Olympiad problems

2012 China Second Round Olympiad, 3

Tags: inequalities

Suppose that $x,y,z\in [0,1]$. Find the maximal value of the expression \[\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}.\]

1924 Eotvos Mathematical Competition, 1

Tags: number theory , geometry , triangle inequality , inequalities , geometric inequalities

Let $a, b, c$ be fìxed natural numbers. Suppose that, for every positive integer n, there is a triangle whose sides have lengths $a^n$, $b^n$, and $c^n$ respectively. Prove that these triangles are isosceles.

2009 Indonesia MO, 2

Tags: function , inequalities

Find the lowest possible values from the function \[ f(x) \equal{} x^{2008} \minus{} 2x^{2007} \plus{} 3x^{2006} \minus{} 4x^{2005} \plus{} 5x^{2004} \minus{} \cdots \minus{} 2006x^3 \plus{} 2007x^2 \minus{} 2008x \plus{} 2009\] for any real numbers $ x$.

2007 Ukraine Team Selection Test, 8

Tags: algebra , polynomial , inequalities

$ F(x)$ is polynomial with real coefficients. $ F(x) \equal{} x^{4}\plus{}a_{1}x^{3}\plus{}a_{2}x^{2}\plus{}a_{1}x^{1}\plus{}a_{0}$. $ M$ is local maximum and $ m$ is minimum. Prove that $ \frac{3}{10}(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}< M\minus{}m < 3(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}$

1992 All Soviet Union Mathematical Olympiad, 566

Tags: algebra , inequalities

Show that for any real numbers $x, y > 1$, we have $$\frac{x^2}{y - 1}+ \frac{y^2}{x - 1} \ge 8$$

2009 District Round (Round II), 2

Tags: inequalities , trigonometry , geometry

in a right-angled triangle $ABC$ with $\angle C=90$,$a,b,c$ are the corresponding sides.Circles $K.L$ have their centers on $a,b$ and are tangent to $b,c$;$a,c$ respectively,with radii $r,t$.find the greatest real number $p$ such that the inequality $\frac{1}{r}+\frac{1}{t}\ge p(\frac{1}{a}+\frac{1}{b})$ always holds.

2016 Tuymaada Olympiad, 7

Tags: inequalities

For every $x$, $y$, $z>{3\over 2}$ prove the inequality $$ x^{24} + \root 5\of {y^{60}+z^{40}} \geq \left(x^4 y^3 + {1\over 3} y^2 z^2 + {1\over 9} x^3 z^3 \right)^2. $$

2014 Contests, 1

Tags: algebra , inequality , inequalities

Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.

2010 AIME Problems, 14

Tags: floor function , search , inequalities , function , algebra , logarithm

For each positive integer n, let $ f(n) \equal{} \sum_{k \equal{} 1}^{100} \lfloor \log_{10} (kn) \rfloor$. Find the largest value of n for which $ f(n) \le 300$. [b]Note:[/b] $ \lfloor x \rfloor$ is the greatest integer less than or equal to $ x$.

2009 Moldova Team Selection Test, 4

Tags: inequalities

let $ x, y, z$ be real number in the interval $ [\frac12;2]$ and $ a, b, c$ a permutation of them. Prove the inequality: $ \dfrac{60a^2\minus{}1}{4xy\plus{}5z}\plus{}\dfrac{60b^2\minus{}1}{4yz\plus{}5x}\plus{}\dfrac{60c^2\minus{}1}{4zx\plus{}5y}\geq 12$

1999 Canada National Olympiad, 5

Tags: algebra , inequalities , three variable inequality , calculus

Let $ x$, $ y$, and $ z$ be non-negative real numbers satisfying $ x \plus{} y \plus{} z \equal{} 1$. Show that \[ x^2 y \plus{} y^2 z \plus{} z^2 x \leq \frac {4}{27} \] and find when equality occurs.

