Olympiad problems

2022 Serbia National Math Olympiad, P2

Tags: inequalities

Let $a$, $b$ and $c$ be positive real numbers and $a^3+b^3+c^3=3$. Prove $$\frac{1}{3-2a}+\frac{1}{3-2b}+\frac{1}{3-2c}\geq 3$$

PEN G Problems, 17

Tags: irrational number , logarithm , inequalities

Suppose that $p, q \in \mathbb{N}$ satisfy the inequality \[\exp(1)\cdot( \sqrt{p+q}-\sqrt{q})^{2}<1.\] Show that $\ln \left(1+\frac{p}{q}\right)$ is irrational.

2022 Kyiv City MO Round 2, Problem 1

Tags: ukraine , inequalities , algebra

Positive reals $x, y, z$ satisfy $$\frac{xy+1}{x+1} = \frac{yz+1}{y+1} = \frac{zx+1}{z+1}$$ Do they all have to be equal? [i](Proposed by Oleksii Masalitin)[/i]

PEN A Problems, 80

Tags: inequalities , divisibility theory , polynomial , algebra

Find all pairs of positive integers $m, n \ge 3$ for which there exist infinitely many positive integers $a$ such that \[\frac{a^{m}+a-1}{a^{n}+a^{2}-1}\] is itself an integer.

2014 Middle European Mathematical Olympiad, 2

Tags: algebra , inequalities , function

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that \[ xf(xy) + xyf(x) \ge f(x^2)f(y) + x^2y \] holds for all $x,y \in \mathbb{R}$.

2003 Czech And Slovak Olympiad III A, 6

Tags: inequalities

a，b，c>0，abc=1，prove that(a/b)+(b/c)+(c/a)≥a+b+c.

MathLinks Contest 7th, 5.1

Tags: product of roots , inequalities , vieta , algebra , absolute value , polynomial

Find all real polynomials $ g(x)$ of degree at most $ n \minus{} 3$, $ n\geq 3$, knowing that all the roots of the polynomial $ f(x) \equal{} x^n \plus{} nx^{n \minus{} 1} \plus{} \frac {n(n \minus{} 1)}2 x^{n \minus{} 2} \plus{} g(x)$ are real.

2001 Bosnia and Herzegovina Team Selection Test, 5

Tags: algebra , inequalities

Let $n$ be a positive integer, $n \geq 1$ and $x_1,x_2,...,x_n$ positive real numbers such that $x_1+x_2+...+x_n=1$. Does the following inequality hold $$\sum_{i=1}^{n} {\frac{x_i}{1-x_1\cdot...\cdot x_{i-1} \cdot x_{i+1} \cdot ... x_n}} \leq \frac{1}{1-\left(\frac{1}{n}\right)^{n-1}} $$

2024 China National Olympiad, 4

Tags: inequalities , algebra

Let $a_1, a_2, \ldots, a_{2023}$ be nonnegative real numbers such that $a_1 + a_2 + \ldots + a_{2023} = 100$. Let $A = \left \{ (i,j) \mid 1 \leqslant i \leqslant j \leqslant 2023, \, a_ia_j \geqslant 1 \right\}$. Prove that $|A| \leqslant 5050$ and determine when the equality holds. [i]Proposed by Yunhao Fu[/i]

2011 Brazil National Olympiad, 3

Tags: inequalities , parallelogram , geometry , trigonometry

Prove that, for all convex pentagons $P_1 P_2 P_3 P_4 P_5$ with area 1, there are indices $i$ and $j$ (assume $P_7 = P_2$ and $P_6 = P_1$) such that: \[ \text{Area of} \ \triangle P_i P_{i+1} P_{i+2} \le \frac{5 - \sqrt 5}{10} \le \text{Area of} \ \triangle P_j P_{j+1} P_{j+2}\]

2017 Saint Petersburg Mathematical Olympiad, 5

Tags: inequalities

Let $x,y,z>0 $ and $\sqrt{xyz}=xy+yz+zx$. Prove that$$x+y+z\leq \frac{1}{3}.$$

2007 Irish Math Olympiad, 5

Tags: inequalities

Let $ r$ and $ n$ be nonnegative integers such that $ r \le n$. $ (a)$ Prove that: $ \frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r}$ is an integer. $ (b)$ Prove that: $ \displaystyle\sum_{r\equal{}0}^{[n/2]}\frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r}<2^{n\minus{}2}$ for all $ n \ge 9$.

