Olympiad problems

1982 National High School Mathematics League, 8

Tags: inequalities

$a,b$ are two different positive real numbers, then which one is the largest? $$A=(a+\frac{1}{a})(b+\frac{1}{b}), B=(\sqrt{ab}+\frac{1}{\sqrt{ab}})^2, C=(\frac{a+b}{2}+\frac{2}{a+b})^2.$$ $\text{(A)}A\qquad\text{(B)}B\qquad\text{(C)}C\qquad\text{(D)}$Not sure.

1983 All Soviet Union Mathematical Olympiad, 354

Tags: inequalities , algebra , digit

Natural number $k$ has $n$ digits in its decimal notation. It was rounded up to tens, then the obtained number was rounded up to hundreds, and so on $(n-1)$ times. Prove that the obtained number $m$ satisfies inequality $m < \frac{18k}{13}$. (Examples of rounding: $191\to190\to 200, 135\to140\to 100$.)

1988 China Team Selection Test, 1

Tags: quadratic , inequalities

Suppose real numbers $A,B,C$ such that for all real numbers $x,y,z$ the following inequality holds: \[A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.\] Find the necessary and sufficient condition $A,B,C$ must satisfy (expressed by means of an equality or an inequality).

2007 Iran Team Selection Test, 1

Tags: polynomial , inequalities , algebra

Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]

2004 All-Russian Olympiad Regional Round, 11.7

Tags: algebra , trigonometry , inequalities

For what natural numbers $n$ for any numbers $a, b , c$, which are values of the angles of an acute triangle, the following inequality is true: $$\sin na + \sin nb + \sin nc < 0?$$

2013 Irish Math Olympiad, 10

Tags: inequalities

Let $a,b,c $ be real numbers and let $x=a+b+c,y=a^2+b^2+c^2,z=a^3+b^3+c^3$ and $S=2x^3-9xy+9z .$ $(a)$ Prove that $S$ is unchanged when $a,b,c$ are replaced by $a+t,b+t,c+t $ , respectively , for any real number $t$. $(b)$ Prove that $ (3y-x^2)^3\ge S^2 .$

2015 Regional Olympiad of Mexico Southeast, 3

Tags: algebra , perimeter , geometry , inequalities

If $T(n)$ is the numbers of triangles with integers sizes(not congruent with each other) with it´s perimeter is equal to $n$, prove that: $$T(2012)<T(2015)$$ $$T(2013)=T(2016)$$

2010 China Girls Math Olympiad, 8

Tags: calculus , integration , inequalities , modular arithmetic , number theory

Determine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1,m_2, n_1, n_2$ such that $a=m_1^2+n_1^2$, $a^2=m_2^2+n_2^2$, and $m_1-n_1=m_2-n_2.$

2008 Korea Junior Math Olympiad, 2

Tags: inequalities

Let $x,y\in\mathbb{R}$ such that $x>2, y>3$. Find the minimum value of $\frac{(x+y)^2}{\sqrt{x^2-4}+\sqrt{y^2-9}}$

2018 Pan-African Shortlist, A4

Tags: inequalities , algebra

Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that $$ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd. $$ Show that $$ \frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12 $$ and that $-12$ is the maximum.

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]

1982 AMC 12/AHSME, 29

Tags: inequalities , calculus , symmetry , function , algebra , domain

Let $ x$,$ y$, and $ z$ be three positive real numbers whose sum is $ 1$. If no one of these numbers is more than twice any other, then the minimum possible value of the product $ xyz$ is $ \textbf{(A)}\ \frac{1}{32}\qquad \textbf{(B)}\ \frac{1}{36}\qquad \textbf{(C)}\ \frac{4}{125}\qquad \textbf{(D)}\ \frac{1}{127}\qquad \textbf{(E)}\ \text{none of these}$

2002 Federal Competition For Advanced Students, Part 1, 2

Tags: inequalities

Find the greatest real number $C$ such that, for all real numbers $x$ and $y \neq x$ with $xy = 2$ it holds that \[\frac{((x + y)^2 - 6)((x - y)^2 + 8)}{(x-y)^2}\geq C.\] When does equality occur?

