Olympiad problems

Novosibirsk Oral Geo Oly VII, 2019.2

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]

2007 Sharygin Geometry Olympiad, 11

Tags: ratio , distance , geometry

A boy and his father are standing on a seashore. If the boy stands on his tiptoes, his eyes are at a height of $1$ m above sea-level, and if he seats on father’s shoulders, they are at a height of $2$ m. What is the ratio of distances visible for him in two eases? (Find the answer to $0,1$, assuming that the radius of Earth equals $6000$ km.)

1987 Tournament Of Towns, (143) 4

Tags: chessboard , sum , distance , geometry , algebra

On a chessboard a square is chosen . The sum of the squares of distances from its centre to the centre of all black squares is designated by $a$ and to the centre of all white squares by $b$. Prove that $a = b$. (A. Andj ans, Riga)

2017 Hanoi Open Mathematics Competitions, 13

Tags: inequalities , distance , geometric inequalities , geometry

Let $ABC$ be a triangle. For some $d>0$ let $P$ stand for a point inside the triangle such that $|AB| - |P B| \ge d$, and $|AC | - |P C | \ge d$. Is the following inequality true $|AM | - |P M | \ge d$, for any position of $M \in BC $?

2009 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , distance , geometric inequality

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 $XZ + Y Z \le 1$

May Olympiad L2 - geometry, 2018.4

Tags: geometry , parallelogram , distance

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 $AM$.

2011 Tournament of Towns, 1

Tags: point , combinatorial geometry , combinatorics , distance

Pete has marked several (three or more) points in the plane such that all distances between them are different. A pair of marked points $A,B$ will be called unusual if $A$ is the furthest marked point from $B$, and $B$ is the nearest marked point to $A$ (apart from $A$ itself). What is the largest possible number of unusual pairs that Pete can obtain?

1948 Moscow Mathematical Olympiad, 142

Tags: distance , combinatorics , combinatorial geometry

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?

1998 China Team Selection Test, 2

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

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 $S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}$.

2008 Balkan MO Shortlist, G8

Tags: geometric inequality , geometry , max , 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 $(AP, BP, CP) \ge \sqrt{d_a^2+d_b^2+d_c^2}$

2021 Sharygin Geometry Olympiad, 9.4

Tags: circumradius , distance , geometry , combinatorial geometry

Define the distance between two triangles to be the closest distance between two vertices, one from each triangle. Is it possible to draw five triangles in the plane such that for any two of them, their distance equals the sum of their circumradii?

2024 Israel TST, P3

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

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.

2020 Tuymaada Olympiad, 4

Tags: combinatorics , distance

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 $g(k) > 2k + 2020$.

1999 Mexico National Olympiad, 4

Tags: combinatorial geometry , distance , board , combinatorics

An $8 \times 8$ board is divided into unit squares. Ten of these squares have their centers marked. Prove that either there exist two marked points on the distance at most $\sqrt2$, or there is a point on the distance $1/2$ from the edge of the board.

2005 Sharygin Geometry Olympiad, 10.5

Tags: tangent , distance , circles , 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 $AA'$.

1999 Cono Sur Olympiad, 5

Tags: combinatorial geometry , square , distance , combinatorics

Give a square of side $1$. Show that for each finite set of points of the sides of the square you can find a vertex of the square with the following property: the arithmetic mean of the squares of the distances from this vertex to the points of the set is greater than or equal to $3/4$.

2018 India IMO Training Camp, 1

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

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.

2001 Kazakhstan National Olympiad, 8

Tags: combinatorial geometry , distance , geometry

There are $ n \geq4 $ points on the plane, the distance between any two of which is an integer. Prove that there are at least $ \frac {1} {6} $ distances, each of which is divisible by $3$.

2011 NZMOC Camp Selection Problems, 5

Tags: distance , square , geometry , collinear

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 Oral Moscow Geometry Olympiad, 4

Tags: geometry , orthocenter , circumcenter , distance

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)

1959 Putnam, B5

Tags: geometry , 3d geometry , sphere , distance

Find the equation of the smallest sphere which is tangent to both of the lines $$\begin{pmatrix} x\\y\\z \end{pmatrix} =\begin{pmatrix} t+1\\ 2t+4\\ -3t +5 \end{pmatrix},\;\;\;\begin{pmatrix} x\\y\\z \end{pmatrix} =\begin{pmatrix} 4t-12\\ -t+8\\ t+17 \end{pmatrix}.$$

2010 Thailand Mathematical Olympiad, 8

Tags: number theory , distance

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 $d(m, n) \le 36$.

1981 All Soviet Union Mathematical Olympiad, 324

Tags: distance , combinatorial geometry , geometry , rectangle

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.

1980 Bundeswettbewerb Mathematik, 2

Tags: geometry , distance , triangle , bisector

In a triangle $ABC$, the bisectors of angles $A$ and $B$ meet the opposite sides of the triangle at points $D$ and $E$, respectively. A point $P$ is arbitrarily chosen on the line $DE$. Prove that the distance of $P$ from line $AB$ equals the sum or the difference of the distances of $P$ from lines $AC$ and $BC$.