Olympiad problems

2018 China National Olympiad, 1

Let $n$ be a positive integer. Let $A_n$ denote the set of primes $p$ such that there exists positive integers $a,b$ satisfying $$\frac{a+b}{p} \text{ and } \frac{a^n + b^n}{p^2}$$ are both integers that are relatively prime to $p$. If $A_n$ is finite, let $f(n)$ denote $|A_n|$. a) Prove that $A_n$ is finite if and only if $n \not = 2$. b) Let $m,k$ be odd positive integers and let $d$ be their gcd. Show that $$f(d) \leq f(k) + f(m) - f(km) \leq 2 f(d).$$

2005 IMO Shortlist, 3

Tags: inequalities , algebra , calculus

Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.

2010 Indonesia TST, 2

Tags: polynomial , inequalities , algebra

Consider a polynomial with coefficients of real numbers $ \phi(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ with three positive real roots. Assume that $ \phi(0)<0$, prove that \[ 2b^3\plus{}9a^2d\minus{}7abc \le 0.\] [i]Hery Susanto, Malang[/i]

1997 Baltic Way, 4

Tags: inequalities

Prove that the arithmetic mean $a$ of $x_1,\ldots ,x_n$ satisfies \[ (x_1-a)^2+\ldots +(x_n-a)^2\le \frac{1}{2}(|x_1-a|+\ldots +|x_n-a|)^2\]

PEN D Problems, 15

Tags: congruence , modular arithmetic , inequalities

Let $n_{1}, \cdots, n_{k}$ and $a$ be positive integers which satify the following conditions:[list][*] for any $i \neq j$, $(n_{i}, n_{j})=1$, [*] for any $i$, $a^{n_{i}} \equiv 1 \pmod{n_i}$, [*] for any $i$, $n_{i}$ does not divide $a-1$. [/list] Show that there exist at least $2^{k+1}-2$ integers $x>1$ with $a^{x} \equiv 1 \pmod{x}$.

2014 Saudi Arabia Pre-TST, 3.2

Tags: algebra , min , inequalities

Let $x, y$ be positive real numbers. Find the minimum of $$x^2 + xy +\frac{y^2}{2}+\frac{2^6}{x + y}+\frac{3^4}{x^3}$$

2013 Tuymaada Olympiad, 7

Tags: inequalities , geometry , 3d geometry , tetrahedron , sphere

Points $A_1$, $A_2$, $A_3$, $A_4$ are the vertices of a regular tetrahedron of edge length $1$. The points $B_1$ and $B_2$ lie inside the figure bounded by the plane $A_1A_2A_3$ and the spheres of radius $1$ and centres $A_1$, $A_2$, $A_3$. Prove that $B_1B_2 < \max\{B_1A_1, B_1A_2, B_1A_3, B_1A_4\}$. [i] A. Kupavsky [/i]

1999 IMC, 2

Tags: inequalities , rearrangement inequality

Does there exist a bijective map $f:\mathbb{N} \rightarrow \mathbb{N}$ so that $\sum^{\infty}_{n=1}\frac{f(n)}{n^2}$ is finite?

2011 Federal Competition For Advanced Students, Part 2, 2

Tags: inequalities

Let $k$ and $n$ be positive integers. Show that if $x_j$ ($1\leqslant j\leqslant n$) are real numbers with $\sum_{j=1}^n\frac{1}{x_j^{2^k}+k}=\frac{1}{k}$, then \[\sum_{j=1}^n\frac{1}{x_j^{2^{k+1}}+k+2}\leqslant\frac{1}{k+1}\mbox{.}\]

1998 Romania National Olympiad, 2

Tags: complex number , inequalities , algebra

Let $a \ge1$ be a real number and $z$ be a complex number such that $| z + a | \le a$ and $|z^2+ a | \le a$. Show that $| z | \le a$.

