Olympiad problems

1971 IMO, 1

All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$. [b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length; [b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.

2010 District Olympiad, 3

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

Consider the cube $ABCDA'B'C'D'$. The bisectors of the angles $\angle A' C'A$ and $\angle A' AC'$ intersect $AA'$ and $A'C$ in the points $P$, respectively $S$. The point $M$ is the foot of the perpendicular from $A'$ on $CP$ , and $N$ is the foot of the perpendicular from $A'$ to $AS$. Point $O$ is the center of the face $ABB'A'$ a) Prove that the planes $(MNO)$ and $(AC'B)$ are parallel. b) Calculate the distance between these planes, knowing that $AB = 1$.

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 $AC$.

2013 Sharygin Geometry Olympiad, 8

Tags: geometry , distance , circles

Three cyclists ride along a circular road with radius $1$ km counterclockwise. Their velocities are constant and different. Does there necessarily exist (in a sufficiently long time) a moment when all the three distances between cyclists are greater than $1$ km? by V. Protasov

2014 Junior Balkan Team Selection Tests - Romania, 3

Tags: point , combinatorial geometry , combinatorics , distance

Consider six points in the interior of a square of side length $3$. Prove that among the six points, there are two whose distance is less than $2$.

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.

2006 German National Olympiad, 4

Tags: triangle , distance , segment , geometry

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 $|EC| -|ED| \geq 1.$

1986 Spain Mathematical Olympiad, 1

Tags: distance , algebra , set

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 $202/100$ .

1954 Putnam, A2

Tags: square , distance

Consider any five points in the interior of square $S$ of side length $1$. Prove that at least one of the distances between these points is less than $\sqrt{2} \slash 2.$ Can this constant be replaced by a smaller number?

1987 All Soviet Union Mathematical Olympiad, 454

Tags: circumcenter , angle bisector , geometry , circumcircle , distance

Vertex $B$ of the $\angle ABC$ lies out the circle, and the $[BA)$ and $[BC)$ beams intersect it. Point $K$ belongs to the intersection of the $[BA)$ beam and the circumference. Chord $KP$ is orthogonal to the angle bisector of $\angle ABC$ . Line $(KP)$ intersects the beam $BC$ in the point $M$. Prove that the segment $[PM]$ is twice as long as the distance from the circle centre to the angle bisector of $\angle ABC$ .

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]

1994 Tuymaada Olympiad, 6

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

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.

2019 Tuymaada Olympiad, 2

Tags: geometry , trapezoid , circumcenter , midpoint , 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 $D$.

1950 Moscow Mathematical Olympiad, 177

Tags: algebra , combinatorics , distance , time

In a country, one can get from some point $A$ to any other point either by walking, or by calling a cab, waiting for it, and then being driven. Every citizen always chooses the method of transportation that requires the least time. It turns out that the distances and the traveling times are as follows: $1$ km takes $10$ min, $2$ km takes $15$ min, $3$ km takes $17.5 $ min. We assume that the speeds of the pedestrian and the cab, and the time spent waiting for cabs, are all constants. How long does it take to reach a point which is $6$ km from $A$?

1989 All Soviet Union Mathematical Olympiad, 500

Tags: geometry , combinatorics , distance

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

1947 Moscow Mathematical Olympiad, 132

Tags: geometry , locus , equidistant , distance

Given line $AB$ and point $M$. Find all lines in space passing through $M$ at distance $d$.

2005 Sharygin Geometry Olympiad, 11.3

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

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 $ABCD$.

1971 IMO Longlists, 28

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

All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$. [b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length; [b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.

1997 Nordic, 3

Tags: ratio , distance , geometry , point set

Let $A, B, C$, and $D$ be four different points in the plane. Three of the line segments $AB, AC, AD, BC, BD$, and $CD$ have length $a$. The other three have length $b$, where $b > a$. Determine all possible values of the quotient $\frac{b}{a}$. .

2011 BAMO, 3

Tags: arithmetic sequence , set , 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 $S = \{a, a+d, a+2d, a+3d, ..., a+kd, ...\}$.

1969 IMO Shortlist, 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 $r.$

2008 Postal Coaching, 2

Tags: geometry , perpendicular , distance , geometric inequalities

Let $ABC$ be a triangle, $AD$ be the altitude from $A$ on to $BC$. Draw perpendiculars $DD_1$ and $DD_2$ from $D$ on to $AB$ and $AC$ respectively and let $p(A)$ be the length of the segment $D_1D_2$. Similarly define $p(B)$ and $p(C)$. Prove that $\frac{p(A)p(B)p(C)}{s^3}\le \frac18$ , where s is the semi-perimeter of the triangle $ABC$.

2018 India PRMO, 7

Tags: geometry , hexagon , distance , circumradius

A point $P$ in the interior of a regular hexagon is at distances $8,8,16$ units from three consecutive vertices of the hexagon, respectively. If $r$ is radius of the circumscribed circle of the hexagon, what is the integer closest to $r$?

1949-56 Chisinau City MO, 23

Tags: distance , geometry , angle

Inside the angle $ABC$ of $60^o$, point $O$ is selected, which is located at distances from the sides of the angle $a$ and $b$, respectively. Determine the distance from the top of the angle to this point.

2013 Oral Moscow Geometry Olympiad, 6

Tags: geometry , minimal , minimum , distance , circumcenter , circumcircle

Let $ABC$ be a triangle. On its sides $AB$ and $BC$ are fixed points $C_1$ and $A_1$, respectively. Find a point $ P$ on the circumscribed circle of triangle $ABC$ such that the distance between the centers of the circumscribed circles of the triangles $APC_1$ and $CPA_1$ is minimal.