Olympiad problems

1998 Tournament Of Towns, 4

Among all sets of real numbers $\{ x_1 , x_2 , ... , x_{20} \}$ from the open interval $(0, 1 )$ such that $$x_1x_2...x_{20}= ( 1 - x_1 ) ( 1 -x_2 ) ... (1 - x_{20} )$$ find the one for which $x_1 x_2... x_{20}$ is maximal. (A Cherniatiev)

2001 India IMO Training Camp, 3

Tags: induction , polynomial , algebra , inequalities

Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$. Prove that \[\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1\]

2024 Belarus - Iran Friendly Competition, 1.2

Tags: algebra , inequalities

Given $n \geq 2$ positive real numbers $x_1 \leq x_2 \leq \ldots \leq x_n$ satisfying the equalities $$x_1+x_2+\ldots+x_n=4n$$ $$\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}=n$$ Prove that $\frac{x_n}{x_1} \geq 7+4\sqrt{3}$

2011 Kurschak Competition, 3

Tags: combinatorial geometry , inequalities

Given $2n$ points and $3n$ lines on the plane. Prove that there is a point $P$ on the plane such that the sum of the distances of $P$ to the $3n$ lines is less than the sum of the distances of $P$ to the $2n$ points.

2019 Jozsef Wildt International Math Competition, W. 60

Tags: inequalities , tetrahedron , inradius , exradius , 3d geometry

In all tetrahedron $ABCD$ holds [list=1] [*] $(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(h_a-r)^2}{(h_a^n-r^n)(h_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}$ [*] $(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(r_a-r)^2}{(r_a^n-r^n)(r_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}$ [/list] for all $n\in \mathbb{N}^*$

1931 Eotvos Mathematical Competition, 3

Tags: geometric inequality , inequalities , geometry

Let $A$ and $B$ be two given points, distance $1 $ apart. Determine a point $P$ on the line $AB$ such that $$\frac{1}{1 + AP}+\frac{1}{1 + BP}$$ is a maximum.

2008 Harvard-MIT Mathematics Tournament, 8

Tags: geometry , inequalities , geometric transformation , reflection , triangle inequality

Let $ ABC$ be an equilateral triangle with side length 2, and let $ \Gamma$ be a circle with radius $ \frac {1}{2}$ centered at the center of the equilateral triangle. Determine the length of the shortest path that starts somewhere on $ \Gamma$, visits all three sides of $ ABC$, and ends somewhere on $ \Gamma$ (not necessarily at the starting point). Express your answer in the form of $ \sqrt p \minus{} q$, where $ p$ and $ q$ are rational numbers written as reduced fractions.

2006 Cezar Ivănescu, 3

Tags: algebra , inequalities

[b]a)[/b] Given two positive reals $ x,y, $ prove that $ \min\left( x,1/x+y,1/y \right)\le\sqrt 2. $ and determine when equality holds. [b]b)[/b] Find all triplets of real numbers $ (a,b,c) $ having the property that for every triplet of real numbers $ (x,y,z) , $ the following equality holds: $$ |ax+by+cz|+|bx+cy+az|+|cx+ay+bz|=|x|+|y|+|z| $$

2012 Regional Competition For Advanced Students, 1

Tags: inequalities

Prove that the inequality \[ a + a^3 - a^4 - a^6 < 1\] holds for all real numbers $a$.

2000 Mongolian Mathematical Olympiad, Problem 2

Tags: vector , inequalities , geometry

Let $n\ge2$. For any two $n$-vectors $\vec x=(x_1,\ldots,x_n)$ and $\vec y=(y_1,\ldots,y_n)$, we define $$f\left(\vec x,\vec y\right)=x_1\overline{y_1}-\sum_{i=2}^nx_i\overline{y_i}.$$Prove that if $f\left(\vec x,\vec x\right)\ge0$, and $f\left(\vec y,\vec y\right)\ge0$, then $\left|f\left(\vec x,\vec y\right)\right|^2\ge f\left(\vec x,\vec x\right)f\left(\vec y,\vec y\right)$.

2009 Purple Comet Problems, 8

Tags: triangle inequality , integration , calculus , inequalities

Find the number of non-congruent scalene triangles whose sides all have integral length, and the longest side has length $11$.

