Olympiad problems

1989 Tournament Of Towns, (203) 1

Tags: algebra , inequalities

The positive numbers $a, b$ and $c$ satisfy $a \ge b \ge c$ and $a + b + c \le 1$ . Prove that $a^2 + 3b^2 + 5c^2 \le 1$ . (F . L . Nazarov)

1984 IMO Shortlist, 20

Tags: inequalities , algebra , logarithm

Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]

2005 Bulgaria Team Selection Test, 4

Tags: function , quadratic , inequalities

Let $a_{i}$ and $b_{i}$, where $i \in \{1,2, \dots, 2005 \}$, be real numbers such that the inequality $(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})$ holds for all $x \in \mathbb{R}$ and all $i \in \{1,2, \dots, 2005 \}$. Find the maximum possible number of positive numbers amongst $a_{i}$ and $b_{i}$, $i \in \{1,2, \dots, 2005 \}$.

2019 Romania National Olympiad, 1

Tags: inequalities , number theory , divisor

Consider $A$, the set of natural numbers with exactly $2019$ natural divisors , and for each $n \in A$, denote $$S_n=\frac{1}{d_1+\sqrt{n}}+\frac{1}{d_2+\sqrt{n}}+...+\frac{1}{d_{2019}+\sqrt{n}}$$ where $d_1,d_2, .., d_{2019}$ are the natural divisors of $n$. Determine the maximum value of $S_n$ when $n$ goes through the set $ A$.

2002 China Team Selection Test, 1

Tags: inequalities , trigonometry , geometry

In acute triangle $ ABC$, show that: $ \sin^3{A}\cos^2{(B \minus{} C)} \plus{} \sin^3{B}\cos^2{(C \minus{} A)} \plus{} \sin^3{C}\cos^2{(A \minus{} B)} \leq 3\sin{A} \sin{B} \sin{C}$ and find out when the equality holds.

2010 Polish MO Finals, 3

Tags: limit , algebra , inequality , inequalities , logarithm

Real number $C > 1$ is given. Sequence of positive real numbers $a_1, a_2, a_3, \ldots$, in which $a_1=1$ and $a_2=2$, satisfy the conditions \[a_{mn}=a_ma_n, \] \[a_{m+n} \leq C(a_m + a_n),\] for $m, n = 1, 2, 3, \ldots$. Prove that $a_n = n$ for $n=1, 2, 3, \ldots$.

2012 IMC, 5

Tags: inequalities , abstract algebra , induction , group theory

Let $c \ge 1$ be a real number. Let $G$ be an Abelian group and let $A \subset G$ be a finite set satisfying $|A+A| \le c|A|$, where $X+Y:= \{x+y| x \in X, y \in Y\}$ and $|Z|$ denotes the cardinality of $Z$. Prove that \[|\underbrace{A+A+\dots+A}_k| \le c^k |A|\] for every positive integer $k$. [i]Proposed by Przemyslaw Mazur, Jagiellonian University.[/i]

2000 Junior Balkan MO, 2

Tags: induction , inequalities , algebra , difference of squares , special factorizations , number theory

Find all positive integers $n\geq 1$ such that $n^2+3^n$ is the square of an integer. [i]Bulgaria[/i]

2020-21 IOQM India, 5

Tags: inequalities , absolute value , algebra

Find the number of integer solutions to $||x| - 2020| < 5$.

2023 CMIMC Team, 2

Tags: team , inequalities , algebra

Real numbers $x$ and $y$ satisfy \begin{align*} x^2 + y^2 &= 2023 \\ (x-2)(y-2) &= 3. \end{align*} Find the largest possible value of $|x-y|$. [i]Proposed by Howard Halim[/i]

2010 Postal Coaching, 5

Tags: rearrangement inequality , inequalities

For any positive real numbers $a, b, c$, prove that \[\sum_{cyclic} \frac{(b + c)(a^4 - b^2 c^2 )}{ab + 2bc + ca} \ge 0\]

2018 China Team Selection Test, 6

Tags: combinatorics , inequalities

Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ subsets of a set of size $n$. Prove that $$ \sum_{i=1}^{m} \sum_{j=1}^{m}|A_i|\cdot |A_i \cap A_j|\geq \frac{1}{mn}\left(\sum_{i=1}^{m}|A_i|\right)^3.$$

2016 India Regional Mathematical Olympiad, 2

Tags: inequalities , algebra

Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$ Prove that $abc \le \frac{1}{8}$.

2005 China National Olympiad, 4

Tags: inequalities , calculus , function , derivative , logarithm , algebra

The sequence $\{a_n\}$ is defined by: $a_1=\frac{21}{16}$, and for $n\ge2$,\[ 2a_n-3a_{n-1}=\frac{3}{2^{n+1}}. \]Let $m$ be an integer with $m\ge2$. Prove that: for $n\le m$, we have\[ \left(a_n+\frac{3}{2^{n+3}}\right)^{\frac{1}{m}}\left(m-\left(\frac{2}{3}\right)^{{\frac{n(m-1)}{m}}}\right)<\frac{m^2-1}{m-n+1}. \]

2016 Estonia Team Selection Test, 8

Tags: inequalities , algebra

Let $x, y$ and $z$ be positive real numbers such that $x + y + z = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ . Prove that $xy + yz + zx \ge 3$.

