Olympiad problems

the 13th XMO, P9

Tags: inequalities , algebra

Find the maximum value of $\lambda ,$ such that for $\forall x,y\in\mathbb R_+$ satisfying $2x-y=2x^3+y^3,x^2+\lambda y^2\leqslant 1.$

2018 Romania Team Selection Tests, 1

Tags: inequalities , n-variable inequality

Find the least number $ c$ satisfyng the condition $\sum_{i=1}^n {x_i}^2\leq cn$ and all real numbers $x_1,x_2,...,x_n$ are greater than or equal to $-1$ such that $\sum_{i=1}^n {x_i}^3=0$

MOAA Individual Speed General Rounds, 2021.4

Tags: speed , algebra , inequalities

Let $a$, $b$, and $c$ be real numbers such that $0\le a,b,c\le 5$ and $2a + b + c = 10$. Over all possible values of $a$, $b$, and $c$, determine the maximum possible value of $a + 2b + 3c$. [i]Proposed by Andrew Wen[/i]

2004 France Team Selection Test, 2

Tags: inequalities , geometry , incenter , trigonometry , inradius , perimeter , cyclic quadrilateral

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

OMMC POTM, 2022 9

Tags: least common multiple , inequalities , number theory

For positive integers $a_1 < a_2 < \dots < a_n$ prove that $$\frac{1}{\operatorname{lcm}(a_1, a_2)}+\frac{1}{\operatorname{lcm}(a_2, a_3)}+\dots+\frac{1}{\operatorname{lcm}(a_{n-1}, a_n)} \leq 1-\frac{1}{2^{n-1}}.$$ [i]Proposed by Evan Chang (squareman), USA[/i]

2023 Brazil Undergrad MO, 1

Tags: inequalities , real analysis , function , algebra

Let $p$ be the [i]potentioral[/i] function, from positive integers to positive integers, defined by $p(1) = 1$ and $p(n + 1) = p(n)$, if $n + 1$ is not a perfect power and $p(n + 1) = (n + 1) \cdot p(n)$, otherwise. Is there a positive integer $N$ such that, for all $n > N,$ $p(n) > 2^n$?

1989 IMO Longlists, 91

Tags: function , domain , inequalities , algebra

For $ \phi: \mathbb{N} \mapsto \mathbb{Z}$ let us define \[ M_{\phi} \equal{} \{f: \mathbb{N} \mapsto \mathbb{Z}, f(x) > f(\phi(x)), \forall x \in \mathbb{N} \}.\] Prove that if $ M_{\phi_1} \equal{} M_{\phi_2} \neq \emptyset,$ then $ \phi_1 \equal{} \phi_2.$ Does this property remain true if \[ M_{\phi} \equal{} \{f: \mathbb{N} \mapsto \mathbb{N}, f(x) > f(\phi(x)), \forall x \in \mathbb{N} \}?\]

2017 Vietnamese Southern Summer School contest, Problem 2

Tags: fraction , inequalities , cyclic inequality

Let $a,b,c$ be the positive real numbers satisfying $a^2+b^2+c^2=3$. Prove that: $$\frac{a}{b(a+c)}+\frac{b}{c(b+a)}+\frac{c}{a(c+b)}\geq \frac{3}{2}.$$

2013 Harvard-MIT Mathematics Tournament, 29

Tags: hmmt , inequalities

Let $A_1,A_2,\ldots,A_m$ be finite sets of size $2012$ and let $B_1,B_2,\ldots,B_m$ be finite sets of size $2013$ such that $A_i\cap B_j=\emptyset$ if and only if $i=j$. Find the maximum value of $m$.

1999 AIME Problems, 8

Tags: geometry , analytic geometry , parallelogram , inequalities

Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac 12,\frac 13,\frac 16\right).$ The area of $\mathcal{S}$ divided by the area of $\mathcal{T}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

2019 Junior Balkan MO, 2

Tags: algebra , inequalities

Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that $a^4 - 2019a = b^4 - 2019b = c$. Prove that $- \sqrt{c} < ab < 0$.

