Olympiad problems

2001 Singapore Senior Math Olympiad, 2

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.

Ukrainian TYM Qualifying - geometry, IV.7

Tags: min , geometric inequality , inscribed , max , perimeter , geometry

Let $ABCD$ be the quadrilateral whose area is the largest among the quadrilaterals with given sides $a, b, c, d$, and let $PORS$ be the quadrilateral inscribed in $ABCD$ with the smallest perimeter. Find this perimeter.

2017 Novosibirsk Oral Olympiad in Geometry, 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]

2017 Czech-Polish-Slovak Junior Match, 4

Tags: perimeter , right triangle , min , geometry

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 $.

2019 Federal Competition For Advanced Students, P2, 6

Tags: divide , min , number theory

Find the smallest possible positive integer n with the following property: For all positive integers $x, y$ and $z$ with $x | y^3$ and $y | z^3$ and $z | x^3$ always to be true that $xyz| (x + y + z) ^n$. (Gerhard J. Woeginger)

2015 Dutch IMO TST, 5

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

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

1998 Switzerland Team Selection Test, 3

Tags: algebra , min , function

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

1964 Swedish Mathematical Competition, 5

Tags: inequalities , max , min , algebra , trigonometry

$a_1, a_2, ... , a_n$ are constants such that $f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0$ for all $x$. We seek estimates of $a_1$. If $n = 2$, find the smallest and largest possible values of $a_1$. Find corresponding estimates for other values of $n$.

1987 Polish MO Finals, 5

Tags: prime , number theory , product , min

Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).

Kyiv City MO 1984-93 - geometry, 1990.8.2

Tags: geometric inequality , incircle , equilateral , angle , min , geometry

A line passes through the center $O$ of an equilateral triangle $ABC$ and intersects the side $BC$. At what angle wrt $BC$ should this line be drawn this line so that its segment inside the triangle has the smallest possible length?

2003 Swedish Mathematical Competition, 1

Tags: algebra , min , system , system of equations , inequalities

If $x, y, z, w$ are nonnegative real numbers satisfying \[\left\{ \begin{array}{l}y = x - 2003 \\ z = 2y - 2003 \\ w = 3z - 2003 \\ \end{array} \right. \] find the smallest possible value of $x$ and the values of $y, z, w$ corresponding to it.

2016 Peru IMO TST, 4

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

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

2018 Costa Rica - Final Round, A1

Tags: algebra , inequalities , min

If $x \in R-\{-7\}$, determine the smallest value of the expression $$\frac{2x^2 + 98}{(x + 7)^2}$$

2001 Grosman Memorial Mathematical Olympiad, 4

Tags: geometry , triangle , distance , min , perpendicular

The lengths of the sides of triangle $ABC$ are $4,5,6$. For any point $D$ on one of the sides, draw the perpendiculars $DP, DQ$ on the other two sides. What is the minimum value of $PQ$?

1989 Romania Team Selection Test, 3

Tags: geometry , locus , min , hexagon

Let $F$ be the boundary and $M,N$ be any interior points of a triangle $ABC$. Consider the function $f_{M,N}: F \to R$ defined by $f_{M,N}(P) = MP^2 +NP^2$ and let $\eta_{M,N}$ be the number of points $P$ for which $f{M,N}$ attains its minimum. (a) Prove that $1 \le \eta_{M,N} \le 3$. (b) If $M$ is fixed, find the locus of $N$ for which $\eta_{M,N} > 1$. (c) Prove that the locus of $M$ for which there are points $N$ such that $\eta_{M,N} = 3$ is the interior of a tangent hexagon.

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.

2020 Estonia Team Selection Test, 2

Tags: combinatorial geometry , angle , min , combinatorics , geometry

Let $n$ be an integer, $n \ge 3$. Select $n$ points on the plane, none of which are three on the same line. Consider all triangles with vertices at selected points, denote the smallest of all the interior angles of these triangles by the variable $\alpha$. Find the largest possible value of $\alpha$ and identify all the selected $n$ point placements for which the max occurs.

1964 Swedish Mathematical Competition, 1

Tags: min , geometric inequalities , geometry

Find the side lengths of the triangle $ABC$ with area $S$ and $\angle BAC = x$ such that the side $BC$ is as short as possible.

I Soros Olympiad 1994-95 (Rus + Ukr), 9.3

Tags: inequalities , algebra , min

Find the smallest possible value of the expression $$\frac{(a+b) (b + c)}{a + 2b+c}$$ where $a, b, c$ are arbitrary numbers from the interval $[1,2]$.

Denmark (Mohr) - geometry, 2001.3

Tags: geometry , square , min

In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$. [img]https://1.bp.blogspot.com/-WD8WXIE6DK4/XzcC9GYsa6I/AAAAAAAAMXg/vl2OrbAdChEYrRpemYmj6DiOrdOSqj_IgCLcBGAsYHQ/s178/2001%2BMohr%2Bp3.png[/img]

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 $w = 2x - 3y + 4z$.

2013 Junior Balkan Team Selection Tests - Romania, 3

Tags: min , max , algebra , inequalities

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 $a^2 + b^2 + c^2=6$

2016 Auckland Mathematical Olympiad, 4

Tags: number theory , minimum , min

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

Novosibirsk Oral Geo Oly IX, 2017.1

Tags: geometry , grid , min

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]

1994 Italy TST, 1

Tags: geometry , max , chord , perpendicular , min , circles

Given a circle $\gamma$ and a point $P$ inside it, find the maximum and minimum value of the sum of the lengths of two perpendicular chords of $\gamma$ passing through $P$.