Found problems: 180
2012 Oral Moscow Geometry Olympiad, 5
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.
2020 May Olympiad, 3
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).
1976 Vietnam National Olympiad, 5
$L, L'$ are two skew lines in space and $p$ is a plane not containing either line. $M$ is a variable line parallel to $p$ which meets $L$ at $X$ and $L'$ at $Y$. Find the position of $M$ which minimises the distance $XY$. $L''$ is another fixed line. Find the line $M$ which is also perpendicular to $L''$ .
1949-56 Chisinau City MO, 60
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.
2018 Bundeswettbewerb Mathematik, 4
We are given six points in space with distinct distances, no three of them collinear. Consider all triangles with vertices among these points.
Show that among these triangles there is one such that its longest side is the shortest side in one of the other triangles.
Geometry Mathley 2011-12, 6.1
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
2014 Romania National Olympiad, 2
Let $ABCDA'B'C'D'$ be a cube with side $AB = a$. Consider points $E \in (AB)$ and $F \in (BC)$ such that $AE + CF = EF$.
a) Determine the measure the angle formed by the planes $(D'DE)$ and $(D'DF)$.
b) Calculate the distance from $D'$ to the line $EF$.
1971 All Soviet Union Mathematical Olympiad, 147
Given an unit square and some circles inside. Radius of each circle is less than $0.001$, and there is no couple of points belonging to the different circles with the distance between them $0.001$ exactly. Prove that the area, covered by the circles is not greater than $0.34$.
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 $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$
1957 Moscow Mathematical Olympiad, 350
The distance between towns $A$ and $B$ is $999$ km.
At every kilometer of the road that connects $A$ and $B$ a sign shows the distances to $A$ and $B$ as follows:
$\fbox{0-999}$ , $\fbox{1-998}$ ,$\fbox{2-997}$ , $ . . . $ , $\fbox{998-1}$ , $\fbox{999-0}$
How many signs are there, with both distances written with the help of only two distinct digits?
1971 IMO Shortlist, 7
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 $2k=\angle BAC+\angle CAD+\angle DAB$.
1990 All Soviet Union Mathematical Olympiad, 532
If every altitude of a tetrahedron is at least $1$, show that the shortest distance between each pair of opposite edges is more than $2$.
1987 Tournament Of Towns, (145) 2
Α disk of radius $1$ is covered by seven identical disks. Prove that their radii are not less than $\frac12$ .
1995 Tuymaada Olympiad, 8
Inside the triangle $ABC$ a point $M$ is given . Find the points $P,Q$ and $R$ lying on the sides $AB,BC$ and $AC$ respectively and such so that the sum $MP+PQ+QR+RM$ is the smallest.
May Olympiad L2 - geometry, 1995.4
Consider a pyramid whose base is an equilateral triangle $BCD$ and whose other faces are triangles isosceles, right at the common vertex $A$. An ant leaves the vertex $B$ arrives at a point $P$ of the $CD$ edge, from there goes to a point $Q$ of the edge $AC$ and returns to point $B$. If the path you made is minimal, how much is the angle $PQA$ ?
2000 Austrian-Polish Competition, 2
In a unit cube, $CG$ is the edge perpendicular to the face $ABCD$. Let $O_1$ be the incircle of square $ABCD$ and $O_2$ be the circumcircle of triangle $BDG$. Determine min$\{XY|X\in O_1,Y\in O_2\}$.
1998 Mexico National Olympiad, 6
A plane in space is equidistant from a set of points if its distances from the points in the set are equal. What is the largest possible number of equidistant planes from five points, no four of which are coplanar?
2013 Sharygin Geometry Olympiad, 2
Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ meet at points $A$ and $B$. Points $C$ and $D$ on $\omega_1$ and $\omega_2$, respectively, lie on the opposite sides of the line $AB$ and are equidistant from this line. Prove that $C$ and $D$ are equidistant from the midpoint of $O_1O_2$.
2011 Bundeswettbewerb Mathematik, 4
Let $ABCD$ be a tetrahedron that is not degenerate and not necessarily regular, where sides $AD$ and $BC$ have the same length $a$, sides $BD$ and $AC$ have the same length $b$, side $AB$ has length $c_1$ and the side $CD$ has length $c_2$. There is a point $P$ for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities $a, b, c_1$ and $c_2$.
2011 Denmark MO - Mohr Contest, 2
In the octagon below all sides have the length $1$ and all angles are equal.
Determine the distance between the corners $A$ and $B$.
[img]https://1.bp.blogspot.com/-i6TAFDvcQ8w/XzXCRhnV_kI/AAAAAAAAMVw/rKrQMfPYYJIaCwl8hhdVHdqO4fIn8O7cwCLcBGAsYHQ/s0/2011%2BMogh%2Bp2.png[/img]
1993 IMO Shortlist, 2
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 $i.$
2008 Junior Balkan Team Selection Tests - Moldova, 7
In an acute triangle $ABC$, points $A_1, B_1, C_1$ are the midpoints of the sides $BC, AC, AB$, respectively. It is known that $AA_1 = d(A_1, AB) + d(A_1, AC)$, $BB1 = d(B_1, AB) + d(A_1, BC)$, $CC_1 = d(C_1, AC) + d(C_1, BC)$, where $d(X, Y Z)$ denotes the distance from point $X$ to the line $YZ$. Prove, that triangle $ABC$ is equilateral.
2019 Hanoi Open Mathematics Competitions, 13
Find all points inside a given equilateral triangle such that the distances from it to three sides of the given triangle are the side lengths of a triangle.
2018 Romania Team Selection Tests, 2
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 Swedish Mathematical Competition, 5
A semicircular arc and a diameter $AB$ with a length of $2$ are given. Let $O$ be the midpoint of the diameter. On the radius perpendicular to the diameter, we select a point $P$ at the distance $d$ from the midpoint of the diameter $O$, $0 <d <1$. A line through $A$ and $P$ intersects the semicircle at point $C$. Through point $P$ we draw another line at right angle against $AC$ that intersects the semicircle at point $D$. Through point $C$ we draw a line $l_1$, parallel to $PD$ and then a line $l_2$, through $D$ parallel to $PC$. The lines $l_1$ and $l_2$ intersect at point $E$. Show that the distance between $O$ and $E$ is equal to $\sqrt{2- d^2}$