Olympiad problems

2010 Turkey MO (2nd round), 3

Tags: inequalities

Prove that for all $n \in \mathbb{Z^+}$ and for all positive real numbers satisfying $a_1a_2...a_n=1$ \[ \displaystyle\sum_{i=1}^{n} \frac{a_i}{\sqrt{{a_i}^4+3}} \leq \frac{1}{2}\displaystyle\sum_{i=1}^{n} \frac{1}{a_i} \]

2022 Tuymaada Olympiad, 4

Tags: inequality , inequalities

For every positive $a_1, a_2, \dots, a_6$, prove the inequality \[ \sqrt[4]{\frac{a_1}{a_2 + a_3 + a_4}} + \sqrt[4]{\frac{a_2}{a_3 + a_4 + a_5}} + \dots + \sqrt[4]{\frac{a_6}{a_1 + a_2 + a_3}} \ge 2 \]

1994 Taiwan National Olympiad, 2

Tags: inequalities

Let $a,b,c$ are positive real numbers and $\alpha$ be any real number. Denote $f(\alpha)=abc(a^{\alpha}+b^{\alpha}+c^{\alpha}), g(\alpha)=a^{2+\alpha}(b+c-a)+b^{2+\alpha}(-b+c+a)+c^{2+\alpha}(b-c+a)$. Determine $\min{|f(\alpha)-g(\alpha)|}$ and $\max{|f(\alpha)-g(\alpha)|}$, if they are exists.

2011 USAMTS Problems, 3

Tags: algebra , function , induction , floor function , inequalities

Find all integers $b$ such that there exists a positive real number $x$ with \[ \dfrac {1}{b} = \dfrac {1}{\lfloor 2x \rfloor} + \dfrac {1}{\lfloor 5x \rfloor} \] Here, $\lfloor y \rfloor$ denotes the greatest integer that is less than or equal to $y$.

2016 Kosovo National Mathematical Olympiad, 3

Tags: inequalities

If $\alpha $ is an acute angle and $a,b\geq 0$ then show that: $\left( a+\frac{b}{\sin \alpha}\right)\left(b+\frac{a}{\cos \alpha}\right)\geq a^2+b^2+3ab$

1993 China National Olympiad, 6

Tags: induction , function , inequalities

Let $f: (0,+\infty)\rightarrow (0,+\infty)$ be a function satisfying the following condition: for arbitrary positive real numbers $x$ and $y$, we have $f(xy)\le f(x)f(y)$. Show that for arbitrary positive real number $x$ and natural number $n$, inequality $f(x^n)\le f(x)f(x^2)^{\dfrac{1}{2}}\dots f(x^n)^{\dfrac{1}{n}}$ holds.

2009 Croatia Team Selection Test, 1

Tags: inequalities , algebra

Solve in the set of real numbers: \[ 3\left(x^2 \plus{} y^2 \plus{} z^2\right) \equal{} 1, \] \[ x^2y^2 \plus{} y^2z^2 \plus{} z^2x^2 \equal{} xyz\left(x \plus{} y \plus{} z\right)^3. \]

2006 AMC 12/AHSME, 10

Tags: triangle inequality , geometry , perimeter , inequalities

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle? $ \textbf{(A) } 43 \qquad \textbf{(B) } 44 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 46 \qquad \textbf{(E) } 47$

2017 India IMO Training Camp, 1

Tags: inequalities

Let $a,b,c$ be distinct positive real numbers with $abc=1$. Prove that $$\sum_{\text{cyc}} \frac{a^6}{(a-b)(a-c)}>15.$$

2022 Harvard-MIT Mathematics Tournament, 4

Tags: algebra , inequalities

Suppose $n \ge 3$ is a positive integer. Let $a_1 < a_2 < ... < a_n$ be an increasing sequence of positive real numbers, and let $a_{n+1} = a_1$. Prove that $$\sum_{k=1}^{n}\frac{a_k}{a_{k+1}}>\sum_{k=1}^{n}\frac{a_{k+1}}{a_k}$$

2013 ELMO Shortlist, 8

Tags: function , inequalities

Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that \[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]

2014 Miklós Schweitzer, 8

Tags: function , inequalities , real analysis , algebra , functional analysis , polynomial

Let $n\ge 1$ be a fixed integer. Calculate the distance $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$ , where $p$ runs over polynomials of degree less than $n$ with real coefficients and $f$ runs over functions $f(x)= \sum_{k=n}^{\infty} c_k x^k$ defined on the closed interval $[0,1]$ , where $c_k \ge 0$ and $\sum_{k=n}^{\infty} c_k=1$.

