Olympiad problems

1999 USAMO, 4

Tags: function , inequalities

Let $a_{1}, a_{2}, \dots, a_{n}$ ($n > 3$) be real numbers such that \[ a_{1} + a_{2} + \cdots + a_{n} \geq n \qquad \mbox{and} \qquad a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2} \geq n^{2}. \] Prove that $\max(a_{1}, a_{2}, \dots, a_{n}) \geq 2$.

2006 Poland - Second Round, 3

Tags: inequalities

Positive reals $a,b,c$ satisfy $ab+bc+ca=abc$. Prove that: $\frac{a^4+b^4}{ab(a^3+b^3)} + \frac{b^4+c^4}{bc(b^3+c^3)}+\frac{c^4+a^4}{ca(c^3+a^3)} \geq 1$

2006 All-Russian Olympiad Regional Round, 9.3

Tags: inequalities , algebra

It is known that $x^2_1+ x^2_2+...+ x^2_6= 6$ and $x_1 + x_2 +....+ x_6 = 0.$ Prove that $ x_1x_2....x_6 \le \frac12$ . .

2006 Germany Team Selection Test, 3

Tags: induction , inequalities

Let $n$ be a positive integer, and let $b_{1}$, $b_{2}$, ..., $b_{n}$ be $n$ positive reals. Set $a_{1}=\frac{b_{1}}{b_{1}+b_{2}+...+b_{n}}$ and $a_{k}=\frac{b_{1}+b_{2}+...+b_{k}}{b_{1}+b_{2}+...+b_{k-1}}$ for every $k>1$. Prove the inequality $a_{1}+a_{2}+...+a_{n}\leq\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}}$.

2004 Mediterranean Mathematics Olympiad, 1

Tags: prime factorization , factorial , number theory , inequalities

Find all natural numbers $m$ such that \[1! \cdot 3! \cdot 5! \cdots (2m-1)! = \biggl( \frac{m(m+1)}{2}\biggr) !.\]

2002 Mongolian Mathematical Olympiad, Problem 5

Tags: sequence , inequalities

Let $a_0,a_1,\ldots$ be an infinite sequence of positive numbers. Prove that the inequality $1+a_n>\sqrt[n]2a_{n-1}$ holds for infinitely many positive integers $n$.

2017 Philippine MO, 1

Tags: function , number theory , inequalities

Given $n \in \mathbb{N}$, let $\sigma (n)$ denote the sum of the divisors of $n$ and $\phi (n)$ denote the number of integers $n \geq m$ for which $\gcd(m,n) = 1$. Show that for all $n \in \mathbb{N}$, \[\large \frac{1}{\sigma (n)} + \frac{1}{\phi (n)} \geq \frac{2}{n}\] and determine when equality holds.

2011 Junior Balkan MO, 4

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

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 ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

2013 IMAC Arhimede, 6

Tags: sums and products , product , sum , inequalities , algebra

Let $p$ be an odd positive integer. Find all values of the natural numbers $n\ge 2$ for which holds $$\sum_{i=1}^{n} \prod_{j\ne i} (x_i-x_j)^p\ge 0$$ where $x_1,x_2,..,x_n$ are any real numbers.

2013 USAMTS Problems, 4

Tags: inequalities , prime factorization , arithmetic sequence , modular arithmetic , relatively prime , number theory

Bunbury the bunny is hopping on the positive integers. First, he is told a positive integer $n$. Then Bunbury chooses positive integers $a,d$ and hops on all of the spaces $a,a+d,a+2d,\dots,a+2013d$. However, Bunbury must make these choices so that the number of every space that he hops on is less than $n$ and relatively prime to $n$. A positive integer $n$ is called [i]bunny-unfriendly[/i] if, when given that $n$, Bunbury is unable to find positive integers $a,d$ that allow him to perform the hops he wants. Find the maximum bunny-unfriendly integer, or prove that no such maximum exists.

2011 Baltic Way, 6

Tags: inequalities , combinatorics , number theory , analytic geometry , probability , relatively prime

Let $n$ be a positive integer. Prove that the number of lines which go through the origin and precisely one other point with integer coordinates $(x,y),0\le x,y\le n$, is at least $\frac{n^2}{4}$.

1997 All-Russian Olympiad, 1

Tags: quadratic , inequalities , algebra , polynomial

Let $P(x)$ be a quadratic polynomial with nonnegative coeficients. Show that for any real numbers $x$ and $y$, we have the inequality $P(xy)^2 \leqslant P(x^2)P(y^2)$. [i]E. Malinnikova[/i]

2014 Federal Competition For Advanced Students, P2, 5

Tags: algebra , inequalities , three variable inequality , 3-variable inequality

Show that the inequality $(x^2 + y^2z^2) (y^2 + x^2z^2) (z^2 + x^2y^2) \ge 8xy^2z^3$ is valid for all integers $x, y$ and $z$.When does equality apply?

