Olympiad problems

2018 Adygea Teachers' Geometry Olympiad, 1

Can the distances from a certain point on the plane to the vertices of a certain square be equal to $1, 4, 7$, and $8$ ?

1985 All Soviet Union Mathematical Olympiad, 417

Tags: 3d geometry , geometry , circles , distance , cube

The $ABCDA_1B_1C_1D_1$ cube has unit length edges. Find the distance between two circumferences, one of those is inscribed into the $ABCD$ base, and another comes through points $A,C$ and $B_1$ .

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 $PQ$?

2018 India IMO Training Camp, 1

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

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.

2019 Mediterranean Mathematics Olympiad, 4

Tags: geometric inequality , geometry , inequalities , distance

Let $P$ be a point in the interior of an equilateral triangle with height $1$, and let $x,y,z$ denote the distances from $P$ to the three sides of the triangle. Prove that \[ x^2+y^2+z^2 ~\ge~ x^3+y^3+z^3 +6xyz \]

2019 ISI Entrance Examination, 8

Tags: distance , coordinate geometry , real analysis

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 $$d(P_0,Q_0)\leqslant d(P,Q)\quad\forall ~P\in C_1~\text{and}~Q\in C_2.$$

2001 Chile National Olympiad, 3

Tags: geometry , geometric inequality , angle bisector , distance

In a triangle $ \vartriangle ABC $, let $ h_a, h_b $ and $ h_c $ the atlitudes. Let $ D $ be the point where the inner bisector of $ \angle BAC $ cuts to the side $ BC $ and $ d_a $ is the distance from the $ D $ point next to $ AB $. The distances $ d_b $ and $ d_c $ are similarly defined. Show that: $$ \dfrac {3} {2} \le \dfrac {d_a} {h_a} + \dfrac {d_b} {h_b} + \dfrac {d_c} {h_c} $$ For what kind of triangles does the equality hold?

2019 Canada National Olympiad, 1

Tags: circumcenter , distance , geometry

Points $A,B,C$ are on a plane such that $AB=BC=CA=6$. At any step, you may choose any three existing points and draw that triangle's circumcentre. Prove that you can draw a point such that its distance from an previously drawn point is: $(a)$ greater than 7 $(b)$ greater than 2019

2021 Israel TST, 2

Tags: combinatorial geometry , combinatorics , distance

Let $n>1$ be an integer. Hippo chooses a list of $n$ points in the plane $P_1, \dots, P_n$; some of these points may coincide, but not all of them can be identical. After this, Wombat picks a point from the list $X$ and measures the distances from it to the other $n-1$ points in the list. The average of the resulting $n-1$ numbers will be denoted $m(X)$. Find all values of $n$ for which Hippo can prepare the list in such a way, that for any point $X$ Wombat may pick, he can point to a point $Y$ from the list such that $XY=m(X)$.

2001 Tuymaada Olympiad, 8

Tags: algebra , game strategy , speed , distance

Can three persons, having one double motorcycle, overcome the distance of $70$ km in $3$ hours? Pedestrian speed is $5$ km / h and motorcycle speed is $50$ km / h.

2010 Junior Balkan Team Selection Tests - Romania, 1

Tags: equilateral , distance , parallel , geometry

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 $\sqrt {3}$ .

2018 Brazil Team Selection Test, 2

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

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.

2005 Sharygin Geometry Olympiad, 10.3

Tags: circles , parallel , distance , chord , geometry

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 $CD$.

1976 Vietnam National Olympiad, 3

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

$P$ 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.

2014 Nordic, 2

Tags: geometry , distance , locus , equilateral triangle

Given an equilateral triangle, find all points inside the triangle such that the distance from the point to one of the sides is equal to the geometric mean of the distances from the point to the other two sides of the triangle.

2010 Oral Moscow Geometry Olympiad, 2

Tags: geometry , paper , folding , square , distance

Given a square sheet of paper with side $1$. Measure on this sheet a distance of $ 5/6$. (The sheet can be folded, including, along any segment with ends at the edges of the paper and unbend back, after unfolding, a trace of the fold line remains on the paper).

1969 IMO Longlists, 11

Tags: geometry , distance , point set

$(BUL 5)$ Let $Z$ be a set of points in the plane. Suppose that there exists a pair of points that cannot be joined by a polygonal line not passing through any point of $Z.$ Let us call such a pair of points unjoinable. Prove that for each real $r > 0$ there exists an unjoinable pair of points separated by distance $r.$

1948 Moscow Mathematical Olympiad, 151

Tags: midpoint , parallel , distance , geometry , convex

The distance between the midpoints of the opposite sides of a convex quadrilateral is equal to a half sum of lengths of the other two sides. Prove that the first pair of sides is parallel.

2015 JBMO Shortlist, C2

Tags: point , combinatorics , combinatorial geometry , distance

$2015$ points are given in a plane such that from any five points we can choose two points with distance less than $1$ unit. Prove that $504$ of the given points lie on a unit disc.

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.

1957 Putnam, A5

Tags: geometry , combinatorial geometry , distance

Given $n$ points in the plane, show that the largest distance determined by these points cannot occur more than $n$ times.

1987 Polish MO Finals, 1

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

There are $n \ge 2$ points in a square side $1$. Show that one can label the points $P_1, P_2, ... , P_n$ such that $\sum_{i=1}^n |P_{i-1} - P_i|^2 \le 4$, where we use cyclic subscripts, so that $P_0$ means $P_n$.

2017 IMO Shortlist, C2

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

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: distance , circles , 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)

1986 All Soviet Union Mathematical Olympiad, 421

Tags: combinatorics , distance

Certain king of a certain state wants to build $n$ cities and $n-1$ roads, connecting them to provide a possibility to move from every city to every city. (Each road connects two cities, the roads do not intersect, and don't come through another city.) He wants also, to make the shortests distances between the cities, along the roads, to be $1,2,3,...,n(n-1)/2$ kilometres. Is it possible for a) $n=6$ b) $n=1986$ ?