Olympiad problems

2016 Balkan MO Shortlist, A5

Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$. Find the minimum value and the maximum value of the product $abcd$.

1966 All Russian Mathematical Olympiad, 083

Tags: minimum , maximum , game strategy , combinatorics

$20$ numbers are written on the board $1, 2, ... ,20$. Two players are putting signs before the numbers in turn ($+$ or $-$). The first wants to obtain the minimal possible absolute value of the sum. What is the maximal value of the absolute value of the sum that can be achieved by the second player?

2016 Hanoi Open Mathematics Competitions, 6

Tags: interval , minimum , algebra , number theory

Determine the smallest positive number $a$ such that the number of all integers belonging to $(a, 2016a]$ is $2016$.

1953 Moscow Mathematical Olympiad, 234

Tags: number theory , divisible , minimum

Find the smallest number of the form $1...1$ in its decimal expression which is divisible by $\underbrace{\hbox{3...3}}_{\hbox{100}}$,.

2009 Greece JBMO TST, 4

Tags: algebra , system of equations , minimum , sum

Find positive real numbers $x,y,z$ that are solutions of the system $x+y+z=xy+yz+zx$ and $xyz=1$ , and have the smallest possible sum.

Brazil L2 Finals (OBM) - geometry, 2010.6

Tags: integer , geometry , minimum , area of a triangle , geometric inequality

The three sides and the area of a triangle are integers. What is the smallest value of the area of this triangle?

1996 Greece Junior Math Olympiad, 3

Tags: inequalities , minimum , algebra

Determine the minimum value of the expression $2x^4 - 2x^2y^2 + y^4 - 8x^2 + 18$ where $x, y \in R$.

2023 China Northern MO, 4

Tags: algebra , minimum , inequalities

Given the sequence $(a_n) $ satisfies $1=a_1< a_2 < a_3< \cdots<a_n $ and there exist real number $m$ such that $$\displaystyle\sum_{i=1}^{n-1} \sqrt[3]{\frac{a_{i+1}-a_i}{(2+a_i)^4}}\leq m $$ for any positive integer $ n $ not less than 2 . Find the minimum of $m.$

1988 All Soviet Union Mathematical Olympiad, 470

Tags: combinatorics , minimum

There are $21$ towns. Each airline runs direct flights between every pair of towns in a group of five. What is the minimum number of airlines needed to ensure that at least one airline runs direct flights between every pair of towns?

2019 BAMO, A

Tags: algebra , inequalities , minimum

Let $a$ and $b$ be positive whole numbers such that $\frac{4.5}{11}<\frac{a}{b}<\frac{5}{11}$. Find the fraction $\frac{a}{b}$ for which the sum $a+b$ is as small as possible. Justify your answer

2013 Hanoi Open Mathematics Competitions, 2

Tags: function , minimum , algebra

The smallest value of the function $f(x) =|x| +\left|\frac{1 - 2013x}{2013 - x}\right|$ where $x \in [-1, 1] $ is: (A): $\frac{1}{2012}$, (B): $\frac{1}{2013}$, (C): $\frac{1}{2014}$, (D): $\frac{1}{2015}$, (E): None of the above.

2003 May Olympiad, 5

Tags: 3d geometry , cube , minimum , geometry

An ant, which is on an edge of a cube of side $8$, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.