Olympiad problems

2018 Greece JBMO TST, 1

Let $a,b,c,d$ be positive real numbers such that $a^2+b^2+c^2+d^2=4$. Prove that exist two of $a,b,c,d$ with sum less or equal to $2$.

2002 China Team Selection Test, 1

Tags: inequalities , geometry

Given triangle $ ABC$ and $ AB\equal{}c$, $ AC\equal{}b$ and $ BC\equal{}a$ satisfying $ a \geq b \geq c$, $ BE$ and $ CF$ are two interior angle bisectors. $ P$ is a point inside triangle $ AEF$. $ R$ and $ Q$ are the projections of $ P$ on sides $ AB$ and $ AC$. Prove that $ PR \plus{} PQ \plus{} RQ < b$.

2020 USA IMO Team Selection Test, 5

Tags: polynomial , integer polynomial , inequalities , number theory

Find all integers $n \ge 2$ for which there exists an integer $m$ and a polynomial $P(x)$ with integer coefficients satisfying the following three conditions: [list] [*]$m > 1$ and $\gcd(m,n) = 1$; [*]the numbers $P(0)$, $P^2(0)$, $\ldots$, $P^{m-1}(0)$ are not divisible by $n$; and [*]$P^m(0)$ is divisible by $n$. [/list] Here $P^k$ means $P$ applied $k$ times, so $P^1(0) = P(0)$, $P^2(0) = P(P(0))$, etc. [i]Carl Schildkraut[/i]

2017 Kosovo Team Selection Test, 3

Tags: inequalities , algebra

If $a$ and $b$ are positive real numbers with sum $3$, and $x, y, z$ positive real numbers with product $1$, prove that : $(ax+b)(ay+b)(az+b)\geq 27$

2021-IMOC, A6

Tags: inequalities , am-gm , algebra

Let $n$ be some positive integer and $a_1 , a_2 , \dots , a_n$ be real numbers. Denote $$S_0 = \sum_{i=1}^{n} a_i^2 , \hspace{1cm} S_1 = \sum_{i=1}^{n} a_ia_{i+1} , \hspace{1cm} S_2 = \sum_{i=1}^{n} a_ia_{i+2},$$ where $a_{n+1} = a_1$ and $a_{n+2} = a_2.$ 1. Show that $S_0 - S_1 \geq 0$. 2. Show that $3$ is the minimum value of $C$ such that for any $n$ and $a_1 , a_2 , \dots , a_n,$ there holds $C(S_0 - S_1) \geq S_1 - S_2$.

2004 India IMO Training Camp, 3

Tags: inequalities

For $a,b,c$ positive reals find the minimum value of \[ \frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{c^2+a^2}{b^2+ca}. \]

2006 Poland - Second Round, 3

Tags: inequalities

Positive reals $a,b,c$ satisfy $ab+bc+ca=abc$. Prove that: $\frac{a^4+b^4}{ab(a^3+b^3)} + \frac{b^4+c^4}{bc(b^3+c^3)}+\frac{c^4+a^4}{ca(c^3+a^3)} \geq 1$

1997 IMO Shortlist, 19

Tags: inequalities , algebra , n-variable inequality

Let $ a_1\geq \cdots \geq a_n \geq a_{n \plus{} 1} \equal{} 0$ be real numbers. Show that \[ \sqrt {\sum_{k \equal{} 1}^n a_k} \leq \sum_{k \equal{} 1}^n \sqrt k (\sqrt {a_k} \minus{} \sqrt {a_{k \plus{} 1}}). \] [i]Proposed by Romania[/i]

1998 Romania National Olympiad, 1

Tags: algebra , system of equations , inequalities

Let $a$ be a real number and $A = \{(x, y) \in R \times R | \, x + y = a\}$, $B = \{(x,y) \in R \times R | \, x^3 + y^3 < a\}$ . Find all values of $a$ such that $A \cap B = \emptyset$ .

2011 Moldova Team Selection Test, 2

Tags: function , inequalities

Let $x_1, x_2, \ldots, x_n$ be real positive numbers such that $x_1\cdot x_2\cdots x_n=1$. Prove the inequality $\frac1{x_1(x_1+1)}+\frac1{x_2(x_2+1)}+\cdots+\frac1{x_n(x_n+1)}\geq\frac n2$

2010 China Team Selection Test, 1

Tags: inequalities , induction , function , combinatorics

Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings: (1) $\sum_{v\in V} f(v)=|E|$; (2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$. Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.