2001 Saint Petersburg Mathematical Olympiad, 11.4

Tags: inequalities , number theory , greatest common divisor , algebra , gcd , least common multiple

For any two positive integers $n>m$ prove the following inequality: $$[m,n]+[m+1,n+1]\geq \dfrac{2nm}{\sqrt{m-n}}$$ As always, $[x,y]$ means the least common multiply of $x,y$. [I]Proposed by A. Golovanov[/i]

2021 Stars of Mathematics, 4

Tags: inequalities , algebra

Let $k$ be a positive integer, and let $a,b$ and $c$ be positive real numbers. Show that \[a(1-a^k)+b(1-(a+b)^k)+c(1-(a+b+c)^k)<\frac{k}{k+1}.\] [i]* * *[/i]

2007 Bulgaria Team Selection Test, 2

Tags: inequalities , number theory

Let $n,k$ be positive integers such that $n\geq2k>3$ and $A= \{1,2,...,n\}.$ Find all $n$ and $k$ such that the number of $k$-element subsets of $A$ is $2n-k$ times bigger than the number of $2$-element subsets of $A.$

2010 Contests, 4

Tags: inequalities , number theory

Find all integer solutions $(a,b)$ of the equation \[ (a+b+3)^2 + 2ab = 3ab(a+2)(b+2)\]

2004 BAMO, 4

Tags: algebra , sum , inequalities

Suppose one is given $n$ real numbers, not all zero, but such that their sum is zero. Prove that one can label these numbers $a_1, a_2, ..., a_n$ in such a manner that $a_1a_2 + a_2a_3 +...+a_{n-1}a_n + a_na_1 < 0$.

1964 Swedish Mathematical Competition, 5

Tags: trigonometry , min , max , inequalities , algebra

$a_1, a_2, ... , a_n$ are constants such that $f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0$ for all $x$. We seek estimates of $a_1$. If $n = 2$, find the smallest and largest possible values of $a_1$. Find corresponding estimates for other values of $n$.

2021 South East Mathematical Olympiad, 3

Tags: inequalities , algebra

Let $a,b,c\geq 0$ and $a^2+b^2+c^2\leq 1.$ Prove that$$\frac{a}{a^2+bc+1}+\frac{b}{b^2+ca+1}+\frac{c}{c^2+ab+1}+3abc<\sqrt 3$$

2001 IMO Shortlist, 5

Tags: inequalities , algebra , recurrence relation , equation , integer sequence

Find all positive integers $a_1, a_2, \ldots, a_n$ such that \[ \frac{99}{100} = \frac{a_0}{a_1} + \frac{a_1}{a_2} + \cdots + \frac{a_{n-1}}{a_n}, \] where $a_0 = 1$ and $(a_{k+1}-1)a_{k-1} \geq a_k^2(a_k - 1)$ for $k = 1,2,\ldots,n-1$.

2004 USAMO, 1

Tags: inequalities , geometry , incenter , inradius , trigonometry , symmetry , trig identities

Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that \[ \frac{1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|. \] When does equality hold?

2002 JBMO ShortLists, 3

Tags: inequalities

Let $ a,b,c$ be positive real numbers such that $ abc\equal{}\frac{9}{4}$. Prove the inequality: $ a^3 \plus{} b^3 \plus{} c^3 > a\sqrt {b \plus{} c} \plus{} b\sqrt {c \plus{} a} \plus{} c\sqrt {a \plus{} b}$ Jury's variant: Prove the same, but with $ abc\equal{}2$

2002 Irish Math Olympiad, 5

Tags: inequalities

Let $ 0<a,b,c<1$. Prove the inequality: $ \frac{a}{1\minus{}a}\plus{}\frac{b}{1\minus{}b}\plus{}\frac{c}{1\minus{}c} \ge \frac {3 \sqrt[3]{abc}}{1\minus{} \sqrt[3]{abc}}.$ Determine the cases of equality.

2019 Romania Team Selection Test, 1

Tags: inequalities , algebra

Determine the largest value the expression $$ \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j} $$ may achieve, as $ x_1,x_2,x_3,x_4 $ run through the non-negative real numbers, and add up to $ 1. $ Find also the specific values of this numbers that make the above sum achieve the asked maximum.

2012 Today's Calculation Of Integral, 838

Tags: calculus , integration , inequalities , calculus computations

Prove that : $\frac{e-1}{e}<\int_0^1 e^{-x^2}dx<\frac{\pi}{4}.$

2014 China Team Selection Test, 5

Tags: complex number , inequalities

Let $n$ be a given integer which is greater than $1$ . Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1,z_2,\cdots,z_n$ ,have that \[\sum_{k\equal{}1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},\] where $z_{n+1}=z_1$.