Olympiad problems

2015 Swedish Mathematical Competition, 3

Let $a$, $b$, $c$ be positive real numbers. Determine the minimum value of the following expression $$ \frac{a^2+2b^2+4c^2}{b(a+2c)}$$

1998 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: inequalities , min , max , algebra

Given an integer $n > 2$, consider all sequences $x_1,x_2,...,x_n$ of nonnegative real numbers such that $$x_1+ 2x_2 + ... + nx_n = 1.$$ Find the maximum value and the minimum value of $x_1^2+x_2^2+...+x_n^2$ and determine all the sequences $x_1,x_2,...,x_n$ for which these values are obtained.

2020 Adygea Teachers' Geometry Olympiad, 4

Tags: geometry , angle , min , length , tangent , arc , circles

A circle is inscribed in an angle with vertex $O$, touching its sides at points $M$ and $N$. On an arc $MN$ nearest to point $O$, an arbitrary point $P$ is selected. At point $P$, a tangent is drawn to the circle $P$, intersecting the sides of the angle at points $A$ and $B$. Prove that that the length of the segment $AB$ is the smallest when $P$ is its midpoint.

2001 Grosman Memorial Mathematical Olympiad, 4

Tags: triangle , min , perpendicular , distance , geometry

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 $PQ$?

1999 Czech And Slovak Olympiad IIIA, 1

Tags: min , integer polynomial , algebra , fraction

We are allowed to put several brackets in the expression $$\frac{29 : 28 : 27 : 26 :... : 17 : 16}{15 : 14 : 13 : 12 : ... : 3 : 2}$$ always in the same places below each other. (a) Find the smallest possible integer value we can obtain in that way. (b) Find all possible integer values that can be obtained. Remark: in this problem, $$\frac{(29 : 28) : 27 : ... : 16}{(15 : 14) : 13 : ... : 2},$$ is valid position of parenthesis, on the other hand $$\frac{(29 : 28) : 27 : ... : 16}{15 : (14 : 13) : ... : 2}$$ is forbidden.

1975 Chisinau City MO, 97

Tags: algebra , inequalities , min

Find the smallest value of the expression $(x-1) (x -2) (x -3) (x - 4) + 10$.

Ukrainian TYM Qualifying - geometry, IV.7

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

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.

Estonia Open Junior - geometry, 2004.1.2

Tags: min , area , circles , geometry

Diameter $AB$ is drawn to a circle with radius $1$. Two straight lines $s$ and $t$ touch the circle at points $A$ and $B$, respectively. Points $P$ and $Q$ are chosen on the lines $s$ and $t$, respectively, so that the line $PQ$ touches the circle. Find the smallest possible area of the quadrangle $APQB$.

1989 Romania Team Selection Test, 3

Tags: hexagon , locus , geometry , min

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.

2015 Saudi Arabia IMO TST, 3

Tags: inequalities , algebra , min , max

Let $a, b,c$ be positive real numbers satisfying the condition $$(x + y + z) \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)= 10$$ Find the greatest value and the least value of $$T = (x^2 + y^2 + z^2) \left(\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}\right)$$ Trần Nam Dũng

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]

2019 Bundeswettbewerb Mathematik, 2

Tags: min , algebra , inequalities

Determine the smallest possible value of the sum $S (a, b, c) = \frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}$ where $a, b, c$ are three positive real numbers with $a^2 + b^2 + c^2 = 1$

2020 Estonia Team Selection Test, 2

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

Ukraine Correspondence MO - geometry, 2007.9

Tags: geometry , perimeter , min , geometric inequality , equal angles

In triangle $ABC$, the lengths of all sides are integers, $\angle B=2 \angle A$ and $\angle C> 90^o$. Find the smallest possible perimeter of this triangle.

2007 Singapore Senior Math Olympiad, 5

Tags: inequalities , min , max , algebra

Find the maximum and minimum of $x + y$ such that $x + y = \sqrt{2x-1}+\sqrt{4y+3}$

2001 Denmark MO - Mohr Contest, 3

Tags: geometry , min , square

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]

1995 Austrian-Polish Competition, 8

Tags: vector , inequalities , sum , min , algebra

Consider the cube with the vertices at the points $(\pm 1, \pm 1, \pm 1)$. Let $V_1,...,V_{95}$ be arbitrary points within this cube. Denote $v_i = \overrightarrow{OV_i}$, where $O = (0,0,0)$ is the origin. Consider the $2^{95}$ vectors of the form $s_1v_1 + s_2v_2 +...+ s_{95}v_{95}$, where $s_i = \pm 1$. (a) If $d = 48$, prove that among these vectors there is a vector $w = (a, b, c)$ such that $a^2 + b^2 + c^2 \le 48$. (b) Find a smaller $d$ (the smaller, the better) with the same property.

2011 Junior Balkan Team Selection Tests - Moldova, 1

Tags: absolute value , algebra , min

The absolute value of the difference of the solutions of the equation $x^2 + px + q = 0$, with $p, q \in R$, is equal to $4$. Find the solutions of the equation if it is known that $(q + 1) p^2 + q^2$ takes the minimum value.

1968 Swedish Mathematical Competition, 5

Tags: trigonometry , algebra , min , inequalities

Let $a, b$ be non-zero integers. Let $m(a, b)$ be the smallest value of $\cos ax + \cos bx$ (for real $x$). Show that for some $r$, $m(a, b) \le r < 0$ for all $a, b$.

2003 Junior Balkan Team Selection Tests - Romania, 4

Tags: square , geometry , area , min , maximum value , distance

Two unit squares with parallel sides overlap by a rectangle of area $1/8$. Find the extreme values of the distance between the centers of the squares.

1977 Spain Mathematical Olympiad, 7

Tags: algebra , inequalities , min

The numbers $A_1 , A_2 ,... , A_n$ are given. Prove, without calculating derivatives, that the value of $X$ that minimizes the sum $(X - A_1)^2 + (X -A_2)^2 + ...+ (X - A_n)^2$ is precisely the arithmetic mean of the given numbers.

2009 Postal Coaching, 4

Tags: algebra , min , max , number theory , angle

For positive integers $n \ge 3$ and $r \ge 1$, define $$P(n, r) = (n - 2)\frac{r^2}{2} - (n - 4) \frac{r}{2}$$ We call a triple $(a, b, c)$ of natural numbers, with $a \le b \le c$, an $n$-gonal Pythagorean triple if $P(n, a)+P(n, b) = P(n, c)$. (For $n = 4$, we get the usual Pythagorean triple.) (a) Find an $n$-gonal Pythagorean triple for each $n \ge 3$. (b) Consider all triangles $ABC$ whose sides are $n$-gonal Pythagorean triples for some $n \ge 3$. Find the maximum and the minimum possible values of angle $C$.

2019 Junior Balkan Team Selection Tests - Romania, 4

Tags: algebra , min , inequalities

Let $a$ and $b$ be positive real numbers such that $3(a^2+b^2-1) = 4(a+b$). Find the minimum value of the expression $\frac{16}{a}+\frac{1}{b}$ .

2016 Singapore Junior Math Olympiad, 5

Tags: point , combinatorics , combinatorial geometry , line , min

Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $200$ distinct points. (Note that for $3$ distinct points, the minimum number of lines is $3$ and for $4$ distinct points, the minimum is $4$)