Olympiad problems

2008 ITest, 90

For $a,b,c$ positive reals, let \[N=\dfrac{a^2+b^2}{c^2+ab}+\dfrac{b^2+c^2}{a^2+bc}+\dfrac{c^2+a^2}{b^2+ca}.\] Find the minimum value of $\lfloor 2008N\rfloor$.

2000 Greece JBMO TST, 4

Tags: geometry , geometric inequality , inequalities

Let $a,b,c$ be sidelengths with $a\ge b\ge c$ and $s\ge a+1$ where $s$ be the semiperimeter of the triangle. Prove that $$ \frac{s-c}{\sqrt{a}}+\frac{s-b}{\sqrt{c}}+\frac{s-a}{\sqrt{b}}\ge \frac{s-b}{\sqrt{a}}+\frac{s-c}{\sqrt{b}}+\frac{s-a}{\sqrt{c}}$$

2002 IMO, 4

Tags: inequalities , number theory , divisor

Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.

2019 Romania National Olympiad, 2

Tags: algebra , inequalities

If $a,b,c\in(0,\infty)$ such that $a+b+c=3$, then $$\frac{a}{3a+bc+12}+\frac{b}{3b+ca+12}+\frac{c}{3c+ab+12}\le \frac{3}{16}$$

JOM 2015 Shortlist, A5

Tags: inequality , inequalities

Let $ a, b, c $ be the side length of a triangle, with $ ab + bc + ca = 18 $ and $ a, b, c > 1 $. Prove that $$ \sum_{cyc}\frac{1}{(a - 1)^3} > \frac{1}{a + b + c - 3} $$

2019 Azerbaijan BMO TST, 3

Tags: inequalities

Let $ a, b, c$ be positive real numbers such that $ abc = 1. $ Prove that: $$ 2 (a^ 2 + b^ 2 + c^ 2) \left (\frac 1 {a^ 2} + \frac 1{b^ 2}+ \frac 1{c^2}\right)\geq 3(a+ b + c + ab + bc + ca).$$

2023 India National Olympiad, 2

Tags: polynomial , algebra , inequalities

Suppose $a_0,\ldots, a_{100}$ are positive reals. Consider the following polynomial for each $k$ in $\{0,1,\ldots, 100\}$: $$a_{100+k}x^{100}+100a_{99+k}x^{99}+a_{98+k}x^{98}+a_{97+k}x^{97}+\dots+a_{2+k}x^2+a_{1+k}x+a_k,$$where indices are taken modulo $101$, [i]i.e.[/i], $a_{100+i}=a_{i-1}$ for any $i$ in $\{1,2,\dots, 100\}$. Show that it is impossible that each of these $101$ polynomials has all its roots real. [i]Proposed by Prithwijit De[/i]

2012 Austria Beginners' Competition, 3

Tags: algebra , inequalities

Let $a$ and $b$ be two positive real numbers with $a \le 2b \le 4a$. Prove that $4ab \le2 (a^2+ b^2) \le 5 ab$.

1993 Vietnam Team Selection Test, 1

Tags: geometry , incenter , circumcircle , inequalities , euler , ratio , angle bisector

Let $H$, $I$, $O$ be the orthocenter, incenter and circumcenter of a triangle. Show that $2 \cdot IO \geq IH$. When does the equality hold ?

2011 Mathcenter Contest + Longlist, 11

Tags: algebra , inequalities

Let $a,b,c\in R^+$ with $a+b+c=3$. Prove that $$2(ab+bc+ca)\le 5+ abc$$ [i](Real Matrik)[/i]

2013 USA TSTST, 2

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

A finite sequence of integers $a_1, a_2, \dots, a_n$ is called [i]regular[/i] if there exists a real number $x$ satisfying \[ \left\lfloor kx \right\rfloor = a_k \quad \text{for } 1 \le k \le n. \] Given a regular sequence $a_1, a_2, \dots, a_n$, for $1 \le k \le n$ we say that the term $a_k$ is [i]forced[/i] if the following condition is satisfied: the sequence \[ a_1, a_2, \dots, a_{k-1}, b \] is regular if and only if $b = a_k$. Find the maximum possible number of forced terms in a regular sequence with $1000$ terms.

