Olympiad problems

2005 Baltic Way, 2

Let $\alpha$, $\beta$ and $\gamma$ be three acute angles such that $\sin \alpha+\sin \beta+\sin \gamma = 1$. Show that \[\tan^{2}\alpha+\tan^{2}\beta+\tan^{2}\gamma \geq \frac{3}{8}. \]

1987 Vietnam National Olympiad, 1

Tags: inequalities

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be positive real numbers ($ n \ge 2$) whose sum is $ S$. Show that \[ \sum_{i\equal{}1}^n\frac{a_i^{2^{k}}}{\left(S\minus{}a_i\right)^{2^t\minus{}1}}\ge\frac{S^{1\plus{}2^k\minus{}2^t}}{\left(n\minus{}1\right)^{2^t\minus{}1}n^{2^k\minus{}2^t}}\] for any nonnegative integers $ k$, $ t$ with $ k \ge t$. When does equality occur?

2006 Moldova National Olympiad, 10.1

Tags: trigonometry , inequalities , function , geometry

Let $a,b$ be the smaller sides of a right triangle. Let $c$ be the hypothenuse and $h$ be the altitude from the right angle. Fint the maximal value of $\frac{c+h}{a+b}$.

2009 Jozsef Wildt International Math Competition, W. 23

Tags: inequalities

If $x_k \in \mathbb{R}$ ($k=1, 2, \cdots , n$) and $m \in \mathbb{N}$ then [list=1] [*] $\sum \limits_{cyc} \left (x_1^2 -x_1x_2+x_2^2 \right )^m \leq 3^m \sum \limits_{k=1}^n x_k^{2m}$ [*] $\prod \limits_{cyc} \left (x_1^2 -x_1x_2+x_2^2 \right )^m \leq \left (\frac{3^m}{n}\right )^m \left (\sum \limits_{k=1}^n x_k^{2m}\right )^n$ [/list]

2002 Moldova National Olympiad, 4

Tags: inequalities

At least two of the nonnegative real numbers $ a_1,a_2,...,a_n$ aer nonzero. Decide whether $ a$ or $ b$ is larger if $ a\equal{}\sqrt[2002]{a_1^{2002}\plus{}a_2^{2002}\plus{}\ldots\plus{}a_n^{2002}}$ and $ b\equal{}\sqrt[2003]{a_1^{2003}\plus{}a_2^{2003}\plus{}\ldots\plus{}a_n^{2003} }$

1999 Irish Math Olympiad, 3

Tags: inequalities

The sum of positive real numbers $ a,b,c,d$ is $ 1$. Prove that: $ \frac{a^2}{a\plus{}b}\plus{}\frac{b^2}{b\plus{}c}\plus{}\frac{c^2}{c\plus{}d}\plus{}\frac{d^2}{d\plus{}a} \ge \frac{1}{2},$ with equality if and only if $ a\equal{}b\equal{}c\equal{}d\equal{}\frac{1}{4}$.

2020 Bulgaria EGMO TST, 1

Tags: inequalities

The positive integers $a$, $p$, $q$ and $r$ are greater than $1$ and are such that $p$ divides $aqr+1$, $q$ divides $apr+1$ and $r$ divides $apq+1$. Prove that: a) There are infinitely many such quadruples $(a,p,q,r)$. b) For each such quadruple we have $a\geq \frac{pqr-1}{pq+qr+rp}$.

2003 JBMO Shortlist, 6

Tags: inequalities

Parallels to the sides of a triangle passing through an interior point divide the inside of a triangle into $6$ parts with the marked areas as in the figure. Show that $\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\ge \frac{3}{2}$ [img]https://cdn.artofproblemsolving.com/attachments/a/a/b0a85df58f2994b0975b654df0c342d8dc4d34.png[/img]

1991 Cono Sur Olympiad, 3

Tags: algebra , number theory , prime number , inequalities

Given a positive integrer number $n$ ($n\ne 0$), let $f(n)$ be the average of all the positive divisors of $n$. For example, $f(3)=\frac{1+3}{2}=2$, and $f(12)=\frac{1+2+3+4+6+12}{6}=\frac{14}{3}$. [b]a[/b] Prove that $\frac{n+1}{2} \ge f(n)\ge \sqrt{n}$. [b]b[/b] Find all $n$ such that $f(n)=\frac{91}{9}$.

2018 JBMO TST-Turkey, 8

Tags: positive real numbers , triangle inequality , inequalities , algebra

Let $x, y, z$ be positive real numbers such that $\sqrt {x}, \sqrt {y}, \sqrt {z}$ are sides of a triangle and $\frac {x}{y}+\frac {y}{z}+\frac {z}{x}=5$. Prove that $\frac {x(y^2-2z^2)}{z}+\frac {y(z^2-2x^2)}{x}+\frac {z(x^2-2y^2)}{y}\geqslant0$

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$

2013 Tournament of Towns, 5

Tags: algebra , function , inequalities

Do there exist two integer-valued functions $f$ and $g$ such that for every integer $x$ we have (a) $f(f(x)) = x, g(g(x)) = x, f(g(x)) > x, g(f(x)) > x$ ? (b) $f(f(x)) < x, g(g(x)) < x, f(g(x)) > x, g(f(x)) > x$ ?

