Given a regular dodecahedron of side $a$. Take two pairs of opposite faces: $E, E'$ and $F, F'$. For the pair $E, E'$ take the line joining the centers of the faces and take points $A$ and $C$ on the line each a distance $m$ outside one of the faces. Similarly, take $B$ and $D$ on the line joining the centers of $F, F'$ each a distance $m$ outside one of the faces. Show that $ABCD$ is a rectangle and find the ratio of its side lengths.
$ABCD$ is any convex quadrilateral. Squares center $E, F, G, H$ are constructed on the outside of the edges $AB, BC, CD$ and $DA$ respectively. Show that $EG$ and $FH$ are equal and perpendicular.
$ABC$ is a triangle with $\angle A = 90^\circ$. For a point $D$ on the side $BC$, the feet of the perpendiculars to $AB$ and $AC$ are $E$ and $F$. For which point $D$ is $EF$ a minimum?
Each day 289 students are divided into 17 groups of 17. No two students are ever in the same group more than once. What is the largest number of days that this can be done?
An acute-angled triangle has unit area. Show that there is a point inside the triangle whose distance from each of the vertices is at least $\frac{2}{\sqrt[4]{27}}$.
Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place:
Alice: Are you going to cover your keys?
Bob: I would like to, but there are only 7 colours and I have 8 keys.
Alice: Yes, but you could always distinguish a key by noticing that the red key next to the green key was different from the red key next to the blue key.
Bob: You must be careful what you mean by "next to" or "three keys over from" since you can turn the key ring over and the keys are arranged in a circle.
Alice: Even so, you don't need 8 colours.
Problem: What is the smallest number of colours needed to distinguish $n$ keys if all the keys are to be covered.
Let $a, b, c$ and $d$ be odd integers such as $0\lt a\lt b\lt c\lt d$ and $a d = b c$. Show that if $a + d = 2^{k}$ and $b + c = 2^{m}$ for certain integers $k$ and $m$, then $a = 1$.
We have two different points in the plane, $A$ and $O$. For each point $X$ of the plane other than $O$, we denoted by $\alpha(X)$ the angle measurement between $OA$ and $OX$, in radians, and counted counterclockwise from $OA$ $(0 \leq \alpha(X)\lt 2\pi)$. Let $C(X)$ the center circumference $O$ and radius of length $OX + \frac{\alpha(X)}{OX}$. We have a finite number of colors and color each of the dots on the plane with them. Prove that there is a point $Y$ such that $\alpha(Y)\gt 0$ and such that its color appears on the circumference of $C(Y)$.
Let $d$ be the sum of the lengths of all diagonals of a flat convex polygon of $n$ vertexes $(n\gt 3)$, and let $p$ its perimeter. Demonstrate that $$
n-3\lt \frac{2d}{p}\lt \left[\frac{n}{2}\right]\left[\frac{n+1}{2}\right]-2
$$ being $[x]$ the whole part of $x$.
Find a pair of positive integers $a$ and $b$ such that 1) $a b(a+b)$ is not divisible by $7$ ; 2) $(a+b)^{7}-a^{7}-b^{7}$ is divisible by $7^{7}$. justify the answer.
Let $ABCD$ be a convex quadrilateral such that the line $CD$ is tangent to the circle of diameter $AB$. Prove that the line $AB$ is tangent to the circle of diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.
$ABC$ is a triangle and $P$ is any point. The lines $PA$, $PB$, $PC$ cut the circumcircle of $ABC$ again at $A'$, $B'$, $C'$ respectively. Show that there are at most eight points $P$ such that $A'B'C'$ is congruent to $ABC$.
Each weight in a set of $n$ has integral weight and the total weight of the set is $2n$. A balance is initially empty. We then place the weights onto a pan of the balance one at a time. Each time we place the heaviest weight not yet placed. If the pans balance, then we place the weight onto the left pan. Otherwise, we place the weight onto the lighter pan. Show that when all the weights have been placed, the scales will balance. [For example, if the weights are $2$, $2$, $1$, $1$. Then we must place $2$ in the left pan, followed by $2$ in the right pan, followed by $1$ in the left pan, followed by $1$ in the right pan.]
Find all pairs of digits $(b, c)$ such that the number $b \ldots b6c \ldots c4$, where there are $n$ $b$s and $n$ $c$s, is a square for all positive integers $n$.