2012 239 Open Mathematical Olympiad, 4

Tags: inequalities , algebra

For some positive numbers $a$, $b$, $c$ and $d$, we know that $$ \frac{1}{a^3 + 1}+ \frac{1}{b^3 + 1}+ \frac{1}{c^3 + 1} + \frac{1}{d^3 + 1} = 2. $$ Prove that $$ \frac{1 - a}{a^2 - a + 1} + \frac{1-b}{b^2 - b + 1} + \frac{1-c}{c^2 - c + 1} +\frac{1-d}{d^2 - d + 1} \geq 0. $$

2019 Taiwan TST Round 3, 1

Tags: inequalities

Find the maximal value of \[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\] where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$. [i]Proposed by Evan Chen, Taiwan[/i]

2006 India Regional Mathematical Olympiad, 3

Tags: algebra , inequalities

If $ a,b,c$ are three positive real numbers, prove that $ \frac {a^{2}\plus{}1}{b\plus{}c}\plus{}\frac {b^{2}\plus{}1}{c\plus{}a}\plus{}\frac {c^{2}\plus{}1}{a\plus{}b}\ge 3$

2013 Kazakhstan National Olympiad, 1

Tags: inequalities , algebra

Find maximum value of $|a^2-bc+1|+|b^2-ac+1|+|c^2-ba+1|$ when $a,b,c$ are reals in $[-2;2]$.

2021 Switzerland - Final Round, 4

Tags: 4-variable inequality , inequality , algebra , inequalities

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2013 All-Russian Olympiad, 2

Tags: inequalities

Let $a,b,c,d$ be positive real numbers such that $ 2(a+b+c+d)\ge abcd $. Prove that \[ a^2+b^2+c^2+d^2 \ge abcd .\]

2019 Moldova EGMO TST, 4

Tags: inequalities

Let $x,y>0$ be real numbers.Prove that: $$\frac{1}{x^2+y^2} +\frac{1}{x^2}+\frac{1}{y^2}\ge\frac{10}{(x+y)^2}$$ I tried CBS, but it doesn't work... Can you give an idea, please?

2013 Saint Petersburg Mathematical Olympiad, 5

Tags: induction , inequalities

Let $x_1$, ... , $x_{n+1} \in [0,1] $ and $x_1=x_{n+1} $. Prove that \[ \prod_{i=1}^{n} (1-x_ix_{i+1}+x_i^2)\ge 1. \] A. Khrabrov, F. Petrov

1968 IMO Shortlist, 9

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

Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality \[abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.\] Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.

2019 China Second Round Olympiad, 1

Tags: inequalities

Suppose that $a_1,a_2,\cdots,a_{100}\in\mathbb{R}^+$ such that $a_i\geq a_{101-i}\,(i=1,2,\cdots,50).$ Let $x_k=\frac{ka_{k+1}}{a_1+a_2+\cdots+a_k}\,(k=1,2,\cdots,99).$ Prove that $$x_1x_2^2\cdots x_{99}^{99}\leq 1.$$

2019 Jozsef Wildt International Math Competition, W. 48

Tags: inequalities , calculus , convex function , integration , function

Let $f : (0,+\infty) \to \mathbb{R}$ a convex function and $\alpha, \beta, \gamma > 0$. Then $$\frac{1}{6\alpha}\int \limits_0^{6\alpha}f(x)dx\ +\ \frac{1}{6\beta}\int \limits_0^{6\beta}f(x)dx\ +\ \frac{1}{6\gamma}\int \limits_0^{6\gamma}f(x)dx$$ $$\geq \frac{1}{3\alpha +2\beta +\gamma}\int \limits_0^{3\alpha +2\beta +\gamma}f(x)dx\ +\ \frac{1}{\alpha +3\beta +2\gamma}\int \limits_0^{\alpha +3\beta +2\gamma}f(x)dx\ $$ $$+\ \frac{1}{2\alpha +\beta +3\gamma}\int \limits_0^{2\alpha +\beta +3\gamma}f(x)dx$$

2005 France Pre-TST, 2

Tags: inequalities

Let $\omega (n)$ denote the number of prime divisors of the integer $n>1$. Find the least integer $k$ such that the inequality $2^{\omega (n) } \leq k \cdot n^{\frac 1 4}$ holds for all $n > 1.$ Pierre.