Olympiad problems

1969 Spain Mathematical Olympiad, 6

Given a polynomial of real coefficients P(x) , can it be affirmed that for any real value of x is true of one of the following inequalities: $$P(x) \le P(x)^2; \,\,\, P(x) < 1 + P(x)^2; \,\,\,P(x) \le \frac12 +\frac12 P(x)^2.$$ Find a simple general procedure (among the many existing ones) that allows, provided we are given two polynomials $P(x)$ and $Q(x)$ , find another $M(x)$ such that for every value of $x$, at the same time $-M(x) < P(x)<M(x)$ and $-M(x)< Q(x)<M(x)$.

2001 District Olympiad, 4

Tags: limit , induction , inequalities , real analysis

Prove that: a) the sequence $a_n=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+n},\ n\ge 1$ is monotonic. b) there is a sequence $(a_n)_{n\ge 1}\in \{0,1\}$ such that: \[\lim_{n\to \infty} \left(\frac{a_1}{n+1}+\frac{a_2}{n+2}+\ldots +\frac{a_n}{n+n}\right)=\frac{1}{2}\] [i]Radu Gologan[/i]

2007 Today's Calculation Of Integral, 184

Tags: calculus , integration , function , inequalities , real analysis , calculus computations , logarithm

(1) For real numbers $x,\ a$ such that $0<x<a,$ prove the following inequality. \[\frac{2x}{a}<\int_{a-x}^{a+x}\frac{1}{t}\ dt<x\left(\frac{1}{a+x}+\frac{1}{a-x}\right). \] (2) Use the result of $(1)$ to prove that $0.68<\ln 2<0.71.$

1953 AMC 12/AHSME, 47

Tags: inequalities , calculus , logarithm

If $ x$ is greater than zero, then the correct relationship is: $ \textbf{(A)}\ \log (1\plus{}x) \equal{} \frac{x}{1\plus{}x} \qquad\textbf{(B)}\ \log (1\plus{}x) < \frac{x}{1\plus{}x} \\ \textbf{(C)}\ \log(1\plus{}x) > x \qquad\textbf{(D)}\ \log (1\plus{}x) < x \qquad\textbf{(E)}\ \text{none of these}$

2018 Moscow Mathematical Olympiad, 1

Tags: algebra , inequalities

$a_1,a_2,...,a_{81}$ are nonzero, $a_i+a_{i+1}>0$ for $i=1,...,80$ and $a_1+a_2+...+a_{81}<0$. What is sign of $a_1*a_2*...*a_{81}$?

2013 Today's Calculation Of Integral, 870

Tags: calculus , integration , ellipse , hyperbola , geometry , inequalities , conic

Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$ (1) Find all points of intersection of $E$ and $H$. (2) Find the area of the region expressed by the system of inequality \[\left\{ \begin{array}{ll} 3x^2+y^2\leq 3 &\quad \\ xy\geq \frac 34 , &\quad \end{array} \right.\]

2011 China Second Round Olympiad, 3

Tags: inequalities , algebra , logarithm

Let $a,b$ be positive reals such that $\frac{1}{a}+\frac{1}{b}\leq2\sqrt2$ and $(a-b)^2=4(ab)^3$. Find $\log_a b$.

2007 Junior Balkan MO, 1

Tags: algebra , quadratic , function , inequalities

Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.

2019 Tournament Of Towns, 1

Tags: inequalities , number theory , prime , number of divisors , prime factorization , factor , divisor

Let us call the number of factors in the prime decomposition of an integer $n > 1$ the complexity of $n$. For example, [i]complexity [/i] of numbers $4$ and $6$ is equal to $2$. Find all $n$ such that all integers between $n$ and $2n$ have complexity a) not greater than the complexity of $n$. b) less than the complexity of $n$. (Boris Frenkin)

1999 Junior Balkan Team Selection Tests - Romania, 4

Tags: inequalities , geometry

Let be three discs $ D_1,D_2,D_3. $ For each $ i,j\in\{1,2,3\} , $ denote $ a_{ij} $ as being the area of $ D_i\cap D_j. $ If $ x_1,x_2,x_3\in\mathbb{R} $ such that $ x_1x_2x_3\neq 0, $ then $$ a_{11} x_1^2+a_{22} x_2^2+a_{33} x_3^2+2a_{12} x_1x_2+2a_{23 }x_2x_3+2a_{31} x_3x_1>0. $$ [i]Vasile Pop[/i]

2009 Junior Balkan Team Selection Tests - Romania, 4

Tags: inequalities , algebra

Let $a,b,c > 0$ be real numbers with the sum equal to $3$. Show that: $$\frac{a+3}{3a+bc}+\frac{b+3}{3b+ca}+\frac{c+3}{3c+ab} \ge 3$$

1997 India National Olympiad, 4

Tags: perimeter , inequalities , triangle inequality , geometry

In a unit square one hundred segments are drawn from the centre to the sides dividing the square into one hundred parts (triangles and possibly quadruilaterals). If all parts have equal perimetr $p$, show that $\dfrac{14}{10} < p < \dfrac{15}{10}$.