JBMO Geometry Collection, 2011

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

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 ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

1989 National High School Mathematics League, 15

Tags: inequalities

For any positive integer $n$, $a_n>0$, and $\sum_{j=1}^{n}a_j^3=\left(\sum_{j=1}^{n}a_j\right)^2$. Prove that $a_n=n$

2011 Today's Calculation Of Integral, 746

Tags: floor function , inequalities , integration , calculus , calculus computations

Prove the following inequality. \[n^ne^{-n+1}\leq n!\leq \frac 14(n+1)^{n+1}e^{-n+1}.\]

2018 ELMO Shortlist, 3

Tags: inequalities

Let $a, b, c,x, y, z$ be positive reals such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$. Prove that \[a^x+b^y+c^z\ge \frac{4abcxyz}{(x+y+z-3)^2}.\] [i]Proposed by Daniel Liu[/i]

2021-IMOC, A8

Tags: algebra , inequalities , inequality , functional equation , function

Find all functions $f : \mathbb{N} \to \mathbb{N}$ with $$f(x) + yf(f(x)) < x(1 + f(y)) + 2021$$ holds for all positive integers $x,y.$

2015 Saudi Arabia GMO TST, 4

Tags: inequalities , number theory , combinatorics , binomial , remainder

For each positive integer $n$, define $s(n) =\sum_{k=0}^n r_k$, where $r_k$ is the remainder when $n \choose k$ is divided by $3$. Find all positive integers $n$ such that $s(n) \ge n$. Malik Talbi

2013 Princeton University Math Competition, 1

Tags: derivative , calculus , integration , function , inequalities

Prove that \[ \frac{1}{a^2+2} + \frac{1}{b^2+2} + \frac{1}{c^2+2} \le \frac{1}{6ab+c^2} + \frac{1}{6bc+a^2} + \frac{1}{6ca+b^2} \] for all positive real numbers $a$, $b$ and $c$ satisfying $a^2+b^2+c^2=1$.

1937 Eotvos Mathematical Competition, 1

Tags: inequalities , algebra

Let $n$ be a positive integer. Prove that $a_1!a_2! ... a_n! < k!$, where $k$ is an integer which is greater than the sum of the positive integers $a_1, a_2,.., a_n$.

2012 Argentina National Olympiad, 1

Tags: inequalities , system , system of equations , algebra

Determine if there are triplets ($x,y,z)$ of real numbers such that $$\begin{cases} x+y+z=7 \\ xy+yz+zx=11\end{cases}$$ If the answer is affirmative, find the minimum and maximum values of $z$ in such a triplet.

2019 Belarusian National Olympiad, 9.3

Tags: polynomial , function , algebra , inequalities

Positive real numbers $a$ and $b$ satisfy the following conditions: the function $f(x)=x^3+ax^2+2bx-1$ has three different real roots, while the function $g(x)=2x^2+2bx+a$ doesn't have real roots. Prove that $a-b>1$. [i](V. Karamzin)[/i]

2008 Croatia Team Selection Test, 1

Tags: inequalities

Let $ x$, $ y$, $ z$ be positive numbers. Find the minimum value of: $ (a)\quad \frac{x^2 \plus{} y^2 \plus{} z^2}{xy \plus{} yz}$ $ (b)\quad \frac{x^2 \plus{} y^2 \plus{} 2z^2}{xy \plus{} yz}$

2000 Harvard-MIT Mathematics Tournament, 1

Tags: inequalities , absolute value

How many integers $x$ satisfy $|x|+5<7$ and $|x-3|>2$?

2020 Jozsef Wildt International Math Competition, W19

Tags: inequalities

Prove the inequality $$\prod_{k=2}^n\left(1+\frac{k^{p-1}}{1^p+2^p+\ldots+k^p}\right)<e^{\frac{p-1}2}$$ [i]Proposed by Arkady Alt[/i]

1980 Czech And Slovak Olympiad IIIA, 4

Tags: algebra , inequalities

Let $a_1 < a_2< ...< a_n$ are real numbers, $$f(x) = \sum_{i=1}^n|x-a_i|,$$ for $n$ even. Find the minimum of this function.