Olympiad problems

2005 India IMO Training Camp, 3

Tags: inequalities

If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

2014 AMC 12/AHSME, 8

Tags: percent , inequalities

A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1: $10\%$ off the listed price if the listed price is at least $\$50$ Coupon 2: $\$20$ off the listed price if the listed price is at least $\$100$ Coupon 3: $18\%$ off the amount by which the listed price exceeds $\$100$ For which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$? $\textbf{(A) }\$179.95\qquad \textbf{(B) }\$199.95\qquad \textbf{(C) }\$219.95\qquad \textbf{(D) }\$239.95\qquad \textbf{(E) }\$259.95\qquad$

2017 Korea Winter Program Practice Test, 4

Tags: algebra , inequalities

Let $a,b,c,d$ be the area of four faces of a tetrahedron, satisfying $a+b+c+d=1$. Show that $$\sqrt[n]{a^n+b^n+c^n}+\sqrt[n]{b^n+c^n+d^n}+\sqrt[n]{c^n+d^n+a^n}+\sqrt[n]{d^n+a^n+b^n} \le 1+\sqrt[n]{2}$$ holds for all positive integers $n$.

2015 IMC, 2

Tags: number , inequalities

For a positive integer $n$, let $f(n)$ be the number obtained by writing $n$ in binary and replacing every 0 with 1 and vice versa. For example, $n=23$ is 10111 in binary, so $f(n)$ is 1000 in binary, therefore $f(23) =8$. Prove that \[\sum_{k=1}^n f(k) \leq \frac{n^2}{4}.\] When does equality hold? (Proposed by Stephan Wagner, Stellenbosch University)

2014 Vietnam National Olympiad, 2

Tags: inequalities

Find the maximum of \[P=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}\] where $x,y,z$ are positive real numbers.

1998 German National Olympiad, 5

Tags: inequalities , sum , recurrence relation , sequence , algebra

A sequence ($a_n$) is given by $a_0 = 0, a_1 = 1$ and $a_{k+2} = a_{k+1} +a_k$ for all integers $k \ge 0$. Prove that the inequality $\sum_{k=0}^n \frac{a_k}{2^k}< 2$ holds for all positive integers $n$.

2007 Nicolae Păun, 2

Tags: function , inequalities , real analysis , primitivability , convexity , monotone , primitive

Consider a sequence of positive real numbers $ \left( x_n \right)_{n\ge 1} $ and a primitivable function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ [b]a)[/b] Prove that $ f $ is monotonic and continuous if for any natural numbers $ n $ and real numbers $ x, $ the inequality $$ f\left( x+x_n \right)\geqslant f(x) $$ is true. [b]b)[/b] Show that $ f $ is convex if for any natural numbers $ n $ and real numbers $ x, $ the inequality $$ f\left( x+2x_n \right) +f(x)\geqslant 2f\left( x+x_n \right) $$ is true. [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

2007 Baltic Way, 4

Tags: inequalities

Let $a_1,a_2,\ldots ,a_n$ be positive real numbers, and let $S=a_1+a_2 +\ldots +a_n$ . Prove that \[(2S+n)(2S+a_1a_2+a_2a_3+\ldots +a_na_1)\ge 9(\sqrt{a_1a_2}+\sqrt{a_2a_3}+\ldots +\sqrt{a_na_1})^2 \]

1990 IMO Longlists, 59

Tags: inequalities

Given eight real numbers $a_1 \leq a_2 \leq \cdots \leq a_7 \leq a_8$. Let $x = \frac{ a_1 + a_2 + \cdots + a_7 + a_8}{8}$, $y = \frac{ a_1^2 + a_2^2 + \cdots + a_7^2 + a_8^2}{8}$. Prove that \[2 \sqrt{y-x^2} \leq a_8 - a_1 \leq 4 \sqrt{y-x^2}.\]

2007 Middle European Mathematical Olympiad, 1

Tags: inequalities

Let $ a,b,c,d$ be positive real numbers with $ a\plus{}b\plus{}c\plus{}d \equal{} 4$. Prove that \[ a^{2}bc\plus{}b^{2}cd\plus{}c^{2}da\plus{}d^{2}ab\leq 4.\]

2012 Dutch IMO TST, 2

Tags: inequalities

Let $a, b, c$ and $d$ be positive real numbers. Prove that $$\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a} +\frac{d - a}{a + b } \ge 0 $$