2012 Olympic Revenge, 5

Tags: inequalities

Let $x_1,x_2,\ldots ,x_n$ positive real numbers. Prove that: \[\sum_{cyc} \frac{1}{x_i^3+x_{i-1}x_ix_{i+1}} \le \sum_{cyc} \frac{1}{x_ix_{i+1}(x_i+x_{i+1})}\]

2012 Belarus Team Selection Test, 4

Tags: algebra , inequalities

Given $0 < a < b < c$ prove that $$ a^{20}b^{12} + b^{20}c^{12 }+ c^{20}a^{12} <b^{20}a^{12}+ a^{20}c^{12} + c^{20}b^{12} $$ (I. Voronovich)

2014 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: algebra , inequality , inequalities

Let $a$, $b$ and $c$ be positive real numbers such that $ab+bc+ca=1$. Prove the inequality: $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq 3(a+b+c)$$

1993 Abels Math Contest (Norwegian MO), 1b

Tags: inequalities , geometric inequalities

Given a triangle with sides of lengths $a,b,c$, prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}< 2$.

1978 Czech and Slovak Olympiad III A, 2

Tags: inequalities , algebra , parameterization

Determine (at least one) pair of real numbers $k,q$ such that the inequality \[2\left|\sqrt{1-x^2}-kx-q\right|\le\sqrt2-1\] holds for all $x\in[0,1].$

1997 Irish Math Olympiad, 5

Tags: inequalities , number theory

Let $ S$ be the set of odd integers greater than $ 1$. For each $ x \in S$, denote by $ \delta (x)$ the unique integer satisfying the inequality $ 2^{\delta (x)}<x<2^{\delta (x) \plus{}1}$. For $ a,b \in S$, define: $ a \ast b\equal{}2^{\delta (a)\minus{}1} (b\minus{}3)\plus{}a.$ Prove that if $ a,b,c \in S$, then: $ (a)$ $ a \ast b \in S$ and $ (b)$ $ (a \ast b)\ast c\equal{}a \ast (b \ast c)$.

2008 Germany Team Selection Test, 2

Tags: inequalities , circumcircle , trigonometry , geometry

For three points $ X,Y,Z$ let $ R_{XYZ}$ be the circumcircle radius of the triangle $ XYZ.$ If $ ABC$ is a triangle with incircle centre $ I$ then we have: \[ \frac{1}{R_{ABI}} \plus{} \frac{1}{R_{BCI}} \plus{} \frac{1}{R_{CAI}} \leq \frac{1}{\bar{AI}} \plus{} \frac{1}{\bar{BI}} \plus{} \frac{1}{\bar{CI}}.\]

Mathley 2014-15, 4

Tags: algebra , inequalities

Let $S_k$ be the set of all triplets of real numbers $(a, b, c)$ satisfying $a <k (b + c)$, $b <k (c + a)$, and $c <k (a + b)$. For what value of $k$ then $S_k$ is a subset of $\{(a, b, c) | ab + bc + ca> 0\}$ ? Michel Bataille, France

India EGMO 2021 TST, 2

Tags: algebra , inequality , 4-variable inequality , inequalities

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2000 Nordic, 4

Tags: inequalities , functional inequality , calculus

The real-valued function $f$ is defined for $0 \le x \le 1, f(0) = 0, f(1) = 1$, and $\frac{1}{2} \le \frac{ f(z) - f(y)}{f(y) - f(x)} \le 2$ for all $0 \le x < y < z \le 1$ with $z - y = y -x$. Prove that $\frac{1}{7} \le f (\frac{1}{3} ) \le \frac{4}{7}$.

2018 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: inequalities , algebra , logarithm

If numbers $x_1$, $x_2$,...,$x_n$ are from interval $\left( \frac{1}{4},1 \right)$ prove the inequality: $\log _{x_1} {\left(x_2-\frac{1}{4} \right)} + \log _{x_2} {\left(x_3-\frac{1}{4} \right)}+ ... + \log _{x_{n-1}} {\left(x_n-\frac{1}{4} \right)} + \log _{x_n} {\left(x_1-\frac{1}{4} \right)} \geq 2n$

1986 China Team Selection Test, 3

Tags: quadratic , inequalities

Let $x_i,$ $1 \leq i \leq n$ be real numbers with $n \geq 3.$ Let $p$ and $q$ be their symmetric sum of degree $1$ and $2$ respectively. Prove that: i) $p^2 \cdot \frac{n-1}{n}-2q \geq 0$ ii) $\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n}$ for every meaningful $i$.

2012 ELMO Shortlist, 8

Tags: binomial coefficient , algebra , polynomial , ceiling function , inequalities , absolute value , geometric series

Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$. [i]Victor Wang.[/i]

1973 IMO Shortlist, 3

Tags: vector , geometry , geometric inequality , inequalities

Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1.