Olympiad problems

1973 IMO Shortlist, 14

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?

2019 Hanoi Open Mathematics Competitions, 13

Tags: triangle inequality , geometry , equilateral , distance

Find all points inside a given equilateral triangle such that the distances from it to three sides of the given triangle are the side lengths of a triangle.

1999 South africa National Olympiad, 1

Tags: inequalities , perimeter , ceiling function , triangle inequality , geometry

How many non-congruent triangles with integer sides and perimeter 1999 can be constructed?

2010 Tournament Of Towns, 5

Tags: inequalities , triangle inequality

For each side of a given pentagon, divide its length by the total length of all other sides. Prove that the sum of all the fractions obtained is less than 2.

1973 IMO, 1

Tags: geometry , triangle inequality , minimization , triangle , optimization

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?

2003 AMC 12-AHSME, 7

Tags: triangle inequality , inequalities , geometry , perimeter , congruent triangles

How many non-congruent triangles with perimeter $ 7$ have integer side lengths? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

2010 Switzerland - Final Round, 4

Tags: inequalities , algebra , function , triangle inequality , three variable inequality

Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that \[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]

2006 International Zhautykov Olympiad, 3

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

Let $ ABCDEF$ be a convex hexagon such that $ AD \equal{} BC \plus{} EF$, $ BE \equal{} AF \plus{} CD$, $ CF \equal{} DE \plus{} AB$. Prove that: \[ \frac {AB}{DE} \equal{} \frac {CD}{AF} \equal{} \frac {EF}{BC}. \]

2006 Kazakhstan National Olympiad, 1

Tags: combinatorics , triangle inequality

Natural numbers from $1$ to $200$ were divided into $50$ sets. Prove that one of them contains three numbers that are the lengths of the sides of some triangle

2004 France Team Selection Test, 3

Tags: pigeonhole principle , triangle inequality , inequalities , complex number , analytic geometry , geometry

Each point of the plane with two integer coordinates is the center of a disk with radius $ \frac {1} {1000}$. Prove that there exists an equilateral triangle whose vertices belong to distinct disks. Prove that such a triangle has side-length greater than 96.

2003 Tournament Of Towns, 6

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

A trapezoid with bases $AD$ and $BC$ is circumscribed about a circle, $E$ is the intersection point of the diagonals. Prove that $\angle AED$ is not acute.

2018 German National Olympiad, 4

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

a) Let $a,b$ and $c$ be side lengths of a triangle with perimeter $4$. Show that \[a^2+b^2+c^2+abc<8.\] b) Is there a real number $d<8$ such that for all triangles with perimeter $4$ we have \[a^2+b^2+c^2+abc<d \quad\] where $a,b$ and $c$ are the side lengths of the triangle?

2012 Balkan MO, 2

Tags: trigonometry , function , triangle inequality , rearrangement inequality

Prove that \[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\] for all positive real numbers $x,y$ and $z$.

2010 USA Team Selection Test, 3

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

Let $h_a, h_b, h_c$ be the lengths of the altitudes of a triangle $ABC$ from $A, B, C$ respectively. Let $P$ be any point inside the triangle. Show that \[\frac{PA}{h_b+h_c} + \frac{PB}{h_a+h_c} + \frac{PC}{h_a+h_b} \ge 1.\]

2010 Contests, 3

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

Let $h_a, h_b, h_c$ be the lengths of the altitudes of a triangle $ABC$ from $A, B, C$ respectively. Let $P$ be any point inside the triangle. Show that \[\frac{PA}{h_b+h_c} + \frac{PB}{h_a+h_c} + \frac{PC}{h_a+h_b} \ge 1.\]

2005 Sharygin Geometry Olympiad, 14

Tags: inequalities , area , area of a triangle , geometric inequality , triangle inequality , geometry

Let $P$ be an arbitrary point inside the triangle $ABC$. Let $A_1, B_1$ and $C_1$ denote the intersection points of the straight lines $AP, BP$ and $CP$, respectively, with the sides $BC, CA$ and $AB$. We order the areas of the triangles $AB_1C_1,A_1BC_1,A_1B_1C$. Denote the smaller by $S_1$, the middle by $S_2$, and the larger by $S_3$. Prove that $\sqrt{S_1 S_2} \le S \le \sqrt{S_2 S_3}$ ,where $S$ is the area of the triangle $A_1B_1S_1$.

2016 Middle European Mathematical Olympiad, 2

Tags: triangle inequality , combinatorics

There are $n \ge 3$ positive integers written on a board. A [i]move[/i] consists of choosing three numbers $a, b, c$ written from the board such that there exists a non-degenerate non-equilateral triangle with sides $a, b, c$ and replacing those numbers with $a + b - c, b + c - a$ and $c + a - b$. Prove that a sequence of moves cannot be infinite.

1980 IMO, 3

Tags: geometry , 3d geometry , tetrahedron , sphere , triangle inequality , angle

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

1983 Swedish Mathematical Competition, 4

Tags: triangle inequality , concentric , max , triangle area , rectangle , geometry

$C$, $C'$ are concentric circles with radii $R$, $R'$. A rectangle has two adjacent vertices on $C$ and the other two vertices on $C'$. Find its sides if its area is as large as possible.

1999 Brazil Team Selection Test, Problem 4

Tags: inequalities , number theory , induction , function , ring theory , triangle inequality , logarithm

Let Q+ and Z denote the set of positive rationals and the set of inte- gers, respectively. Find all functions f : Q+ → Z satisfying the following conditions: (i) f(1999) = 1; (ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+; (iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.

Kyiv City MO 1984-93 - geometry, 1990.9.4

Tags: triangle inequality , geometry , trigonometry

Let $\alpha, \beta, \gamma$ be the angles of some triangle. Prove that there is a triangle whose sides are equal to $\sin \alpha$, $\sin \beta$, $\sin \gamma$.

1985 Canada National Olympiad, 5

Tags: triangle inequality , induction , inequalities

Let $1 < x_1 < 2$ and, for $n = 1$, 2, $\dots$, define $x_{n + 1} = 1 + x_n - \frac{1}{2} x_n^2$. Prove that, for $n \ge 3$, $|x_n - \sqrt{2}| < 2^{-n}$.

2015 Romania Team Selection Tests, 2

Tags: inequalities , inradius , circles , triangle inequality , geometry

Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.

2007 China Northern MO, 4

Tags: inequalities , area of a triangle , algebra , geometry , inradius , heron's formula , triangle inequality

The inradius of triangle $ ABC$ is $ 1$ and the side lengths of $ ABC$ are all integers. Prove that triangle $ ABC$ is right-angled.

2008 Harvard-MIT Mathematics Tournament, 10

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

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.