Olympiad problems

2012 China Second Round Olympiad, 6

Let $f(x)$ be an odd function on $\mathbb{R}$, such that $f(x)=x^2$ when $x\ge 0$. Knowing that for all $x\in [a,a+2]$, the inequality $f(x+a)\ge 2f(x)$ holds, find the range of real number $a$.

1953 AMC 12/AHSME, 45

Tags: inequalities

The lengths of two line segments are $ a$ units and $ b$ units respectively. Then the correct relation between them is: $ \textbf{(A)}\ \frac{a\plus{}b}{2} > \sqrt{ab} \qquad\textbf{(B)}\ \frac{a\plus{}b}{2} < \sqrt{ab} \qquad\textbf{(C)}\ \frac{a\plus{}b}{2} \equal{} \sqrt{ab}\\ \textbf{(D)}\ \frac{a\plus{}b}{2} \leq \sqrt{ab} \qquad\textbf{(E)}\ \frac{a\plus{}b}{2} \geq \sqrt{ab}$

2024 Argentina National Math Olympiad Level 3, 5

Tags: geometric inequality , algebra , geometry , inequalities

In triangle $ABC$, let $A'$, $B'$ and $C'$ be points on the sides $BC$, $CA$ and $AB$, respectively, such that$$\frac{BA'}{A'C}=\frac{CB'}{B'A}=\frac{AC'}{C'B}.$$ The line parallel to $B'C'$ passing through $A'$ intersects line $AC$ at $P$ and line $AB$ at $Q$. Prove that$$\frac{PQ}{B'C'} \geqslant 2.$$

2016 Peru Cono Sur TST, P2

Tags: polygon , geometric inequality , geometry , inequalities , area

Let $\omega$ be a circle. For each $n$, let $A_n$ be the area of a regular $n$-sided polygon circumscribed to $\omega$ and $B_n$ the area of a regular $n$-sided polygon inscribed in $\omega$ . Try that $3A_{2015} + B_{2015}> 4A_{4030}$

1998 Switzerland Team Selection Test, 9

Tags: algebra , inequalities

If $x$ and $y$ are positive numbers, prove the inequality $\frac{x}{x^4 +y^2 }+\frac{y}{x^2 +y^4} \le \frac{1}{xy}$ .

2012 ELMO Shortlist, 6

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

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]

1989 Romania Team Selection Test, 1

Tags: algebra , inequalities , induction , limit

Prove that $\sqrt {1+\sqrt {2+\ldots +\sqrt {n}}}<2$, $\forall n\ge 1$.

1993 Romania Team Selection Test, 4

Tags: max , binomial , combination , inequalities , combinatorics

For each integer $n > 3$ find all quadruples $(n_1,n_2,n_3,n_4)$ of positive integers with $n_1 +n_2 +n_3 +n_4 = n$ which maximize the expression $$\frac{n!}{n_1!n_2!n_3!n_4!}2^{ {n_1 \choose 2}+{n_2 \choose 2}+{n_3 \choose 2}+{n_4 \choose 2}+n_1n_2+n_2n_3+n_3n_4}$$

2020 USA IMO Team Selection Test, 5

Tags: polynomial , integer polynomial , inequalities , number theory

Find all integers $n \ge 2$ for which there exists an integer $m$ and a polynomial $P(x)$ with integer coefficients satisfying the following three conditions: [list] [*]$m > 1$ and $\gcd(m,n) = 1$; [*]the numbers $P(0)$, $P^2(0)$, $\ldots$, $P^{m-1}(0)$ are not divisible by $n$; and [*]$P^m(0)$ is divisible by $n$. [/list] Here $P^k$ means $P$ applied $k$ times, so $P^1(0) = P(0)$, $P^2(0) = P(P(0))$, etc. [i]Carl Schildkraut[/i]

1991 Baltic Way, 5

Tags: search , inequalities

For any positive numbers $a, b, c$ prove the inequalities \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge \frac{2}{a+b}+\frac{2}{b+c}+\frac{2}{c+a}\ge \frac{9}{a+b+c}.\]

2012 Turkmenistan National Math Olympiad, 1

Tags: trigonometry , inequalities

Find the max and min value of $a\cos^2 x+b\sin x\cos x+c\sin^2 x$.