2020 Moldova EGMO TST, 1

Tags: inequalities , number theory

Let[i] $a,b,c$[/i] be positive integers , such that $A=\frac{a^2+1}{bc}+\frac{b^2+1}{ca}+\frac{c^2+1}{ab}$ is, also, an integer. Proof that $\gcd( a, b, c)\leq\lfloor\sqrt[3]{a+ b+ c}\rfloor$.

2018 Azerbaijan Senior NMO, 5

Tags: inequalities , three variable inequality

Prove that if $x$, $y$, $z$ are positive real numbers and $xyz = 1$ then \[\frac{x^3}{x^2+y}+\frac{y^3}{y^2+z}+\frac{z^3}{z^2+x}\geq \dfrac {3} {2}.\] [i]A. Golovanov[/i]

2010 China Team Selection Test, 2

Tags: inequalities , triangle inequality , floor function , polynomial , algebra

Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$, and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.

2011 Dutch BxMO TST, 4

Tags: algebra , number theory , inequalities

Let $n \ge 2$ be an integer. Let $a$ be the greatest positive integer such that $2^a | 5^n - 3^n$. Let $b$ be the greatest positive integer such that $2^b \le n$. Prove that $a \le b + 3$.

1998 National High School Mathematics League, 2

Tags: inequalities

Let $a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n$ are real numbers in $[1,2]$. If $\sum_{i=1}^{n}a_i^2=\sum_{i=1}^{n}b_i^2$, prove that $$\sum_{i=1}^{n}\frac{a_i^3}{b_i}\leq\frac{17}{10}\sum_{i=1}^{n}a_i^2.$$ Find when the equality holds.

2016 MMATHS, 4

Tags: algebra , inequalities

For real numbers $a, b, c$ with $a + b + c = 3$, prove that $$a^2(b - c)^2 + b^2(c - a)^2 + c^2(a - b)^2 \ge \frac9 2 abc(1 - abc)$$ and state when equality is reached.

2016 Romanian Master of Mathematics, 4

Tags: algebra , inequalities

Let $x$ and $y$ be positive real numbers such that: $x+y^{2016}\geq 1$. Prove that $x^{2016}+y> 1-\frac{1}{100}$

2022 Indonesia TST, A

Tags: function , inequalities , fe , algebra

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying \[ f(a^2) - f(b^2) \leq (f(a)+b)(a-f(b)) \] for all $a,b \in \mathbb{R}$.

2019 Belarusian National Olympiad, 11.6

Tags: geometry , inequalities

The diagonals of the inscribed quadrilateral $ABCD$ intersect at the point $O$. The points $P$, $Q$, $R$, and $S$ are the feet of the perpendiculars from $O$ to the sides $AB$, $BC$, $CD$, and $DA$, respectively. Prove the inequality $BD\ge SP+QR$. [i](A. Naradzetski)[/i]

2018 Bulgaria JBMO TST, 4

Tags: inequalities , combinatorics

Each cell of an infinite table (infinite in all directions) is colored with one of $n$ given colors. All six cells of any $2\times 3$ (or $3 \times 2$) rectangle have different colors. Find the smallest possible value of $n$.

1988 Swedish Mathematical Competition, 6

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

The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_{n+1} = \sqrt{a_n^2 +\frac{1}{a_n}}$ for $n \ge 1$. Prove that there exists $a$ such that $\frac{1}{2} \le \frac{a_n}{n^a} \le 2$ for $n \ge 1$.

1986 IMO Longlists, 1

Tags: inequalities

Let $k$ be one of the integers $2, 3,4$ and let $n = 2^k -1$. Prove the inequality \[1+ b^k + b^{2k} + \cdots+ b^{nk} \geq (1 + b^n)^k\] for all real $b \geq 0.$

1993 APMO, 3

Tags: inequalities , algebra , polynomial , ratio

Let \begin{eqnarray*} f(x) & = & a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \ \ \mbox{and} \\ g(x) & = & c_{n+1} x^{n+1} + c_n x^n + \cdots + c_0 \end{eqnarray*} be non-zero polynomials with real coefficients such that $g(x) = (x+r)f(x)$ for some real number $r$. If $a = \max(|a_n|, \ldots, |a_0|)$ and $c = \max(|c_{n+1}|, \ldots, |c_0|)$, prove that $\frac{a}{c} \leq n+1$.