Olympiad problems

Estonia Open Junior - geometry, 2016.2.5

On the plane three different points $P, Q$, and $R$ are chosen. It is known that however one chooses another point $X$ on the plane, the point $P$ is always either closer to $X$ than the point $Q$ or closer to $X$ than the point $R$. Prove that the point $P$ lies on the line segment $QR$.

2019 Tuymaada Olympiad, 2

Tags: midpoint , geometry , trapezoid , circumcenter , distance

A trapezoid $ABCD$ with $BC // AD$ is given. The points $B'$ and $C'$ are symmetrical to $B$ and $C$ with respect to $CD$ and $AB$, respectively. Prove that the midpoint of the segment joining the circumcentres of $ABC'$ and $B'CD$ is equidistant from $A$ and $D$.

2006 German National Olympiad, 4

Tags: triangle , segment , geometry , distance

Let $D$ be a point inside a triangle $ABC$ such that $|AC| -|AD| \geq 1$ and $|BC|- |BD| \geq 1.$ Prove that for any point $E$ on the segment $AB$, we have $|EC| -|ED| \geq 1.$

2010 Thailand Mathematical Olympiad, 8

Tags: distance , number theory

Define the modulo $2553$ distance $d(x, y)$ between two integers $x, y$ to be the smallest nonnegative integer $d$ equivalent to either $x - y$ or $y - x$ modulo $2553$. Show that, given a set S of integers such that $|S| \ge 70$, there must be $m, n \in S$ with $d(m, n) \le 36$.

1948 Moscow Mathematical Olympiad, 151

Tags: midpoint , distance , convex , parallel , geometry

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.

Geometry Mathley 2011-12, 6.1

Tags: circumcircle , incenter , circumradius , excenter , distance , tangent , geometry

Show that the circumradius $R$ of a triangle $ABC$ equals the arithmetic mean of the oriented distances from its incenter $I$ and three excenters $I_a,I_b, I_c$ to any tangent $\tau$ to its circumcircle. In other words, if $\delta(P)$ denotes the distance from a point $P$ to $\tau$, then with appropriate choices of signs, we have $$\delta(I) \pm \delta_(I_a) \pm \delta_(I_b) \pm \delta_(I_c) = 4R$$ Luis González

1993 IMO Shortlist, 2

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

Show that there exists a finite set $A \subset \mathbb{R}^2$ such that for every $X \in A$ there are points $Y_1, Y_2, \ldots, Y_{1993}$ in $A$ such that the distance between $X$ and $Y_i$ is equal to 1, for every $i.$

2009 Junior Balkan Team Selection Tests - Romania, 4

Tags: distance , geometric inequality , geometry

Consider $K$ a polygon in plane, such that the distance between any two vertices is not greater than $1$. Let $X$ and $Y$ be two points inside $K$. Show that there exist a point $Z$, lying on the border of K, such that $XZ + Y Z \le 1$

2020 Tuymaada Olympiad, 4

Tags: distance , combinatorics

For each positive integer $k$, let $g(k)$ be the maximum possible number of points in the plane such that pairwise distances between these points have only $k$ different values. Prove that there exists $k$ such that $g(k) > 2k + 2020$.

1974 Putnam, B1

Tags: circles , distance

Which configurations of five (not necessarily distinct) points $p_1 ,\ldots, p_5$ on the circle $x^2 +y^2 =1$ maximize the sum of the ten distances $$\sum_{i<j} d(p_i, p_j)?$$

1949-56 Chisinau City MO, 60

Tags: regular tetrahedron , tetrahedron , 3d geometry , geometry , distance

Show that the sum of the distances from any point of a regular tetrahedron to its faces is equal to the height of this tetrahedron.

2006 Oral Moscow Geometry Olympiad, 4

Tags: geometry , symmetric , distance , cyclic quadrilateral

The quadrangle $ABCD$ is inscribed in a circle, the center $O$ of which lies inside it. The tangents to the circle at points $A$ and $C$ and a straight line, symmetric to $BD$ wrt point $O$, intersect at one point. Prove that the products of the distances from $O$ to opposite sides of the quadrilateral are equal. (A. Zaslavsky)

1996 Poland - Second Round, 6

Tags: geometry , distance , parallelepiped , 3d geometry

Prove that every interior point of a parallelepiped with edges $a,b,c$ is on the distance at most $\frac12 \sqrt{a^2 +b^2 +c^2}$ from some vertex of the parallelepiped.

2024 Israel TST, P3

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

For a set $S$ of at least $3$ points in the plane, let $d_{\text{min}}$ denote the minimal distance between two different points in $S$ and $d_{\text{max}}$ the maximal distance between two different points in $S$. For a real $c>0$, a set $S$ will be called $c$-[i]balanced[/i] if \[\frac{d_{\text{max}}}{d_{\text{min}}}\leq c|S|\] Prove that there exists a real $c>0$ so that for every $c$-balanced set of points $S$, there exists a triangle with vertices in $S$ that contains at least $\sqrt{|S|}$ elements of $S$ in its interior or on its boundary.

2009 Kyiv Mathematical Festival, 3

Tags: geometry , minimum , distance

Points $A_1,A_2,...,A_n$ are selected from the equilateral triangle with a side that is equal to $1$. Denote by $d_k$ the least distance from $A_k$ to all other selected points. Prove that $d_1^2+...+d_n^2 \le 3,5$.

2010 Oral Moscow Geometry Olympiad, 2

Tags: geometry , paper , folding , distance , square

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).

1957 Putnam, A5

Tags: geometry , distance , combinatorial geometry

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

1985 Greece National Olympiad, 3

Tags: analytic geometry , number theory , distance , lattice points , geometry

Consider the line (E): $5x-10y+3=0$ . Prove that: a) Line $(E)$ doesn't pass through points with integer coordinates. b) There is no point $A(a_1,a_2)$ with $ a_1,a_2 \in \mathbb{Z}$ with distance from $(E)$ less then $\frac{\sqrt3}{20}$.

Geometry Mathley 2011-12, 4.1

Tags: geometry , circles , distance

Five points $K_i, i = 1, 2, 3, 4$ and $P$ are chosen arbitrarily on the same circle. Denote by $P(i, j)$ the distance from $P$ to the line passing through $K_i$ and $K_j$ . Prove that $$P(1, 2)P(3, 4) = P(1, 4)P(2, 3) = P(1, 3)P(2, 4)$$ Bùi Quang Tuấn

1986 All Soviet Union Mathematical Olympiad, 431

Tags: combinatorial geometry , geometry , dodecagon , distance , sum

Given two points inside a convex dodecagon (twelve sides) situated $10$ cm far from each other. Prove that the difference between the sum of distances, from the point to all the vertices, is less than $1$ m for those points.

2001 Chile National Olympiad, 3

Tags: geometric inequality , angle bisector , distance , geometry

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?

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.

2017 IMO Shortlist, C2

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

2018 Romania Team Selection Tests, 2

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

2006 Spain Mathematical Olympiad, 3

Tags: circumcircle , isosceles , distance , geometry

$ABC$ is an isosceles triangle with $AB = AC$. Let $P$ be any point of a circle tangent to the sides $AB$ in $B$ and to AC in C. Denote $a$, $b$ and $c$ to the distances from $P$ to the sides $BC, AC$ and $AB$ respectively. Prove that: $a^2=bc$