Olympiad problems

2001 Estonia Team Selection Test, 2

Point $X$ is taken inside a regular $n$-gon of side length $a$. Let $h_1,h_2,...,h_n$ be the distances from $X$ to the lines defined by the sides of the $n$-gon. Prove that $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$

2012 Oral Moscow Geometry Olympiad, 5

Tags: sum , distance , geometry , circles , minimum

Inside the circle with center $O$, points $A$ and $B$ are marked so that $OA = OB$. Draw a point $M$ on the circle from which the sum of the distances to points $A$ and $B$ is the smallest among all possible.

2019 Novosibirsk Oral Olympiad in Geometry, 1

Tags: distance , geometry

Lyuba, Tanya, Lena and Ira ran across a flat field. At some point it turned out that among the pairwise distances between them there are distances of $1, 2, 3, 4$ and $5$ meters, and there are no other distances. Give an example of how this could be.

May Olympiad L1 - geometry, 2020.3

Tags: distance , geometry

A clueless ant makes the following route: starting at point $ A $ goes $ 1$ cm north, then $ 2$ cm east, then $ 3$ cm south, then $ 4$ cm west, immediately $ 5$ cm north, continues $ 6$ cm east, and so on, finally $ 41$ cm north and ends in point $ B $. Calculate the distance between $ A $ and $ B $ (in a straight line).

2005 Sharygin Geometry Olympiad, 10.5

Tags: circles , tangent , distance , geometry

Two circles of radius $1$ intersect at points $X, Y$, the distance between which is also equal to $1$. From point $C$ of one circle, tangents $CA, CB$ are drawn to the other. Line $CB$ will cross the first circle a second time at point $A'$. Find the distance $AA'$.

1940 Moscow Mathematical Olympiad, 061

Tags: locus , distance , geometry

Given two lines on a plane, find the locus of all points with the difference between the distance to one line and the distance to the other equal to the length of a given segment.

1986 Spain Mathematical Olympiad, 1

Tags: set , distance , algebra

Define the distance between real numbers $x$ and $y$ by $d(x,y) =\sqrt{([x]-[y])^2+(\{x\}-\{y\})^2}$ . Determine (as a union of intervals) the set of real numbers whose distance from $3/2$ is less than $202/100$ .

2012 Romania National Olympiad, 1

Tags: geometry , distance , square

Let $P$ be a point inside the square $ABCD$ and $PA = 1$, $PB = \sqrt2$ and $PC =\sqrt3$. a) Determine the length of segment $[PD]$. b) Determine the angle $\angle APB$.

1983 Bundeswettbewerb Mathematik, 4

Tags: combinatorics , distance , combinatorial geometry

Let $g$ be a straight line and $n$ a given positive integer. Prove that there are always n different points on g to choose as well as a point not lying on g in such a way that the distance between each two of these $n + 1$ points is an integer.

1976 All Soviet Union Mathematical Olympiad, 220

Tags: distance , combinatorics , combinatorial geometry

There are $50$ exact watches lying on a table. Prove that there exist a certain moment, when the sum of the distances from the centre of the table to the ends of the minute hands is more than the sum of the distances from the centre of the table to the centres of the watches.

2016 Oral Moscow Geometry Olympiad, 3

Tags: right triangle , geometry , distance

A circle with center $O$ passes through the ends of the hypotenuse of a right-angled triangle and intersects its legs at points $M$ and $K$. Prove that the distance from point $O$ to line $MK$ is half the hypotenuse.

1956 Moscow Mathematical Olympiad, 324

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

a) What is the least number of points that can be chosen on a circle of length $1956$, so that for each of these points there is exactly one chosen point at distance $1$, and exactly one chosen point at distance $2$ (distances are measured along the circle)? b) On a circle of length $15$ there are selected $n$ points such that for each of them there is exactly one selected point at distance $1$ from it, and exactly one is selected point at distance $2$ from it. (All distances are measured along the circle.) Prove that $n$ is divisible by $10$.

