Olympiad problems

2012 All-Russian Olympiad, 3

Tags: inequalities

Any two of the real numbers $a_1,a_2,a_3,a_4,a_5$ differ by no less than $1$. There exists some real number $k$ satisfying \[a_1+a_2+a_3+a_4+a_5=2k\]\[a_1^2+a_2^2+a_3^2+a_4^2+a_5^2=2k^2\] Prove that $k^2\ge 25/3$.

1998 All-Russian Olympiad Regional Round, 11.8

Tags: combinatorial number theory , combinatorics , algebra , inequalities , number theory

A sequence $a_1,a_2,\cdots$ of positive integers contains each positive integer exactly once. Moreover for every pair of distinct positive integer $m$ and $n$, $\frac{1}{1998} < \frac{|a_n- a_m|}{|n-m|} < 1998$, show that $|a_n - n | <2000000$ for all $n$.

1954 AMC 12/AHSME, 18

Tags: inequalities

Of the following sets, the one that includes all values of $ x$ which will satisfy $ 2x \minus{} 3 > 7 \minus{} x$ is: $ \textbf{(A)}\ x > 4 \qquad \textbf{(B)}\ x < \frac {10}{3} \qquad \textbf{(C)}\ x \equal{} \frac {10}{3} \qquad \textbf{(D)}\ x > \frac {10}{3} \qquad \textbf{(E)}\ x < 0$

2007 Moldova Team Selection Test, 1

Tags: inequalities , circumcircle , trigonometry , function , triangle inequality , geometry

Let $ABC$ be a triangle and $M,N,P$ be the midpoints of sides $BC, CA, AB$. The lines $AM, BN, CP$ meet the circumcircle of $ABC$ in the points $A_{1}, B_{1}, C_{1}$. Show that the area of triangle $ABC$ is at most the sum of areas of triangles $BCA_{1}, CAB_{1}, ABC_{1}$.

2010 IMAC Arhimede, 6

Tags: algebra , inequalities

Consider real numbers $a, b ,c \ge0$ with $a+b+c=2$. Prove that: $\frac{bc}{\sqrt[4]{3a^2+4}}+\frac{ca}{\sqrt[4]{3b^2+4}}+\frac{ab}{\sqrt[4]{3c^2+4}} \le \frac{2*\sqrt[4] {3}}{3}$

1967 Miklós Schweitzer, 4

Tags: inequalities , logarithm , real analysis

Let $ a_1,a_2,...,a_N$ be positive real numbers whose sum equals $ 1$. For a natural number $ i$, let $ n_i$ denote the number of $ a_k$ for which $ 2^{1-i} \geq a_k \geq 2^{-i}$ holds. Prove that \[ \sum_{i=1}^{\infty} \sqrt{n_i2^{-i}} \leq 4+\sqrt{\log_2 N}.\] [i]L. Leinder[/i]

2009 Postal Coaching, 1

Tags: algebra , min , inequalities

Find the minimum value of the expression $f(a, b, c) = (a + b)^4 + (b + c)^4 + (c + a)^4 - \frac47 (a^4 + b^4 + c^4)$, as $a, b, c$ varies over the set of all real numbers

2014 Saudi Arabia Pre-TST, 4.2

Tags: algebra , inequalities

Given $x \ge 0$, prove that $$\frac{(x^2 + 1)^6}{2^7}+\frac12 \ge x^5 - x^3 + x$$

2007 Indonesia TST, 3

Tags: algebra , inequalities , polynomial

Let $a, b, c$ be positive reals such that $a + b + c = 1$ and $P(x) = 3^{2005}x^{2007 }- 3^{2005}x^{2006} - x^2$. Prove that $P(a) + P(b) + P(c) \le -1$.

1992 IMO Longlists, 11

Tags: function , inequalities , number theory , relatively prime , number theoretic functions

Let $\phi(n,m), m \neq 1$, be the number of positive integers less than or equal to $n$ that are coprime with $m.$ Clearly, $\phi(m,m) = \phi(m)$, where $\phi(m)$ is Euler’s phi function. Find all integers $m$ that satisfy the following inequality: \[\frac{\phi(n,m)}{n} \geq \frac{\phi(m)}{m}\] for every positive integer $n.$

1976 All Soviet Union Mathematical Olympiad, 225

Tags: inequalities , vector , algebra

Given $4$ vectors $a,b,c,d$ in the plane, such that $a+b+c+d=0$. Prove the following inequality: $$|a|+|b|+|c|+|d| \ge |a+d|+|b+d|+|c+d|$$

1981 All Soviet Union Mathematical Olympiad, 320

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

A pupil has tried to make a copy of a convex polygon, drawn inside the unit circle. He draw one side, from its end -- another, and so on. Having finished, he has noticed that the first and the last vertices do not coincide, but are situated $d$ units of length far from each other. The pupil draw angles precisely, but made relative error less than $p$ in the lengths of sides. Prove that $d < 4p$.