2002 Greece National Olympiad, 3

Tags: circumcircle , trigonometry , inequalities , function , geometry

In a triangle $ABC$ we have $\angle C>10^0$ and $\angle B=\angle C+10^0.$We consider point $E$ on side $AB$ such that $\angle ACE=10^0,$ and point $D$ on side $AC$ such that $\angle DBA=15^0.$ Let $Z\neq A$ be a point of interection of the circumcircles of the triangles $ABD$ and $AEC.$Prove that $\angle ZBA>\angle ZCA.$

2000 IMO Shortlist, 1

Tags: inequalities , three variable inequality , ineqstd

Let $ a, b, c$ be positive real numbers so that $ abc \equal{} 1$. Prove that \[ \left( a \minus{} 1 \plus{} \frac 1b \right) \left( b \minus{} 1 \plus{} \frac 1c \right) \left( c \minus{} 1 \plus{} \frac 1a \right) \leq 1. \]

2013 Ukraine Team Selection Test, 5

Tags: inequalities , algebra

For positive $x, y$, and $z$ that satisfy the condition $xyz = 1$, prove the inequality $$\sqrt[3]{\frac{x+y}{2z}}+\sqrt[3]{\frac{y+z}{2x}}+\sqrt[3]{\frac{z+x}{2y}}\le \frac{5(x+y+z)+9}{8}$$

2002 Czech-Polish-Slovak Match, 6

Tags: polynomial , inequalities , cauchy inequality , algebra

Let $n \ge 2$ be a fixed even integer. We consider polynomials of the form \[P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + 1\] with real coefficients, having at least one real roots. Find the least possible value of $a^2_1 + a^2_2 + \cdots + a^2_{n-1}$.

2011 Silk Road, 3

Tags: inequalities , trigonometry

For all $a,b,c\in \bb{R}^+ $ such that $a+b+c=1$ and $ ( \frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} )(a-bc)(b-ac)(c-ab)\le M \cdot abc$. Find min $M$

2019 Romania Team Selection Test, 1

Tags: inequalities , minimum , algebra

Let be a natural number $ n\ge 3. $ Find $$ \inf_{\stackrel{ x_1,x_2,\ldots ,x_n\in\mathbb{R}_{>0}}{1=P\left( x_1,x_2,\ldots ,x_n\right)}}\sum_{i=1}^n\left( \frac{1}{x_i} -x_i \right) , $$ where $ P\left( x_1,x_2,\ldots ,x_n\right) :=\sum_{i=1}^n \frac{1}{x_i+n-1} , $ and find in which circumstances this infimum is attained.

2021 China Team Selection Test, 4

Tags: number theory function , number theory , floor function , inequalities , inequality

Proof that $$ \sum_{m=1}^n5^{\omega (m)} \le \sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2 \le \sum_{m=1}^n5^{\Omega (m)} .$$

2016 Iran Team Selection Test, 4

Tags: inequalities , algebra , multivariate polynomial , maximization , n-variable inequality

Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.

2017 CHKMO, Q4

Tags: inequalities , algebra

Find the smallest possible value of the nonnegative number $\lambda$ such that the inequality $$\frac{a+b}{2}\geq\lambda \sqrt{ab}+(1-\lambda )\sqrt{\frac{a^2+b^2}{2}}$$ holds for all positive real numbers $a, b$.

2000 Singapore Team Selection Test, 3

Tags: combinatorial geometry , combinatorics , geometric inequalities , inequalities

There are $n$ blue points and $n$ red points on a straight line. Prove that the sum of all distances between pairs of points of the same colour is less than or equal to the sum of all distances between pairs of points of different colours

1996 Polish MO Finals, 3

Tags: inequalities

$a_i, x_i$ are positive reals such that $a_1 + a_2 + ... + a_n = x_1 + x_2 + ... + x_n = 1$. Show that \[ 2 \sum_{i<j} x_ix_j \leq \frac{n-2}{n-1} + \sum \frac{a_ix_i ^2}{1-a_i} \] When do we have equality?

2016 Thailand Mathematical Olympiad, 2

Tags: bijection , sum , inequalities

Let $M$ be a positive integer, and $A = \{1, 2,... , M + 1\}$. Show that if $f$ is a bijection from $A$ to $A$ then $\sum_{n=1}^{M} \frac{1}{f(n) + f(n + 1)} > \frac{M}{M + 3}$

2025 Romania EGMO TST, P2

Tags: divisible , number theory , inequalities , subset

Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$