Found problems: 180
Given a triangle $ ABC $ and a point $ M $ inside it. Prove that $$
\min \{MA, MB, MC\} + MA + MB + MC <AB + BC + AC. $$
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?
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 $2$.
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 $BC$.
Prove that there exists a set $S$ of $3^{1000}$ points in the plane such that for each point $P$ in $S$, there are at least $2000$ points in $S$ whose distance to $P$ is exactly $1$ inch.
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.
Let $P_1$, $P_2$, $P_3$, $P_4$ and $P_5$ be five different points which are inside $D$ or on the border of figure $D$. Let $M=min\left\{P_iP_j \mid i \neq j\right\}$ be minimal distance between different points $P_i$. For which configuration of points $P_i$, value $M$ is at maximum, if :
$a)$ $D$ is unit square
$b)$ $D$ is equilateral triangle with side equal $1$
$c)$ $D$ is unit circle, circle with radius $1$
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
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?
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.
$L, L'$ 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 $L''$ .
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 $3$.
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 $i.$
Consider the equilateral triangle $ABC$ with center of gravity $G$. Let $M$ be a point, inside the triangle and $O$ be the midpoint of the segment $MG$. Three segments go through $M$, each parallel to one side of the triangle and with the ends on the other two sides of the given triangle.
a) Show that $O$ is at equal distances from the midpoints of the three segments considered.
b) Show that the midpoints of the three segments are the vertices of an equilateral triangle.
The lengths of the sides of triangle $ABC$ are $4,5,6$. For any point $D$ on one of the sides, draw the perpendiculars $DP, DQ$ on the other two sides. What is the minimum value of $PQ$?
Consider two equilateral triangles $ABC$ and $MNP$ with the property that $AB \parallel MN, BC \parallel NP$ and $CA \parallel PM$ , so that the surfaces of the triangles intersect after a convex hexagon. The distances between the three pairs of parallel lines are at most equal to $1$. Show that at least one of the two triangles has the side at most equal to $\sqrt {3}$ .
The distances from the centroid $G$ of a triangle $ABC$ to its sides $a,b,c$ are denoted $g_a,g_b,g_c$ respectively. Let $r$ be the inradius of the triangle. Prove that:
a) $g_a,g_b,g_c \ge \frac{2}{3}r$
b) $g_a+g_b+g_c \ge 3r$
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 $(AP, BP, CP) \ge \sqrt{d_a^2+d_b^2+d_c^2}$
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 $\frac{1}{\sqrt3}$
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 $AM$.
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.
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.
In a village, the maximum distance between two houses is $M$ and the minimum distance is $m$. Prove that if the village has $6$ houses, then $\frac{M}{m} \ge \sqrt3$.
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 $.
Do there exist points $A, B, C, D$ in space, such that $AB = CD = 8, AC = BD = 10$, and $AD = BC = 13$?