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 $a_{1}+a_{2}+...+a_{n}\leq\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}}$.

2013 Online Math Open Problems, 25

Tags: geometry , perimeter , function , calculus , inequalities , geometric transformation

Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$. [i]Proposed by Evan Chen[/i]

2016 Iran MO (2nd Round), 1

Tags: inequalities , algebra

If $0<a\leq b\leq c$ prove that $$\frac{(c-a)^2}{6c}\leq \frac{a+b+c}{3}-\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$$

1997 All-Russian Olympiad, 1

Tags: quadratic , polynomial , inequalities , algebra

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]

2011 239 Open Mathematical Olympiad, 3

Tags: inequalities

Positive reals $a,b,c,d$ satisfy $a+b+c+d=4$. Prove that $\sum_{cyc}\frac{a}{a^3 + 4} \le \frac{4}{5}$

2010 Victor Vâlcovici, 1

Tags: inequalities , trigonometry

[b]a)[/b] Let be real numbers $ s,t\ge 0 $ and $ a,b\ge 1. $ Show that for any real $ x, $ it holds: $$ a^{s\sin x+t\cos x}b^{s\cos x+t\sin x}\le 10^{(s+t)\sqrt{\text{tg}^2 a+\text{tg}^2 b}} $$ [b]b)[/b] For $ a,b>0 $ is the above inequality still true? [i]Ilie Diaconu[/i]

1998 Austrian-Polish Competition, 1

Tags: inequalities , algebra

Let $x_1, x_2,y _1,y_2$ be real numbers such that $x_1^2 + x_2^2 \le 1$. Prove the inequality $$(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)$$

2005 Slovenia Team Selection Test, 6

Tags: inequalities , algebra

Let $a,b,c > 0$ and $ab+bc+ca = 1$. Prove the inequality $3\sqrt[3]{\frac{1}{abc} +6(a+b+c) }\le \frac{\sqrt[3]3}{abc}$

2000 All-Russian Olympiad, 4

Tags: inequalities , algebra , combinatorics , sequence

Let $a_1, a_2, \cdots, a_n$ be a sequence of nonnegative integers. For $k=1,2,\cdots,n$ denote \[ m_k = \max_{1 \le l \le k} \frac{a_{k-l+1} + a_{k-l+2} + \cdots + a_k}{l}. \] Prove that for every $\alpha > 0$ the number of values of $k$ for which $m_k > \alpha$ is less than $\frac{a_1+a_2+ \cdots +a_n}{\alpha}.$

1986 IMO Longlists, 35

Tags: triangle inequality , inequalities

Establish the maximum and minimum values that the sum $|a| + |b| + |c|$ can have if $a, b, c$ are real numbers such that the maximum value of $|ax^2 + bx + c|$ is $1$ for $-1 \leq x \leq 1.$

2012 Estonia Team Selection Test, 5

Tags: inequalities , max , algebra

Let $x, y, z$ be positive real numbers whose sum is $2012$. Find the maximum value of $$ \frac{(x^2 + y^2 + z^2)(x^3 + y^3 + z^3)}{(x^4 + y^4 + z^4)}$$

2019 Azerbaijan Junior NMO, 4

Tags: inequalities , semiperimeter

Prove that, for any triangle with side lengths $a,b,c$, the following inequality holds $$\frac{a}{(b+c)^2}+\frac{b}{(c+a)^2}+\frac{c}{(a+b)^2}\geq\frac9{8p}$$ ($p$ denotes the semiperimeter of a triangle)

2019 Mediterranean Mathematics Olympiad, 2

Tags: inequalities , algebra , sequence

Let $m_1<m_2<\cdots<m_s$ be a sequence of $s\ge2$ positive integers, none of which can be written as the sum of (two or more) distinct other numbers in the sequence. For every integer $r$ with $1\le r<s$, prove that \[ r\cdot m_r+m_s ~\ge~ (r+1)(s-1). \] (Proposed by Gerhard Woeginger, Austria)