2007 IMO Shortlist, 3

Tags: function , algebra , calculus , inequalities

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

2000 Austrian-Polish Competition, 9

Tags: algebra , inequalities

If three nonnegative reals $a$, $b$, $c$ satisfy $a+b+c=1$, prove that $2 \leq \left(1-a^{2}\right)^{2}+\left(1-b^{2}\right)^{2}+\left(1-c^{2}\right)^{2}\leq \left(1+a\right)\left(1+b\right)\left(1+c\right)$.

2011 JBMO Shortlist, 6

Tags: inequalities , triangle inequality , geometry , rectangle , vector , trigonometry

Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that \[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\] If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

2023 District Olympiad, P3

Tags: complex number , inequalities

Let $n\geqslant 2$ be an integer. Determine all complex numbers $z{}$ which satisfy \[|z^{n+1}-z^n|\geqslant|z^{n+1}-1|+|z^{n+1}-z|.\]

2010 Contests, 2

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

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]

2005 Taiwan National Olympiad, 1

Tags: function , inequalities

Let $a,b,c$ be three positive real numbers such that $abc=1$. Prove that: \[ 1+\frac{3}{a+b+c}\ge{\frac{6}{ab+bc+ca}} . \]

2020 Regional Olympiad of Mexico Center Zone, 2

Tags: algebra , inequalities

Let $a$, $b$ and $c$ be positive real numbers, prove that \[\frac{2a^2 b^2}{a^5+b^5}+\frac{2b^2 c^2}{b^5+c^5}+\frac{2c^2 a^2}{c^5+a^5}\le\frac{a+b}{2ab}+\frac{b+c}{2bc}+\frac{c+a}{2ca}\]

1952 AMC 12/AHSME, 45

Tags: inequalities

If $ a$ and $ b$ are two unequal positive numbers, then: $ \textbf{(A)}\ \frac {2ab}{a \plus{} b} > \sqrt {ab} > \frac {a \plus{} b}{2} \qquad\textbf{(B)}\ \sqrt {ab} > \frac {2ab}{a \plus{} b} > \frac {a \plus{} b}{2}$ $ \textbf{(C)}\ \frac {2ab}{a \plus{} b} > \frac {a \plus{} b}{2} > \sqrt {ab} \qquad\textbf{(D)}\ \frac {a \plus{} b}{2} > \frac {2ab}{a \plus{} b} > \sqrt {ab}$ $ \textbf{(E)}\ \frac {a \plus{} b}{2} > \sqrt {ab} > \frac {2ab}{a \plus{} b}$

2020 Regional Olympiad of Mexico Southeast, 5

Tags: circumcircle , inequality , tangent , inequalities , geometry

Let $ABC$ an acute triangle with $\angle BAC\geq 60^\circ$ and $\Gamma$ it´s circumcircule. Let $P$ the intersection of the tangents to $\Gamma$ from $B$ and $C$. Let $\Omega$ the circumcircle of the triangle $BPC$. The bisector of $\angle BAC$ intersect $\Gamma$ again in $E$ and $\Omega$ in $D$, in the way that $E$ is between $A$ and $D$. Prove that $\frac{AE}{ED}\leq 2$ and determine when equality holds.

2019 Iran MO (3rd Round), 1

Tags: inequalities

$a,b$ and $c$ are positive real numbers so that $\sum_{\text{cyc}} (a+b)^2=2\sum_{\text{cyc}} a +6abc$. Prove that $$\sum_{\text{cyc}} (a-b)^2\leq\left|2\sum_{\text{cyc}} a -6abc\right|.$$

2011 Morocco National Olympiad, 4

Tags: inequalities , geometry , trapezoid , trigonometry

The diagonals of a trapezoid $ ABCD $ whose bases are $ [AB] $ and $ [CD] $ intersect at $P.$ Prove that \[S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA},\] Where $S_{XYZ} $ denotes the area of $\triangle XYZ $.

1999 Tuymaada Olympiad, 4

Tags: inequalities

Prove the inequality \[ {x\over y^2-z}+{y\over z^2-x}+{z\over x^2-y} > 1, \] where $2 < x, y, z < 4.$ [i]Proposed by A. Golovanov[/i]

2014 Thailand Mathematical Olympiad, 5

Tags: min , inequalities , algebra

Determine the maximal value of $k$ such that the inequality $$\left(k +\frac{a}{b}\right) \left(k + \frac{b}{c}\right)\left(k + \frac{c}{a}\right) \le \left( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\right) \left( \frac{b}{a}+ \frac{c}{b}+ \frac{a}{c}\right)$$ holds for all positive reals $a, b, c$.