1987 Traian Lălescu, 1.1

Tags: inequalities

Let be a natural number $ n. $ Show that: [b]a.[/b] $ \left( 1+1/n \right)^k<1+k/n+k^2/n^2, $ for all naturals $ k $ with $ 1\le k\le n. $ [b]b.[/b] $ 3^n\cdot n!>(1+n)^n. $

1986 AMC 12/AHSME, 29

Tags: inequalities , calculus , triangle inequality , integration

Two of the altitudes of the scalene triangle $ABC$ have length $4$ and $12$. If the length of the third altitude is also an integer, what is the biggest it can be? $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ \text{none of these} $

2007 South East Mathematical Olympiad, 1

Tags: function , algebra , inequalities

Let $f(x)$ be a function satisfying $f(x+1)-f(x)=2x+1 (x \in \mathbb{R})$.In addition, $|f(x)|\le 1$ holds for $x\in [0,1]$. Prove that $|f(x)|\le 2+x^2$ holds for $x \in \mathbb{R}$.

1983 National High School Mathematics League, 6

Tags: inequalities

Let $a,b,c,d,m,n$ be positive real numbers. $P=\sqrt{ab}+\sqrt{cd},Q=\sqrt{ma+nc}\cdot\sqrt{\frac{b}{m}+\frac{d}{n}}$. Then $\text{(A)}P\geq Q\qquad\text{(B)}P\leq Q\qquad\text{(C)}P<Q\qquad\text{(D)}$Not sure

2019 Durer Math Competition Finals, 1

Tags: inequalities , algebra , sequence , sum

Let $a_o,a_1,a_2,..,a_ n$ be a non-decreasing sequence of $n+1$ real numbers where $a_0 = 0$ and for every $j > i $ we have $a_j - a_i \le j - i$. Show that $$\left (\sum_{i=0}^n a_i \right )^2 \ge \sum_{i=0}^n a_i^3$$

2012 IMAC Arhimede, 6

Tags: algebra , inequalities , three variable inequality

Let $a,b,c$ be positive real numbers that satisfy the condition $a + b + c = 1$. Prove the inequality $$\frac{a^{-3}+b}{1-a}+\frac{b^{-3}+c}{1-b}+\frac{c^{-3}+a}{1-c}\ge 123$$

PEN I Problems, 6

Tags: floor function , inequalities

Prove that for all positive integers $n$, \[\lfloor \sqrt{n}+\sqrt{n+1}+\sqrt{n+2}\rfloor =\lfloor \sqrt{9n+8}\rfloor.\]

2013 Turkey Team Selection Test, 3

Tags: trigonometry , inequalities

For all real numbers $x,y,z$ such that $-2\leq x,y,z \leq 2$ and $x^2+y^2+z^2+xyz = 4$, determine the least real number $K$ satisfying \[\dfrac{z(xz+yz+y)}{xy+y^2+z^2+1} \leq K.\]

1999 Brazil National Olympiad, 3

Tags: inequalities , rectangle , geometry , graph theory , combinatorics

How many coins can be placed on a $10 \times 10$ board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board?

2020 Jozsef Wildt International Math Competition, W56

Tags: inequalities

If $p_k>0,a_k\ge2~(k=1,2,\ldots,n)$ and $$S_n=\sum_{k=1}^na_k,A_n=\prod_{\text{cyc}}a_1^{p_2+p_3+\ldots+p^n},B_n=\prod_{k=1}^na_k^{p_k},$$ then prove that $$\sum_{k=1}^np_k\log_{S_n-a_k}a_k\ge\left(\sum_{k=1}^np_k\right)\log_{A_n}B_n$$ [i]Proposed by Mihály Bencze[/i]

2019 Kosovo National Mathematical Olympiad, 3

Tags: inequalities , algebra

Show that for any non-negative real numbers $a,b,c,d$ such that $a^2+b^2+c^2+d^2=1$ the following inequality hold: $$a+b+c+d-1\geq 16abcd$$ When does equality hold?

2008 Moldova Team Selection Test, 2

Tags: inequalities , function

Let $ a_1,\ldots,a_n$ be positive reals so that $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le\frac n2$. Find the minimal value of $ \sqrt{a_1^2\plus{}\frac1{a_2^2}}\plus{}\sqrt{a_2^2\plus{}\frac1{a_3^2}}\plus{}\ldots\plus{}\sqrt{a_n^2\plus{}\frac1{a_1^2}}$.

PEN P Problems, 11

Tags: inequalities , additive number theory

For each positive integer $n$, let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $f(4)=4$, because the number $4$ can be represented in the following four ways: \[4, 2+2, 2+1+1, 1+1+1+1.\] Prove that, for any integer $n \geq 3$, \[2^{n^{2}/4}< f(2^{n}) < 2^{n^{2}/2}.\]