3)(b^2 + 3)(c^2 + 3)(d^2 + 3) + k(a - 1)(b - 1)(c - 1)(d - 1) \ge 0\] holds for all real numbers $a$, $b$, $c$, and $d$.

1996 IMO Shortlist, 9

Tags: pythagorean theorem , geometry , perimeter , geometric inequality , inequalities

In the plane, consider a point $ X$ and a polygon $ \mathcal{F}$ (which is not necessarily convex). Let $ p$ denote the perimeter of $ \mathcal{F}$, let $ d$ be the sum of the distances from the point $ X$ to the vertices of $ \mathcal{F}$, and let $ h$ be the sum of the distances from the point $ X$ to the sidelines of $ \mathcal{F}$. Prove that $ d^2 \minus{} h^2\geq\frac {p^2}{4}.$

2014 Serbia JBMO TST, 1

Tags: inequalities

For $a, b, c, d, e$ in the interval $[0,1]$, prove that $(1+a+b+c+d+e)^2=>4(a^2+b^2+c^2+d^2+e^2)$

2009 All-Russian Olympiad Regional Round, 11.3

Tags: algebra , inequalities , trigonometry

Prove that $$x\cos x \le \frac{\pi^2}{16}$$ for $0 \le x \le \frac{\pi}{2}$

2009 Princeton University Math Competition, 4

Tags: inequalities , tetrahedron , geometry , 3d geometry

Tetrahedron $ABCD$ has sides of lengths, in increasing order, $7, 13, 18, 27, 36, 41$. If $AB=41$, then what is the length of $CD$?

2002 AIME Problems, 13

Tags: inequalities , number theory , ratio , relatively prime , triangle inequality , geometry

In triangle $ABC,$ point $D$ is on $\overline{BC}$ with $CD=2$ and $DB=5,$ point $E$ is on $\overline{AC}$ with $CE=1$ and $EA=3,$ $AB=8,$ and $\overline{AD}$ and $\overline{BE}$ intersect at $P.$ Points $Q$ and $R$ lie on $\overline{AB}$ so that $\overline{PQ}$ is parallel to $\overline{CA}$ and $\overline{PR}$ is parallel to $\overline{CB}.$ It is given that the ratio of the area of triangle $PQR$ to the area of triangle $ABC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

1999 Singapore Senior Math Olympiad, 3

Tags: algebra , increasing , decreasing , subsequence , sequence , inequalities

Let $\{a_1,a_2,...,a_{100}\}$ be a sequence of $100$ distinct real numbers. Show that there exists either an increasing subsequence $a_{i_1}<a_{i_2}<...<a_{i_{10}}$ $(i_1<i_2<...<i_{10})$ of $10$ numbers, or a decreasing subsequence $ a_{j_1}>a_{j_2}>...>a_{j_{12}}$ $(j_1<j_2<...<j_{12})$ of $12$ numbers, or both.

2008 Junior Balkan Team Selection Tests - Moldova, 9

Tags: inequalities , algebra

Find all triplets $ (x,y,z)$, that satisfy: $ \{\begin{array}{c}\ \ x^2 - 2x - 4z = 3\ y^2 - 2y - 2x = - 14 \ z^2 - 4y - 4z = - 18 \end{array}$