2004 Federal Math Competition of S&M, 3

Tags: inequalities

If $a,b,c$ are positive numbers such that $abc = 1$, prove the inequality $\frac{1}{\sqrt{b+\frac{1}{a}+\frac{1}{2}}} + \frac{1}{\sqrt{c+\frac{1}{b}+\frac{1}{2}}} + \frac{1}{\sqrt{a+\frac{1}{c}+\frac{1}{2}}} \geq \sqrt{2}$

2008 IMO Shortlist, 2

Tags: inequalities , algebra , vieta , calculus

[b](a)[/b] Prove that \[\frac {x^{2}}{\left(x \minus{} 1\right)^{2}} \plus{} \frac {y^{2}}{\left(y \minus{} 1\right)^{2}} \plus{} \frac {z^{2}}{\left(z \minus{} 1\right)^{2}} \geq 1\] for all real numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$. [b](b)[/b] Prove that equality holds above for infinitely many triples of rational numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$. [i]Author: Walther Janous, Austria[/i]

2019 New Zealand MO, 8

Tags: algebra , inequalities

Suppose that $x_1, x_2, x_3, . . . x_n$ are real numbers between $0$ and $ 1$ with sum $s$. Prove that $$\prod_{i=1}^{n} \frac{x_i}{s + 1 - x_i} + \prod_{i=1}^{n} (1 - x_i) \le 1.$$

2017 China Team Selection Test, 5

Tags: inequalities

Given integer $m\geq2$,$x_1,...,x_m$ are non-negative real numbers,prove that:$$(m-1)^{m-1}(x_1^m+...+x_m^m)\geq(x_1+...+x_m)^m-m^mx_1...x_m$$and please find out when the equality holds.

1985 Austrian-Polish Competition, 7

Tags: inequalities , maximization , ratio , four variables

Find an upper bound for the ratio $$\frac{x_1x_2+2x_2x_3+x_3x_4}{x_1^2+x_2^2+x_3^2+x_4^2}$$ over all quadruples of real numbers $(x_1,x_2,x_3,x_4)\neq (0,0,0,0)$. [i]Note.[/i] The smaller the bound, the better the solution.

2015 IFYM, Sozopol, 1

Tags: parallelogram , inequalities , trigonometry , trig identities , law of cosines , geometry

Let ABCD be a convex quadrilateral such that $AB + CD = \sqrt{2}AC$ and $BC + DA = \sqrt{2}BD$. Prove that ABCD is a parallelogram.

2009 Today's Calculation Of Integral, 398

Tags: calculus , integration , inequalities , geometry , search , calculus computations

In $ xyz$ space, find the volume of the solid expressed by the sytem of inequality: $ 0\leqq x\leqq 1,\ 0\leqq y\leqq 1,\ 0\leqq z\leqq 1$ $ x^2 \plus{} y^2 \plus{} z^2 \minus{} 2xy \minus{} 1\geqq 0$

2009 Canada National Olympiad, 3

Tags: calculus , inequalities

Define $f(x,y,z)=\frac{(xy+yz+zx)(x+y+z)}{(x+y)(y+z)(z+x)}$. Determine the set of real numbers $r$ for which there exists a triplet of positive real numbers satisfying $f(x,y,z)=r$.

2011 AMC 10, 2

Tags: inequalities

Josanna's test scores to date are 90, 80, 70, 60, and 85. Her goal is to raise her test average at least 3 points with her next test. What is the minimum test score she would need to accomplish this goal? $ \textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95 $

2001 Baltic Way, 12

Tags: inequalities

Let $a_1, a_2,\ldots , a_n$ be positive real numbers such that $\sum_{i=1}^na_i^3=3$ and $\sum_{i=1}^na_i^5=5$. Prove that $\sum_{i=1}^na_i>\frac{3}{2}$.

2010 China Team Selection Test, 2

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

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 Hanoi Open Mathematics Competitions, 5

Tags: algebra , maximum , inequalities

Let $a, b, c$ be positive integers such that $a + 2b +3c = 100$. Find the greatest value of $M = abc$

2004 China Team Selection Test, 1

Tags: inequalities , circumcircle , geometry

Find the largest value of the real number $ \lambda$, such that as long as point $ P$ lies in the acute triangle $ ABC$ satisfying $ \angle{PAB}\equal{}\angle{PBC}\equal{}\angle{PCA}$, and rays $ AP$, $ BP$, $ CP$ intersect the circumcircle of triangles $ PBC$, $ PCA$, $ PAB$ at points $ A_1$, $ B_1$, $ C_1$ respectively, then $ S_{A_1BC}\plus{} S_{B_1CA}\plus{} S_{C_1AB} \geq \lambda S_{ABC}$.

2017 Kyiv Mathematical Festival, 4

Tags: inequalities

Real numbers $x,y$ are such that $x^2\ge y$ and $y^2\ge x.$ Prove that $\frac{x}{y^2+1}+\frac{y}{x^2+1}\le1.$

2010 Junior Balkan MO, 4

Tags: geometry , rectangle , rotation , inequalities , combinatorics

A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.