Found problems: 180
2009 Kyiv Mathematical Festival, 5
Assume that a triangle $ABC$ satisfies the following property:
For any point from the triangle, the sum of distances from $D$ to the lines $AB,BC$ and $CA$ is less than $1$.
Prove that the area of the triangle is less than or equal to $\frac{1}{\sqrt3}$
2000 Kazakhstan National Olympiad, 8
Given a triangle $ ABC $ and a point $ M $ inside it. Prove that $$
\min \{MA, MB, MC\} + MA + MB + MC <AB + BC + AC. $$
1947 Moscow Mathematical Olympiad, 132
Given line $AB$ and point $M$. Find all lines in space passing through $M$ at distance $d$.
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?
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.
2015 Abels Math Contest (Norwegian MO) Final, 3
The five sides of a regular pentagon are extended to lines $\ell_1, \ell_2, \ell_3, \ell_4$, and $\ell_5$.
Denote by $d_i$ the distance from a point $P$ to $\ell_i$.
For which point(s) in the interior of the pentagon is the product $d_1d_2d_3d_4d_5$ maximal?
1978 Vietnam National Olympiad, 6
Given a rectangular parallelepiped $ABCDA'B'C'D'$ with the bases $ABCD, A'B'C'D'$, the edges $AA',BB', CC',DD'$ and $AB = a,AD = b,AA' = c$. Show that there exists a triangle with the sides equal to the distances from $A,A',D$ to the diagonal $BD'$ of the parallelepiped. Denote those distances by $m_1,m_2,m_3$. Find the relationship between $a, b, c,m_1,m_2,m_3$.
2021 Israel TST, 2
Let $n>1$ be an integer. Hippo chooses a list of $n$ points in the plane $P_1, \dots, P_n$; some of these points may coincide, but not all of them can be identical. After this, Wombat picks a point from the list $X$ and measures the distances from it to the other $n-1$ points in the list. The average of the resulting $n-1$ numbers will be denoted $m(X)$.
Find all values of $n$ for which Hippo can prepare the list in such a way, that for any point $X$ Wombat may pick, he can point to a point $Y$ from the list such that $XY=m(X)$.
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]
2017 Sharygin Geometry Olympiad, P22
Let $P$ be an arbitrary point on the diagonal $AC$ of cyclic quadrilateral $ABCD$, and $PK, PL, PM, PN, PO$ be the perpendiculars from $P$ to $AB, BC, CD, DA, BD$ respectively. Prove that the distance from $P$ to $KN$ is equal to the distance from $O$ to $ML$.
2005 Abels Math Contest (Norwegian MO), 2a
In an aquarium there are nine small fish. The aquarium is cube shaped with a side length of two meters and is completely filled with water. Show that it is always possible to find two small fish with a distance of less than $\sqrt3$ meters.
1983 Bundeswettbewerb Mathematik, 4
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.
May Olympiad L2 - geometry, 2018.4
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 $AM$.
Geometry Mathley 2011-12, 4.1
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
1948 Moscow Mathematical Olympiad, 142
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?
2005 Sharygin Geometry Olympiad, 10.3
Two parallel chords $AB$ and $CD$ are drawn in a circle with center $O$.
Circles with diameters $AB$ and $CD$ intersect at point $P$.
Prove that the midpoint of the segment $OP$ is equidistant from lines $AB$ and $CD$.
2009 Postal Coaching, 5
Let $ABCD$ be a quadrilateral that has an incircle with centre $O$ and radius $r$. Let $P = AB \cap CD$, $Q = AD \cap BC$, $E = AC \cap BD$. Show that $OE \cdot d = r^2$, where $d$ is the distance of $O$ from $PQ$.
1986 Spain Mathematical Olympiad, 1
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 $202/100$ .
2013 Sharygin Geometry Olympiad, 8
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
Durer Math Competition CD Finals - geometry, 2019.C5
$A, B, C, D$ are four distinct points such that triangles $ABC$ and $CBD$ are both equilateral. Find as many circles as you can, which are equidistant from the four points. How can these circles be constructed?
[i]Remark: The distance between a point $P$ and a circle c is measured as follows: we join $P$ and the centre of the circle with a straight line, and measure how much we need to travel along thisline (starting from $P$) to hit the perimeter of the circle. If $P$ is an internal point of the circle, the distance is the length of the shorter such segment. The distance between a circle and itscentre is the radius of the circle.[/i]
1998 China Team Selection Test, 2
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 $S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}$.
2002 Moldova Team Selection Test, 4
Let $C$ be the circle with center $O(0,0)$ and radius $1$, and $A(1,0), B(0,1)$ be points on the circle. Distinct points $A_1,A_2, ....,A_{n-1}$ on $C$ divide the smaller arc $AB$ into $n$ equal parts ($n \ge 2$). If $P_i$ is the orthogonal projection of $A_i$ on $OA$ ($i =1, ... ,n-1$), find all values of $n$ such that $P_1A^{2p}_1 +P_2A^{2p}_2 +...+P_{n-1}A^{2p}_{n-1}$ is an integer for every positive integer $p$.
2010 Chile National Olympiad, 5
Consider a line $ \ell $ in the plane and let $ B_1, B_2, B_3 $ be different points in $ \ell$. Let $ A $ be a point that is not in $ \ell$. Show that there is $ P, Q $ in $ {B_1, B_2, B_3} $ with $ P \ne Q $ so that the distance from $ A $ to $ \ell$ is greater than the distance from $ P $ to the line that passes through $ A $ and $ Q $.
1949-56 Chisinau City MO, 23
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.
2008 Postal Coaching, 2
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 $ABC$.