2005 IMC, 3

Tags: integration , inequalities

3) $f$ cont diff, $R\rightarrow ]0,+\infty[$, prove $|\int_{0}^{1}f^{3}-{f(0)}^{2}\int_{0}^{1}f| \leq \max_{[0,1]} |f'|(\int_{0}^{1}f)^{2}$

2006 Iran MO (3rd Round), 1

Tags: floor function , inequalities , ceiling function , combinatorics

Let $A$ be a family of subsets of $\{1,2,\ldots,n\}$ such that no member of $A$ is contained in another. Sperner’s Theorem states that $|A|\leq{n\choose{\lfloor\frac{n}{2}\rfloor}}$. Find all the families for which the equality holds.

2008 IMO Shortlist, 5

Tags: algebra , inequalities

Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that \[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\] [i]Proposed by Pavel Novotný, Slovakia[/i]

2016 Balkan MO Shortlist, A7

Tags: inequalities , number theory

Find all integers $n\geq 2$ for which there exist the real numbers $a_k, 1\leq k \leq n$, which are satisfying the following conditions: \[\sum_{k=1}^n a_k=0, \sum_{k=1}^n a_k^2=1 \text{ and } \sqrt{n}\cdot \Bigr(\sum_{k=1}^n a_k^3\Bigr)=2(b\sqrt{n}-1), \text{ where } b=\max_{1\leq k\leq n} \{a_k\}.\]

2019 Saudi Arabia JBMO TST, 1

Tags: sum , radical , inequalities

Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove that $$\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c} \ge 2\sqrt2 \left( \sqrt{\frac{1-a}{a}}+\sqrt{\frac{1-b}{b}}+\sqrt{\frac{1-c}{c}}\right)$$

2024 China National Olympiad, 2

Tags: inequalities , algebra

Find the largest real number $c$ such that $$\sum_{i=1}^{n}\sum_{j=1}^{n}(n-|i-j|)x_ix_j \geq c\sum_{j=1}^{n}x^2_i$$ for any positive integer $n $ and any real numbers $x_1,x_2,\dots,x_n.$

2002 Rioplatense Mathematical Olympiad, Level 3, 4

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

Let $a, b$ and $c$ be positive real numbers. Show that $\frac{a+b}{c^2}+ \frac{c+a}{b^2}+ \frac{b+c}{a^2}\ge \frac{9}{a+b+c}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

2015 China Team Selection Test, 2

Tags: inequalities

Let $a_1,a_2,a_3, \cdots ,a_n$ be positive real numbers. For the integers $n\ge 2$, prove that\[ \left (\frac{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}{\sum_{j=1}^{n}a_j} \right )^{\frac{1}{n}}+\frac{\left (\prod_{i=1}^{n}a_i \right )^{\frac{1}{n}}}{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}\le \frac{n+1}{n}\]

1985 Austrian-Polish Competition, 3

Tags: quadrilateral , geometric inequalities , sum , area , geometry , inequalities

In a convex quadrilateral of area $1$, the sum of the lengths of all sides and diagonals is not less than $4+\sqrt 8$. Prove this.

2018 Saudi Arabia JBMO TST, 2

Tags: inequalities

Let $a, b, c$ be reals which satisfy $a+b+c+ab+bc+ac+abc=>7$, prove that $$\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2}+\sqrt{c^2+a^2+2}=>6$$

1997 Israel National Olympiad, 3

Tags: product , number theory , inequalities , prime

Let $n?$ denote the product of all primes smaller than $n$. Prove that $n? > n$ holds for any natural number $n > 3$.

2009 Danube Mathematical Competition, 5

Tags: sum , permutation , inequalities , function , combinatorics

Let $\sigma, \tau$ be two permutations of the quantity $\{1, 2,. . . , n\}$. Prove that there is a function $f: \{1, 2,. . . , n\} \to \{-1, 1\}$ such that for any $1 \le i \le j \le n$, we have $\left|\sum_{k=i}^{j} f(\sigma (k)) \right| \le 2$ and $\left|\sum_{k=i}^{j} f(\tau (k))\right| \le 2$

2013 JBMO TST - Turkey, 5

Tags: inequalities

Let $a, b, c ,d$ be real numbers greater than $1$ and $x, y$ be real numbers such that \[ a^x+b^y = (a^2+b^2)^x \quad \text{and} \quad c^x+d^y = 2^y(cd)^{y/2} \] Prove that $x<y$.

2019 IFYM, Sozopol, 7

Tags: algebra , inequalities

Let $a, b, c$ be positive real numbers such that $abc=8$. Prove that \[ \frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} +\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3} \]