Olympiad problems

2009 Sharygin Geometry Olympiad, 3

The cirumradius and the inradius of triangle $ABC$ are equal to $R$ and $r, O, I$ are the centers of respective circles. External bisector of angle $C$ intersect $AB$ in point $P$. Point $Q$ is the projection of $P$ to line $OI$. Find distance $OQ.$ (A.Zaslavsky, A.Akopjan)

1989 All Soviet Union Mathematical Olympiad, 500

Tags: distance , geometry , combinatorics

An insect is on a square ceiling side $1$. The insect can jump to the midpoint of the segment joining it to any of the four corners of the ceiling. Show that in $8$ jumps it can get to within $1/100$ of any chosen point on the ceiling

2016 ASMT, 5

Tags: analytic geometry , distance , geometry

Plane $A$ passes through the points $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$. Plane $B$ is parallel to plane $A$, but passes through the point $(1, 0, 1)$. Find the distance between planes $A$ and $B$.

2015 Bundeswettbewerb Mathematik Germany, 4

Tags: perpendicular , distance , incenter , geometry , circumcircle

Let $ABC$ be a triangle, such that its incenter $I$ and circumcenter $U$ are distinct. For all points $X$ in the interior of the triangle let $d(X)$ be the sum of distances from $X$ to the three (possibly extended) sides of the triangle. Prove: If two distinct points $P,Q$ in the interior of the triangle $ABC$ satisfy $d(P)=d(Q)$, then $PQ$ is perpendicular to $UI$.

1976 Vietnam National Olympiad, 3

Tags: distance , geometric inequality , area of a triangle , tetrahedron , geometry

$P$ is a point inside the triangle $ABC$. The perpendicular distances from $P$ to the three sides have product $p$. Show that $p \le \frac{ 8 S^3}{27abc}$, where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.

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 $PQ$?

1993 Chile National Olympiad, 1

Tags: geometry , path , distance , square

There are four houses, located on the vertices of a square. You want to draw a road network, so that you can go from any house to any other. Prove that the network formed by the diagonals is not the shortest. Find a shorter network.

2013 Abels Math Contest (Norwegian MO) Final, 2

Tags: geometry , combinatorial geometry , combinatorics , distance

In a triangle $T$, all the angles are less than $90^o$, and the longest side has length $s$. Show that for every point $p$ in $T$ we can pick a corner $h$ in $T$ such that the distance from $p$ to $h$ is less than or equal to $s/\sqrt3$.

2018 Brazil Team Selection Test, 2

Tags: sorting , extremal combinatorics , radius , string , combinatorics , distance

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

2009 Oral Moscow Geometry Olympiad, 5

Tags: circles , distance , geometry

A treasure is buried at some point on a round island with a radius of $1$ km. On the coast of the island there is a mathematician with a device that indicates the direction to the treasure when the distance to the treasure does not exceed $500$ m. In addition, the mathematician has a map of the island, on which he can record all his movements, perform measurements and geometric constructions. The mathematician claims that he has an algorithm for how to get to the treasure after walking less than $4$ km. Could this be true? (B. Frenkin)

1999 Spain Mathematical Olympiad, 5

Tags: inequalities , centroid , geometric inequality , distance , geometry

The distances from the centroid $G$ of a triangle $ABC$ to its sides $a,b,c$ are denoted $g_a,g_b,g_c$ respectively. Let $r$ be the inradius of the triangle. Prove that: a) $g_a,g_b,g_c \ge \frac{2}{3}r$ b) $g_a+g_b+g_c \ge 3r$

1987 Austrian-Polish Competition, 5

Tags: distance , partition , combinatorics , combinatorial geometry

The Euclidian three-dimensional space has been partitioned into three nonempty sets $A_1,A_2,A_3$. Show that one of these sets contains, for each $d > 0$, a pair of points at mutual distance $d$.

2005 Sharygin Geometry Olympiad, 11.3

Tags: ratio , diagonal , distance , geometry , cyclic quadrilateral

Inside the inscribed quadrilateral $ABCD$ there is a point $K$, the distances from which to the sides $ABCD$ are proportional to these sides. Prove that $K$ is the intersection point of the diagonals of $ABCD$.

2019 ISI Entrance Examination, 8

Tags: real analysis , coordinate geometry , distance

Consider the following subsets of the plane:$$C_1=\Big\{(x,y)~:~x>0~,~y=\frac1x\Big\} $$ and $$C_2=\Big\{(x,y)~:~x<0~,~y=-1+\frac1x\Big\}$$ Given any two points $P=(x,y)$ and $Q=(u,v)$ of the plane, their distance $d(P,Q)$ is defined by $$d(P,Q)=\sqrt{(x-u)^2+(y-v)^2}$$ Show that there exists a unique choice of points $P_0\in C_1$ and $Q_0\in C_2$ such that $$d(P_0,Q_0)\leqslant d(P,Q)\quad\forall ~P\in C_1~\text{and}~Q\in C_2.$$

1998 China Team Selection Test, 2

Tags: point set , combinatorics , distance , geometry , combinatorial geometry

Let $n$ be a natural number greater than 2. $l$ is a line on a plane. There are $n$ distinct points $P_1$, $P_2$, …, $P_n$ on $l$. Let the product of distances between $P_i$ and the other $n-1$ points be $d_i$ ($i = 1, 2,$ …, $n$). There exists a point $Q$, which does not lie on $l$, on the plane. Let the distance from $Q$ to $P_i$ be $C_i$ ($i = 1, 2,$ …, $n$). Find $S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}$.