2010 Romania Team Selection Test, 2

Tags: function , parameterization , induction , inequalities , rearrangement inequality , algebra

Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by \[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \] is a decreasing function. [i]Dan Marinescu et al.[/i]

2001 China Team Selection Test, 2

Tags: trigonometry , inequality , sequence , inequalities , analysis

Let ${a_n}$ be a non-increasing sequence of positive numbers. Prove that if for $n \ge 2001$, $na_{n} \le 1$, then for any positive integer $m \ge 2001$ and $x \in \mathbb{R}$, the following inequality holds: $\left | \sum_{k=2001}^{m} a_{k} \sin kx \right | \le 1 + \pi$

2018 Korea USCM, 6

Tags: inequalities , real analysis , derivative , function

Suppose a continuous function $f:[0,1]\to\mathbb{R}$ is differentiable on $(0,1)$ and $f(0)=1$, $f(1)=0$. Then, there exists $0<x_0<1$ such that $$|f'(x_0)| \geq 2018 f(x_0)^{2018}$$

1953 Czech and Slovak Olympiad III A, 3

Tags: inequalities

Prove that the inequality $$\left(a_1+\cdots+a_n\right)\left(\frac{1}{a_1}+\cdots+\frac{1}{a_n}\right)\ge n^2$$ holds for any positive numbers $a_1,\ldots,a_n$ and determine when equality occurs.

2007 iTest Tournament of Champions, 4

Tags: inequalities

Find the smallest positive integer $k$ such that \[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\] for some ordered triple of positive integers $(a,b,c)$.

2014 ELMO Shortlist, 1

Tags: inequalities , trigonometry , calculus , topology , three variable inequality

In a non-obtuse triangle $ABC$, prove that \[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]

2005 All-Russian Olympiad Regional Round, 11.7

Tags: inequalities , number theory

11.7 Let $N$ be a number of perfect squares from $\{1,2,...,10^{20}\}$, which 17-th digit from the end is 7, and $M$ be a number of perfect squares from $\{1,2,...,10^{20}\}$, which 17-th digit from the end is 8. Compare $M$ and $N$. ([i]A. Golovanov[/i])

2020 Bulgaria Team Selection Test, 5

Tags: function , induction , algebra , inequalities , additive function , functional equation

Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$. Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$

2008 Alexandru Myller, 3

Tags: integral , function , real analysis , inequalities , calculus , integration

Find the nondecreasing functions $ f:[0,1]\rightarrow\mathbb{R} $ that satisfy $$ \left| \int_0^1 f(x)e^{nx} dx\right|\le 2008 , $$ for any nonnegative integer $ n. $ [i]Mihai Piticari[/i]

2011 Saudi Arabia BMO TST, 2

Tags: algebra , inequalities

Let $a_1,a_2,..., a_n$ be real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $|a_1| + |a_2 | + ... + |a_n | = 1$. Prove that $$ |a_1 + 2a_2 + ... + na_n | \le \frac{n-1}{2} $$

1985 IMO Longlists, 23

Tags: floor function , inequalities , relatively prime , number theory

Let $\mathbb N = {1, 2, 3, . . .}$. For real $x, y$, set $S(x, y) = \{s | s = [nx+y], n \in \mathbb N\}$. Prove that if $r > 1$ is a rational number, there exist real numbers $u$ and $v$ such that \[S(r, 0) \cap S(u, v) = \emptyset, S(r, 0) \cup S(u, v) = \mathbb N.\]

2019 Junior Balkan Team Selection Tests - Romania, 3

Tags: inequality , inequalities

Real numbers $a,b,c,d$ such that $|a|>1$ , $|b|>1$ , $|c|>1$ , $|d|>1$ and $ab(c+d)+dc(a+b)+a+b+c+d=0$ then prove that $\frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}+\frac{1}{d-1} >0$