Found problems: 180
Let $S$ be a point outside of the plane of the parallelogram $ABCD$, such that the triangles $SAB$, $SBC$, $SCD$ and $SAD$ are equivalent.
a) Prove that $ABCD$ is a rhombus.
b) If the distance from $S$ to the plane $(A, B, C, D)$ is $12$, $BD = 30$ and $AC = 40$, compute the distance from the projection of the point $S$ on the plane $(A, B, C, D)$ to the plane $(S,B,C)$ .
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.
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 $ABC$.
$A, B, C, D$ 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]
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.
$P$ is a point inside the triangle $ABC$. The perpendicular distances from $P$ to the three sides have product $p$. Show that $p \le \frac{ 8 S^3}{27abc}$, where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.
A regular pentagon of side length $1$ is given. Determine the smallest $r$ for which the pentagon can be covered by five discs of radius $r$ and justify your answer.
Given a square grid where the distance between two adjacent grid points is $1$. Can the distance between two grid points be $\sqrt5, \sqrt6, \sqrt7$ or $\sqrt{2007}$ ?
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?
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
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.
In a village there are only $10$ houses, arranged in a circle of a radius $r$ meters. Each has is the same distance from each of the two closest houses. Every year on Sunday of Pascoa, the village priest makes the Easter visit, leaving the parish house (point $A$) and following the path described in Figure 1. This year the priest decided to take the path represented in the Figure 2. Prove that this year the priest will walk another $10r$ meters.
[img]https://cdn.artofproblemsolving.com/attachments/a/9/a6315f4a63f28741ca6fbc75c19a421eb1da06.png[/img]
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 $c_2$.
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 $A$?
Plane $A$ passes through the points $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$. Plane $B$ is parallel to plane $A$, but passes through the point $(1, 0, 1)$. Find the distance between planes $A$ and $B$.
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 $PQ$.
Given two points inside a convex dodecagon (twelve sides) situated $10$ cm far from each other. Prove that the difference between the sum of distances, from the point to all the vertices, is less than $1$ m for those points.
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
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).
The distance between any two of six given points in the plane is at least $1$. Prove that the distance between some two points is at least $\sqrt{\frac{5+\sqrt5}{2}}$
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 $S = \{a, a+d, a+2d, a+3d, ..., a+kd, ...\}$.
Given five points inside an equilateral triangle of side length $2$, show that there are two points whose distance from each other is at most $ 1$.
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 $p$.
A treasure is buried at some point on a round island with a radius of $1$ km. On the coast of the island there is a mathematician with a device that indicates the direction to the treasure when the distance to the treasure does not exceed $500$ m. In addition, the mathematician has a map of the island, on which he can record all his movements, perform measurements and geometric constructions. The mathematician claims that he has an algorithm for how to get to the treasure after walking less than $4$ km. Could this be true?
(B. Frenkin)
The distance between the midpoints of the opposite sides of a convex quadrilateral is equal to a half sum of lengths of the other two sides. Prove that the first pair of sides is parallel.