2010 Junior Balkan Team Selection Tests - Romania, 1

Tags: geometry , equilateral , parallel , distance

Consider two equilateral triangles $ABC$ and $MNP$ with the property that $AB \parallel MN, BC \parallel NP$ and $CA \parallel PM$ , so that the surfaces of the triangles intersect after a convex hexagon. The distances between the three pairs of parallel lines are at most equal to $1$. Show that at least one of the two triangles has the side at most equal to $\sqrt {3}$ .

1981 All Soviet Union Mathematical Olympiad, 324

Tags: rectangle , distance , geometry , combinatorial geometry

Six points are marked inside the $3\times 4$ rectangle. Prove that there is a pair of marked points with the distance between them not greater than $\sqrt5$.

2017 Yasinsky Geometry Olympiad, 4

Tags: geometry , construction , distance

Three points are given on the plane. With the help of compass and ruler construct a straight line in this plane, which will be equidistant from these three points. Explore how many solutions have this construction.

2019 Novosibirsk Oral Olympiad in Geometry, 2

Tags: geometry , equilateral , distance

Kikoriki live on the shores of a pond in the form of an equilateral triangle with a side of $600$ m, Krash and Wally live on the same shore, $300$ m from each other. In summer, Dokko to Krash walk $900$ m, and Wally to Rosa - also $900$ m. Prove that in winter, when the pond freezes and it will be possible to walk directly on the ice, Dokko will walk as many meters to Krash as Wally to Rosa. [url=https://en.wikipedia.org/wiki/Kikoriki]about Kikoriki/GoGoRiki / Smeshariki [/url]

1998 Bosnia and Herzegovina Team Selection Test, 1

Tags: maximum and minimum , distance , circles , geometry

Let $P_1$, $P_2$, $P_3$, $P_4$ and $P_5$ be five different points which are inside $D$ or on the border of figure $D$. Let $M=min\left\{P_iP_j \mid i \neq j\right\}$ be minimal distance between different points $P_i$. For which configuration of points $P_i$, value $M$ is at maximum, if : $a)$ $D$ is unit square $b)$ $D$ is equilateral triangle with side equal $1$ $c)$ $D$ is unit circle, circle with radius $1$

2018 Bundeswettbewerb Mathematik, 4

Tags: combinatorics , geometry , 3d geometry , triangle , distance

We are given six points in space with distinct distances, no three of them collinear. Consider all triangles with vertices among these points. Show that among these triangles there is one such that its longest side is the shortest side in one of the other triangles.

2002 Singapore Senior Math Olympiad, 2

Tags: geometry , distance , product , incircle , circumcircle

The vertices of a triangle inscribed in a circle are the points of tangency of a triangle circumscribed about the circle. Prove that the product of the perpendicular distances from any point on the circle to the sides of the inscribed triangle is the same as the product of the perpendicular distances from the same point to the sides of the circumscribed triangle.

2011 BAMO, 3

Tags: set , arithmetic sequence , distance , algebra

Let $S$ be a finite, nonempty set of real numbers such that the distance between any two distinct points in $S$ is an element of $S$. In other words, $|x-y|$ is in $S$ whenever $x \ne y$ and $x$ and $y$ are both in $S$. Prove that the elements of $S$ may be arranged in an arithmetic progression. This means that there are numbers $a$ and $d$ such that $S = \{a, a+d, a+2d, a+3d, ..., a+kd, ...\}$.

2009 Swedish Mathematical Competition, 5

Tags: semicircle , distance , geometry

A semicircular arc and a diameter $AB$ with a length of $2$ are given. Let $O$ be the midpoint of the diameter. On the radius perpendicular to the diameter, we select a point $P$ at the distance $d$ from the midpoint of the diameter $O$, $0 <d <1$. A line through $A$ and $P$ intersects the semicircle at point $C$. Through point $P$ we draw another line at right angle against $AC$ that intersects the semicircle at point $D$. Through point $C$ we draw a line $l_1$, parallel to $PD$ and then a line $l_2$, through $D$ parallel to $PC$. The lines $l_1$ and $l_2$ intersect at point $E$. Show that the distance between $O$ and $E$ is equal to $\sqrt{2- d^2}$

2005 Sharygin Geometry Olympiad, 10.3

Tags: circles , geometry , chord , parallel , distance

Two parallel chords $AB$ and $CD$ are drawn in a circle with center $O$. Circles with diameters $AB$ and $CD$ intersect at point $P$. Prove that the midpoint of the segment $OP$ is equidistant from lines $AB$ and $CD$.