2018 Serbia Team Selection Test, 2

Tags: inequality , algebra , inequalities

Let $n$ be a fixed positive integer and let $x_1,\ldots,x_n$ be positive real numbers. Prove that $$x_1\left(1-x_1^2\right)+x_2\left(1-(x_1+x_2)^2\right)+\cdots+x_n\left(1-(x_1+...+x_n)^2\right)<\frac{2}{3}.$$

1986 Vietnam National Olympiad, 2

Tags: function , inequalities

Find all $ n > 1$ such that the inequality \[ \sum_{i\equal{}1}^nx_i^2\ge x_n\sum_{i\equal{}1}^{n\minus{}1}x_i\] holds for all real numbers $ x_1$, $ x_2$, $ \ldots$, $ x_n$.

2011 Bogdan Stan, 3

Tags: inequalities , cauchy schwarz

Prove that $$ a+b+c>\left( \sqrt\alpha +\sqrt\beta +\sqrt\gamma \right)^2, $$ for all positive real numbers $ a,b,c,\alpha ,\beta ,\gamma $ that are under the condition $$ abc>\alpha bc+\beta ac+\gamma ab. $$ [i]Țuțescu Lucian[/i] and [i]Chiriță Aurel[/i]

2023 ISI Entrance UGB, 6

Tags: inequalities , recurrence relation

Let $\{u_n\}_{n \ge 1}$ be a sequence of real numbers defined as $u_1 = 1$ and \[ u_{n+1} = u_n + \frac{1}{u_n} \text{ for all $n \ge 1$.}\] Prove that $u_n \le \frac{3\sqrt{n}}{2}$ for all $n$.

2006 Victor Vâlcovici, 1

Tags: inequalities

Let be two nonnegative real numbers $ a,b, $ not both $ 0, $ satisfying $ \frac{1}{2a+b} +\frac{1}{a+2b} =1. $ Prove the following inequalities and explain the equality cases for [b]a),b).[/b] [b]a)[/b] $ 4/3\le a+b\le 3/2 $ [b]b)[/b] $ 8/9\le a^2+b^2\le 9/4 $ [b]c)[/b] $ ab<1/2 $ [i]Laurențiu Panaitopol[/i]

2007 Indonesia TST, 3

Tags: inequalities , floor function , diophantine equation , number theory

For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.

1992 Flanders Math Olympiad, 4

Tags: trigonometry , parameterization , inequalities , cyclic quadrilateral , geometry

Let $A,B,P$ positive reals with $P\le A+B$. (a) Choose reals $\theta_1,\theta_2$ with $A\cos\theta_1 + B\cos\theta_2=P$ and prove that \[ A\sin\theta_1 + B\sin\theta_2 \le \sqrt{(A+B-P)(A+B+P)} \] (b) Prove equality is attained when $\theta_1=\theta_2=\arccos\left(\dfrac{P}{A+B}\right)$. (c) Take $A=\dfrac{1}{2}xy, B=\dfrac{1}{2}wz$ and $P=\dfrac14 \left(x^2+y^2-z^2-w^2\right)$ with $0<x\le y\le x+z+w$, $z,w>0$ and $z^2+w^2<x^2+y^2$. Show that we can translate (a) and (b) into the following theorem: from all quadrilaterals with (ordered) sidelenghts $(x,y,z,w)$, the cyclical one has the greatest area.

2013 ELMO Shortlist, 5

Tags: function , calculus , derivative , 3-variable inequality , logarithm , inequalities

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]

1975 Dutch Mathematical Olympiad, 3

Tags: inequalities , sum , algebra

Given are the real numbers $x_1,x_2,...,x_n$ and $t_1,t_2,...,t_n$ for which holds: $\sum_{i=1}^n x_i = 0$. Prove that $$\sum_{i=1}^n \left( \sum_{j=1}^n (t_i-t_j)^2x_ix_j \right)\le 0.$$

2006 National Olympiad First Round, 16

Tags: inequalities

How many positive integer tuples $ (x_1,x_2,\dots, x_{13})$ are there satisfying the inequality $x_1+x_2+\dots + x_{13}\leq 2006$? $ \textbf{(A)}\ \frac{2006!}{13!1993!} \qquad\textbf{(B)}\ \frac{2006!}{14!1992!} \qquad\textbf{(C)}\ \frac{1993!}{12!1981!} \qquad\textbf{(D)}\ \frac{1993!}{13!1980!} \qquad\textbf{(E)}\ \text{None of above} $