2013 Albania Team Selection Test, 2

Tags: rearrangement inequality , inequalities

Let $a,b,c,d$ be positive real numbers such that $abcd=1$.Find with proof that $x=3 $ is the minimal value for which the following inequality holds: \[a^x+b^x+c^x+d^x\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}\]

2016 Saint Petersburg Mathematical Olympiad, 2

Tags: algebra , inequalities , positive real

Given the positive numbers $x_1, x_2,..., x_n$, such that $x_i \le 2x_j$ with $1 \le i < j \le n$. Prove that there are positive numbers $y_1\le y_2\le...\le y_n$, such that $x_k \le y_k \le 2x_k$ for all $k=1,2,..., n$

1998 India Regional Mathematical Olympiad, 3

Tags: inequalities

Prove that for every natural number $n > 1$ \[ \frac{1}{n+1} \left( 1 + \frac{1}{3} +\frac{1}{5} + \ldots + \frac{1}{2n-1} \right) > \frac{1}{n} \left( \frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2n} \right) . \]

2018 Balkan MO Shortlist, G3

Tags: geometry , geometric inequality , inequalities , semiperimeter , maximum value , minimum

Let $P$ be an interior point of triangle $ABC$. Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of $$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right)$$ taking into consideration all possible choices of triangle $ABC$ and of point $P$. by Elton Bojaxhiu, Albania

2018 District Olympiad, 2

Tags: algebra , inequalities

Let $a,b,c \in [1, \infty)$. Prove that: $$\frac{a\sqrt{b}}{a+b}+\frac{b\sqrt{c}}{b+c}+\frac{c\sqrt{b}}{c+a}+\frac32 \le a+b+c$$

2007 Switzerland - Final Round, 7

Tags: inequalities , algebra , radical

Let $a, b, c$ be nonnegative real numbers with arithmetic mean $m =\frac{a+b+c}{3}$ . Provethat $$\sqrt{a+\sqrt{b + \sqrt{c}}} +\sqrt{b+\sqrt{c + \sqrt{a}}} +\sqrt{c +\sqrt{a + \sqrt{b}}}\le 3\sqrt{m+\sqrt{m + \sqrt{m}}}.$$

1961 IMO Shortlist, 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 $\geq 2$

2024 VJIMC, 3

Tags: inequalities , graph , degree , combinatorics

Let $n$ be a positive integer and let $G$ be a simple undirected graph on $n$ vertices. Let $d_i$ be the degree of its $i$-th vertex, $i = 1, \dots , n$. Denote $\Delta=\max d_i$. Prove that if \[\sum_{i=1}^n d_i^2>n\Delta(n-\Delta),\] then $G$ contains a triangle.

2019 Jozsef Wildt International Math Competition, W. 24

Tags: inequalities

If $a$, $b$, $c > 0$, prove that$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{a+b}{a+b+2c}+\frac{b+c}{2a+b+c}+\frac{c+a}{a+2b+c}$$

2012 ELMO Shortlist, 6

Tags: floor function , algebra , polynomial , vieta , quadratic , inequalities , number theory

Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

VI Soros Olympiad 1999 - 2000 (Russia), 8.7

Tags: algebra , inequalities

Prove that for any positive real $x$ and $y$, holds the inequality $$\frac{1}{(x+y)^2}+\frac{1}{x^2}+\frac{1}{y^2} \ge \frac{9}{4xy}$$

2009 Silk Road, 1

Tags: inequalities

Prove that, abc≤1 and a,b,c>0 \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq 1+ \frac{6}{a+b+c} \]

1972 AMC 12/AHSME, 10

Tags: inequalities

For $x$ real, the inequality $1\le |x-2|\le 7$ is equivalent to $\textbf{(A) }x\le 1\text{ or }x\ge 3\qquad\textbf{(B) }1\le x\le 3\qquad\textbf{(C) }-5\le x\le 9\qquad$ $\textbf{(D) }-5\le x\le 1\text{ or }3\le x\le 9\qquad \textbf{(E) }-6\le x\le 1\text{ or }3\le x\le 10$

2010 China Team Selection Test, 3

Tags: floor function , inequalities , algorithm , combinatorics

Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying (1) for each $n_i$, its digits belong to the set $\{1,2\}$; (2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right. Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.