Olympiad problems

2004 Estonia National Olympiad, 2

The positive differences $a_i-a_j$ of five different positive integers $a_1, a_2, a_3, a_4, a_5$ are all different (there are altogether $10$ such differences). Find the least possible value of the largest number among the $a_i$.

2007 Portugal MO, 6

Tags: geometry , min , max , distance

In a village, the maximum distance between two houses is $M$ and the minimum distance is $m$. Prove that if the village has $6$ houses, then $\frac{M}{m} \ge \sqrt3$.

2005 Bosnia and Herzegovina Junior BMO TST, 1

Tags: min , max , algebra , inequalities

Non-negative real numbers $x, y, z$ satisfy the following relations: $3x + 5y + 7z = 10$ and $x + 2y + 5z = 6$. Find the minimum and maximum of $w = 2x - 3y + 4z$.

Estonia Open Junior - geometry, 2009.2.1

Tags: rectangle , distance , sum , min , geometry

A Christmas tree must be erected inside a convex rectangular garden and attached to the posts at the corners of the garden with four ropes running at the same height from the ground. At what point should the Christmas tree be placed, so that the sum of the lengths of these four cords is as small as possible?

2011 Junior Balkan Team Selection Tests - Romania, 4

Tags: min , algebra , absolute value

Let $m$ be a positive integer. Determine the smallest positive integer $n$ for which there exist real numbers $x_1, x_2,...,x_n \in (-1, 1)$ such that $|x_1| + |x_2| +...+ |x_n| = m + |x_1 + x_2 + ... + x_n|$.

2017 Purple Comet Problems, 27

Tags: algebra , min

Find the minimum value of $4(x^2 + y^2 + z^2 + w^2) + (xy - 7)^2 + (yz - 7)^2 + (zw - 7)^2 + (wx - 7)^2$ as $x, y, z$, and $w$ range over all real numbers.

2005 Thailand Mathematical Olympiad, 21

Tags: trigonometry , algebra , min , inequalities

Compute the minimum value of $cos(a-b) + cos(b-c) + cos(c-a)$ as $a,b,c$ ranges over the real numbers.

2013 Junior Balkan Team Selection Tests - Romania, 3

Tags: min , max , inequalities , algebra

Find the minimum and the maximum value of the expression $\sqrt{4 -a^2} +\sqrt{4 -b^2} +\sqrt{4 -c^2}$ where $a,b, c$ are positive real numbers satisfying the condition $a^2 + b^2 + c^2=6$

2019 Hanoi Open Mathematics Competitions, 9

Tags: algebra , inequalities , min

Let $a$ and $b$ be positive real numbers with $a > b$. Find the smallest possible values of $$S = 2a +3 +\frac{32}{(a - b)(2b +3)^2}$$

2002 Estonia National Olympiad, 1

Tags: algebra , min

Peeter, Juri, Kati and Mari are standing at the entrance of a dark tunnel. They have one torch and none of them dares to be in the tunnel without it, but the tunnel is so narrow that at most two people can move together. It takes $1$ minute for Peeter, $2$ minutes for Juri, $5$ for Kati and $10$ for Mari to pass the tunnel. Find the minimum time in which they can all pass through the tunnel.

2015 Dutch IMO TST, 5

Tags: positive integer , number theory , divide , min , max

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

2015 Saudi Arabia JBMO TST, 4

Tags: combinatorics , coloring , min

The numbers $1,2,...,64 $ are written on the unit squares of a table $8 \times 8$. The two smallest numbers of every row are marked black and the two smaller numbers of every comlumn are marked white. Prove or disprove that there are at least k numbers on the table that are marked both black and white when: a) $k=3$ b) $k=4$ c) $k=5$ .

2001 Singapore Senior Math Olympiad, 2

Tags: inequalities , min , algebra

Let $n$ be a positive integer, and let $f(n) =1^n + 2^{n-1} + 3^{n-2}+ 4^{n-3}+... + (n-1)^2 + n^1$ Find the smallest possible value of $\frac{f(n+2)}{f(n)}$ .Justify your answer.

2017 Czech-Polish-Slovak Junior Match, 4

Tags: geometry , perimeter , min , right triangle

Given is a right triangle $ABC$ with perimeter $2$, with $\angle B=90^o$ . Point $S$ is the center of the excircle to the side $AB$ of the triangle and $H$ is the intersection of the heights of the triangle $ABS$ . Determine the smallest possible length of the segment $HS $.

1998 Switzerland Team Selection Test, 3

Tags: function , min , algebra

Given positive numbers $a,b,c$, find the minimum of the function $f(x) = \sqrt{a^2 +x^2} +\sqrt{(b-x)^2 +c^2}$.

2000 Argentina National Olympiad, 6

Tags: geometry , paper , folding , min , area

You have an equilateral paper triangle of area $9$ and fold it in two, following a straight line that passes through the center of the triangle and does not contain any vertex of the triangle. Thus there remains a quadrilateral in which the two pieces overlap, and three triangles without overlaps. Determine the smallest possible value of the quadrilateral area of the overlay.

Novosibirsk Oral Geo Oly IX, 2017.1

Tags: grid , min , geometry

Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . [img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]

2014 Saudi Arabia Pre-TST, 3.2

Tags: algebra , min , inequalities

Let $x, y$ be positive real numbers. Find the minimum of $$x^2 + xy +\frac{y^2}{2}+\frac{2^6}{x + y}+\frac{3^4}{x^3}$$

2016 Auckland Mathematical Olympiad, 4

Tags: number theory , min , minimum

Find the smallest positive value of $36^k - 5^m$, where $k$ and $m$ are positive integers.

2012 Belarus Team Selection Test, 1

Tags: algebra , sum , min , irrational , polynomial , cubic , cubic equation

A cubic trinomial $x^3 + px + q$ with integer coefficients $p$ and $q$ is said to be [i]irrational [/i] if it has three pairwise distinct real irrational roots $a_1,a_2, a_3$ Find all irrational cubic trinomials for which the value of $|a_1| + [a_2| + |a_3|$ is the minimal possible. (E. Barabanov)

2015 Dutch IMO TST, 5

Tags: positive integer , number theory , divide , min , max

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

1996 Israel National Olympiad, 6

Tags: inequalities , min , algebra

Let $x,y,z$ be real numbers with $|x|,|y|,|z| > 2$. What is the smallest possible value of $|xyz+2(x+y+z)|$ ?

2020 Greece JBMO TST, 4

Tags: algebra , sum , product , inequalities , min

Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, . . . , 8 \}$ with $A \cup B = X$ and $A \cap B = \emptyset$. Let $P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$. Find the minimum possible value of sum $P_A +P_B$. PS. It is a variation of [url=https://artofproblemsolving.com/community/c6h2267998p17621980]JBMO Shortlist 2019 A3 [/url]

Cono Sur Shortlist - geometry, 1993.14

Tags: centroid , geometric inequality , min , geometry

Prove that the sum of the squares of the distances from a point $P$ to the vertices of a triangle $ABC$ is minimum when $ P$ is the centroid of the triangle.

2016 Peru IMO TST, 4

Tags: positive integer , number theory , divide , min , max

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers