Olympiad problems

2006 Bulgaria Team Selection Test, 2

Tags: algebra , inequalities

Prove that if $a,b,c>0,$ then \[ \frac{ab}{3a+4b+5c}+\frac{bc}{3b+4c+5a}+\frac{ca}{3c+4a+5b}\le \frac{a+b+c}{12}. \] [i] Nikolai Nikolov[/i]

2005 Morocco TST, 2

Tags: inequalities

Let $a,b,c$ be positive real numbers. Prove the inequality \[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c+\frac{4(a-b)^2}{a+b+c}.\] When does equality occur?

2000 USA Team Selection Test, 1

Tags: inequalities

Let $a, b, c$ be nonnegative real numbers. Prove that \[ \frac{a+b+c}{3} - \sqrt[3]{abc} \leq \max\{(\sqrt{a} - \sqrt{b})^2, (\sqrt{b} - \sqrt{c})^2, (\sqrt{c} - \sqrt{a})^2\}. \]

2012 China Northern MO, 6

Tags: function , inequalities , logarithm

Prove that\[(1+\frac{1}{3})(1+\frac{1}{3^2})\cdots(1+\frac{1}{3^n})< 2.\]

2004 Croatia Team Selection Test, 2

Tags: inequalities

Prove that if $a,b,c$ are positive numbers with $abc=1$, then \[\frac{a}{b} +\frac{b}{c} + \frac{c}{a} \ge a + b + c. \]

2013 Cuba MO, 7

Tags: algebra , inequalities

Let $x, y, z$ be positive real numbers whose sum is $2013$. Find the maximum possible value of $$\frac{(x^2+y^2+z^2)(x^3+y^3+z^3)}{ (x^4+y^4+z^4)}.$$

2011 Junior Balkan MO, 4

Tags: rectangle , inequalities , trigonometry , vector , triangle inequality , geometry

Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that \[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\] If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

2010 AMC 10, 23

Tags: probability , ellipse , inequalities , conic

Each of 2010 boxes in a line contains a single red marble, and for $ 1 \le k \le 2010$, the box in the $ kth$ position also contains $ k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $ P(n)$ be the probability that Isabella stops after drawing exactly $ n$ marbles. What is the smallest value of $ n$ for which $ P(n) < \frac {1}{2010}$? $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 63 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 201 \qquad \textbf{(E)}\ 1005$

2019 Thailand Mathematical Olympiad, 5

Tags: algebra , inequalities

Let $a,b,c$ be positive reals such that $abc=1$. Prove the inequality $$\frac{4a-1}{(2b+1)^2} + \frac{4b-1}{(2c+1)^2} + \frac{4c-1}{(2a+1)^2}\geqslant 1.$$

2012 Cuba MO, 4

Tags: algebra , inequalities

Let $x, y, z$ be positive reals. Prove that $$\frac{xz}{x^2 + xy + y^2 + 6z^2} + \frac{zx}{z^2 + zy + y^2 + 6x^2} + \frac{xy}{x^2 + xz + z^2 + 6y^2} \le \frac13$$

2018 Slovenia Team Selection Test, 3

Tags: inequalities , algebra

Let $a$, $b$ and $c$ be positive real numbers satisfying $abc=1$. Prove that the following inequality holds: $$\frac{a+b+c}{3}\geq\frac{a}{a^2b+2}+\frac{b}{b^2c+2}+\frac{c}{c^2a+2}.$$

2011 Bosnia Herzegovina Team Selection Test, 2

Tags: inequalities

Let $a, b, c$ be positive reals such that $a+b+c=1$. Prove that the inequality \[a \sqrt[3]{1+b-c} + b\sqrt[3]{1+c-a} + c\sqrt[3]{1+a-b} \leq 1\] holds.

2005 Romania National Olympiad, 3

Tags: real analysis , calculus , integration , function , limit , inequalities

Let $f:[0,\infty)\to(0,\infty)$ a continous function such that $\lim_{n\to\infty} \int^x_0 f(t)dt$ exists and it is finite. Prove that \[ \lim_{x\to\infty} \frac 1{\sqrt x} \int^x_0 \sqrt {f(t)} dt = 0. \] [i]Radu Miculescu[/i]

1969 Czech and Slovak Olympiad III A, 4

Tags: complex number , algebra , absolute value , inequalities

Determine all complex numbers $z$ such that \[\Bigl|z-\bigl|z+|z|\bigr|\Bigr|-|z|\sqrt3\ge0\] and draw the set of all such $z$ in complex plane.

2019 Jozsef Wildt International Math Competition, W. 51

Tags: inequalities

Let $a$, $b$, $c$, $d$, $e$ be real strictly positive real numbers such that $abcde = 1$. Then is true the following inequality:$$\frac{de}{a(b+1)}+\frac{ea}{b(c+1)}+\frac{ab}{c(d+1)}+\frac{bc}{d(e+1)}+\frac{cd}{e(a+1)}\geq \frac{5}{2}$$

2016 IFYM, Sozopol, 1

Tags: function , algebra , triangle inequality , inequalities

Find all functions $f: \mathbb{R}^+\rightarrow \mathbb{R}^+$ with the following property: $a,b,$ and $c$ are lengths of sides of a triangle, if and only if $f(a),f(b),$ and $f(c)$ are lengths of sides of a triangle.

1976 IMO, 1

Tags: geometry , trigonometry , inequalities , area of a triangle , geometric inequality

In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.

2021 Polish Junior MO Finals, 2

Tags: inequalities , geometry , triangle geometry

Point $M$ is the midpoint of the hypotenuse $AB$ of a right angled triangle $ABC$. Points $P$ and $Q$ lie on segments $AM$ and $MB$ respectively and $PQ=CQ$. Prove that $AP\leq 2\cdot MQ$.

1961 Polish MO Finals, 2

Tags: inequalities , algebra

Prove that if $ a + b = 1 $, then $$ a^5 + b^5 \geq \frac{1}{16}$$

2017 Thailand Mathematical Olympiad, 8

Tags: sides of a triangle , minimum , right triangle , inequalities , algebra

Let $a, b, c$ be side lengths of a right triangle. Determine the minimum possible value of $\frac{a^3 + b^3 + c^3}{abc}$.

2011 Abels Math Contest (Norwegian MO), 3a

Tags: inequalities , algebra , sequence

The positive numbers $a_1, a_2,...$ satisfy $a_1 = 1$ and $(m+n)a_{m+n }\le a_m +a_n$ for all positive integers $m$ and $n$. Show that $\frac{1}{a_{200}} > 4 \cdot 10^7$ . .

2023 China Northern MO, 5

Tags: inequalities , combinatorics , graph theory

Given a finite graph $G$, let $f(G)$ be the number of triangles in graph $G$, $g(G)$ be the number of edges in graph $G$, find the minimum constant $c$, so that for each graph $G$, there is $f^ 2(G)\le c \cdot g^3(G)$.

2012 Balkan MO, 3

Tags: induction , floor function , combinatorics , logarithm , inequalities

Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$ Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$

2017 Junior Balkan Team Selection Tests - Romania, 1

Tags: inequalities , algebra

If $a, b, c \in [-1, 1]$ satisfy $a + b + c + abc = 0$, prove that $a^2 + b^2 + c^2 \ge 3(a + b + c)$ . When does the equality hold?

2006 Putnam, B2

Tags: inequalities , pigeonhole principle

Prove that, for every set $X=\{x_{1},x_{2},\dots,x_{n}\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that \[\left|m+\sum_{s\in S}s\right|\le\frac1{n+1}\]