2002 China Team Selection Test, 2

Tags: inequalities , trigonometry , geometry

$ A_1$, $ B_1$ and $ C_1$ are the projections of the vertices $ A$, $ B$ and $ C$ of triangle $ ABC$ on the respective sides. If $ AB \equal{} c$, $ AC \equal{} b$, $ BC \equal{} a$ and $ AC_1 \equal{} 2t AB$, $ BA_1 \equal{} 2rBC$, $ CB_1 \equal{} 2 \mu AC$. Prove that: \[ \frac {a^2}{b^2} \cdot \left( \frac {t}{1 \minus{} 2t} \right)^2 \plus{} \frac {b^2}{c^2} \cdot \left( \frac {r}{1 \minus{} 2r} \right)^2 \plus{} \frac {c^2}{a^2} \cdot \left( \frac {\mu}{1 \minus{} 2\mu} \right)^2 \plus{} 16tr \mu \geq 1 \]

2015 Romania National Olympiad, 2

Tags: algebra , inequalities

The numbers $x, y, z, t, a$ and $b$ are positive integers, so that $xt-yz = 1$ and $$\frac{x}{y} \ge \frac{a}{b} \ge \frac{z}{t} .$$Prove that $$ab \le (x + z) (y +t)$$

1970 Polish MO Finals, 2

Tags: algebra , inequalities

Consider three sequences $(a_n)_{n=1}^{^\infty}$, $(b_n)_{n=1}^{^\infty}$ , $(c_n)_{n=1}^{^\infty}$, each of which has pairwisedistinct terms. Prove that there exist two indices $k$ and $l$ for which $k < l$, $$a_k < a_l , b_k < b_l , \,\,\, and \,\,\, c_k < c_l.$$

2008 JBMO Shortlist, 7

Tags: inequalities , algebra

Let $a, b$ and $c$ be positive real numbers such that $abc = 1$. Prove the inequality $\Big(ab + bc +\frac{1}{ca}\Big)\Big(bc + ca +\frac{1}{ab}\Big)\Big(ca + ab +\frac{1}{bc}\Big)\ge (1 + 2a)(1 + 2b)(1 + 2c)$.

2013 Moldova Team Selection Test, 1

Tags: inequalities , function

Consider real numbers $x,y,z$ such that $x,y,z>0$. Prove that \[ (xy+yz+xz)\left(\frac{1}{x^2+y^2}+\frac{1}{x^2+z^2}+\frac{1}{y^2+z^2}\right) > \frac{5}{2}. \]

1995 Irish Math Olympiad, 1

Tags: induction , logarithm , inequalities

Prove that for every positive integer $ n$, $ n^n \le (n!)^2 \le \left( \frac{(n\plus{}1)(n\plus{}2)}{6} \right) ^n.$

2021 Saudi Arabia Training Tests, 33

Tags: number theory , inequalities

Call a positive integer $x$ to be [i]remote from squares and cubes [/i] if each integer $k$ satisfies both $|x - k^2| > 10^6$ and $|x - k^3| > 10^6$. Prove that there exist infinitely many positive integer $n$ such that $2^n$ is remote from squares and cubes.

2015 USA TSTST, 4

Tags: inequalities , algebra

Let $x$, $y$, and $z$ be real numbers (not necessarily positive) such that $x^4+y^4+z^4+xyz=4$. Show that $x\le2$ and $\sqrt{2-x}\ge\frac{y+z}{2}$. [i]Proposed by Alyazeed Basyoni[/i]

PEN A Problems, 13

Tags: floor function , inequalities , divisibility theory , logarithm

Show that for all prime numbers $p$, \[Q(p)=\prod^{p-1}_{k=1}k^{2k-p-1}\] is an integer.

2009 Baltic Way, 4

Tags: inequalities

Determine all integers $ n>1$ for which the inequality \[ x_1^2\plus{}x_2^2\plus{}\ldots\plus{}x_n^2\ge(x_1\plus{}x_2\plus{}\ldots\plus{}x_{n\minus{}1})x_n\] holds for all real $ x_1,x_2,\ldots,x_n$.

2012 India National Olympiad, 1

Tags: inequalities , trigonometry , function , circumcircle , topology , geometry

Let $ABCD$ be a quadrilateral inscribed in a circle. Suppose $AB=\sqrt{2+\sqrt{2}}$ and $AB$ subtends $135$ degrees at center of circle . Find the maximum possible area of $ABCD$.

2003 Junior Macedonian Mathematical Olympiad, Problem 3

Tags: geometry , geometric inequality , inequalities

Let $ABC$ be a given triangle. The circumcircle of the triangle has radius $R$, the incircle has radius $r$, the longest side of the triangle is $a$, while the shortest altitude is $h$. Show that: $\frac{R}{r} > \frac{a}{h}$.

2008 Balkan MO, 2

Tags: induction , function , cauchy inequality , inequalities

Is there a sequence $ a_1,a_2,\ldots$ of positive reals satisfying simoultaneously the following inequalities for all positive integers $ n$: a) $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le n^2$ b) $ \frac1{a_1}\plus{}\frac1{a_2}\plus{}\ldots\plus{}\frac1{a_n}\le2008$?

2021 XVII International Zhautykov Olympiad, #4

Tags: geometric inequality , geometry , inequalities

Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$. Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$