1993 IMO Shortlist, 3

Tags: inequalities , function , four variables , 4-variable inequality , algebra

Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.

2018 Ramnicean Hope, 3

Tags: complex number , inequalities , complex plane , modulus

Consider a complex number whose affix in the complex plane is situated on the first quadrant of the unit circle centered at origin. Then, the following inequality holds. $$ \sqrt{2} +\sqrt{2+\sqrt{2}} \le |1+z|+|1+z^2|+|1+z^4|\le 6 $$ [i]Costică Ambrinoc[/i]

2010 JBMO Shortlist, 1

Tags: induction , inequalities , prime factorization , number theory

Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

2024 Stars of Mathematics, P3

Tags: inequalities

Fix postive integer $n\geq 2$. Let $a_1,a_2,\dots ,a_n$ be real numbers in the interval $[1,2024]$. Prove that $$\sum_{i=1}^n\frac{1}{a_i}(a_1+a_2+\dots +a_i)>\frac{1}{44}n(n+33).$$ [i]Proposed by Radu-Andrei Lecoiu[/i]

2010 Today's Calculation Of Integral, 662

Tags: calculus , integration , geometric transformation , rotation , inequalities , perimeter , geometry

In $xyz$ space, let $A$ be the solid generated by a rotation of the figure, enclosed by the curve $y=2-2x^2$ and the $x$-axis about the $y$-axis. (1) When the solid is cut by the plane $x=a\ (|a|\leq 1)$, find the inequality which expresses the figure of the cross-section. (2) Denote by $L$ the distance between the point $(a,\ 0,\ 0)$ and the point on the perimeter of the cross-section found in (1), find the maximum value of $L$. (3) Find the volume of the solid by a rotation of the solid $A$ about the $x$-axis. [i]1987 Sophia University entrance exam/Science and Technology[/i]

2015 Cuba MO, 9

Tags: algebra , inequalities

Determine the largest possible value of$ M$ for which it holds that: $$\frac{x}{1 +\dfrac{yz}{x}}+ \frac{y}{1 + \dfrac{zx}{y}}+ \frac{z}{1 + \dfrac{xy}{z}} \ge M,$$ for all real numbers $x, y, z > 0$ that satisfy the equation $xy + yz + zx = 1$.

V Soros Olympiad 1998 - 99 (Russia), 11.7

Tags: inequalities , algebra , trigonometry

Prove that for all positive and admissible values of $x$ the following inequality holds: $$\sin x + arc \sin x>2x$$

2010 Contests, 4

Tags: inequalities

Given $n$ positive real numbers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n \ge 0$ and $x_1^2+x_2^2+\cdots+x_n^2=1$, prove that \[\frac{x_1}{\sqrt{1}}+\frac{x_2}{\sqrt{2}}+\cdots+\frac{x_n}{\sqrt{n}}\ge 1.\]

2006 Hanoi Open Mathematics Competitions, 9

Tags: minimum , inequalities , algebra

What is the smallest possible value of $x^2 + y^2 - x -y - xy$?

2009 China Team Selection Test, 2

Tags: induction , ratio , inequalities

Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1,$ then $ (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.$

2010 Turkey MO (2nd round), 3

Tags: inequalities , analytic geometry , combinatorics

Let $K$ be the set of all sides and diagonals of a convex $2010-gon$ in the plane. For a subset $A$ of $K,$ if every pair of line segments belonging to $A$ intersect, then we call $A$ as an [i]intersecting set.[/i] Find the maximum possible number of elements of union of two [i]intersecting sets.[/i]

2011 ELMO Shortlist, 5

Tags: geometry , inequalities

Given positive reals $x,y,z$ such that $xy+yz+zx=1$, show that \[\sum_{\text{cyc}}\sqrt{(xy+kx+ky)(xz+kx+kz)}\ge k^2,\]where $k=2+\sqrt{3}$. [i]Victor Wang.[/i]

1981 Romania Team Selection Tests, 3.

Tags: binomial coefficient , algebra , inequalities

Let $n>r\geqslant 3$ be two integers and $d$ be a positive integer such that $nd\geqslant \dbinom{n+r}{r+1}$. Show that \[(n-t)(d-t)>\dbinom{n-t+r}{r+1},\] for $t=1,2,\ldots,n-1$ [i]Vasile Brânzănescu[/i]

1998 AIME Problems, 9

Tags: probability , analytic geometry , geometry , inequalities

Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ mintues. The probability that either one arrives while the other is in the cafeteria is $40 \%,$ and $m=a-b\sqrt{c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$