2022/2023 Tournament of Towns, P5

Tags: distance , geometry

The distance between any two of five given points exceeds 2. Is it true that the distance between some two of these points exceeds 3 if these five points are in a) the plane; and b) three-dimensional space? [i]Alexey Tolpygo[/i]

2008 Oral Moscow Geometry Olympiad, 4

Tags: orthocenter , circumcenter , distance , geometry

Angle $A$ in triangle $ABC$ is equal to $120^o$. Prove that the distance from the center of the circumscribed circle to the orthocenter is equal to $AB + AC$. (V. Protasov)

2011 NZMOC Camp Selection Problems, 5

Tags: distance , square , collinear , geometry

Let a square $ABCD$ with sides of length $1$ be given. A point $X$ on $BC$ is at distance $d$ from $C$, and a point $Y$ on $CD$ is at distance $d$ from $C$. The extensions of: $AB$ and $DX$ meet at $P$, $AD$ and $BY$ meet at $Q, AX$ and $DC$ meet at $R$, and $AY$ and $BC$ meet at $S$. If points $P, Q, R$ and $S$ are collinear, determine $d$.

2008 Balkan MO Shortlist, G8

Tags: max , geometric inequality , geometry , distance

Let $P$ be a point in the interior of a triangle $ABC$ and let $d_a,d_b,d_c$ be its distances to $BC,CA,AB$ respectively. Prove that max $(AP, BP, CP) \ge \sqrt{d_a^2+d_b^2+d_c^2}$

1954 Moscow Mathematical Olympiad, 271

Tags: 3d geometry , geometric inequality , distance , geometry

Do there exist points $A, B, C, D$ in space, such that $AB = CD = 8, AC = BD = 10$, and $AD = BC = 13$?

1994 Tuymaada Olympiad, 6

Tags: right triangle , speed , geometry , distance , minimum

In three houses $A,B$ and $C$, forming a right triangle with the legs $AC=30$ and $CB=40$, live three beetles $a,b$ and $c$, capable of moving at speeds of $2, 3$ and $4$, respectively. Suppose that you simultaneously release these bugs from point $M$ and mark the time after which beetles reach their homes. Find on the plane such a point $M$, where is the last time to reach the house a bug would be minimal.

2001 Tuymaada Olympiad, 8

Tags: speed , game strategy , algebra , 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.

May Olympiad L2 - geometry, 2018.4

Tags: distance , parallelogram , geometry

In a parallelogram $ABCD$, let $M$ be the point on the $BC$ side such that $MC = 2BM$ and let $N$ be the point of side $CD$ such that $NC = 2DN$. If the distance from point $B$ to the line $AM$ is $3$, calculate the distance from point $N$ to the line $AM$.

1977 Vietnam National Olympiad, 6

Tags: 3-dimensional geometry , distance , geometry , maximum , minimum

The planes $p$ and $p'$ are parallel. A polygon $P$ on $p$ has $m$ sides and a polygon $P'$ on $p'$ has $n$ sides. Find the largest and smallest distances between a vertex of $P$ and a vertex of $P'$.

1948 Moscow Mathematical Olympiad, 142

Tags: distance , combinatorial geometry , combinatorics

Find all possible arrangements of $4$ points on a plane, so that the distance between each pair of points is equal to either $a$ or $b$. For what ratios of $a : b$ are such arrangements possible?

1990 Romania Team Selection Test, 9

Tags: distance , geometry , combinatorial geometry

The distance between any two of six given points in the plane is at least $1$. Prove that the distance between some two points is at least $\sqrt{\frac{5+\sqrt5}{2}}$

2021 Sharygin Geometry Olympiad, 8.4

Tags: circumcenter , distance , geometry

Let $A_1$ and $C_1$ be the feet of altitudes $AH$ and $CH$ of an acute-angled triangle $ABC$. Points $A_2$ and $C_2$ are the reflections of $A_1$ and $C_1$ about $AC$. Prove that the distance between the circumcenters of triangles $C_2HA_1$ and $C_1HA_2$ equals $AC$.

2019 Canada National Olympiad, 1

Tags: circumcenter , geometry , distance

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