Problems
Think about the symmetries of an equilateral triangle. Is applying rotation by \(120^{\circ}\), and then reflecting in the vertical median the same as applying these two symmetries the other way around?
Let \(n\ge3\) be a positive integer. A regular \(n\)-gon is a polygon with \(n\) sides where every side has the same length, and every angle is the same. For example, a regular \(3\)-gon is an equilateral triangle, and a regular \(4\)-gon is a square.
What symmetries does a regular \(n\)-gon have, and how many?
The set of symmetries of an object (e.g. a square) form an object called a group. We can formally define a group \(G\) as follows.
A is a non-empty set \(G\) with a binary operation \(*\) satisfying the following axioms (you can think of them as rules). A binary operation takes two elements of \(G\) and gives another element of \(G\).
Closure: For all \(g\) and \(h\) in \(G\), \(g*h\) is also in \(G\).
Identity: There is an element \(e\) of \(G\) such that \(e*g=g=g*e\) for all \(g\) in \(G\).
Associativity: For all \(g\), \(h\) and \(k\) in \(G\), \((g*h)*k=g*(h*k)\).
Inverses: For every \(g\) in \(G\), there exists a \(g^{-1}\) in \(G\) such that \(g*g^{-1}=e\).
Prove that the symmetries of the ‘reduce-reuse-recycle’ symbol form a group.
Consider the triangle \(BCD\), inscribed in a circle with center \(A\), the segments \(EF\), \(FG\), \(EG\) are tangent to the circle at the points \(C\), \(D\), \(B\) respectively. Prove that the Euler line of the triangle \(BCD\) passes through the center of the circle circumscribed around the triangle \(EFG\).
Show that there are infinitely many composite numbers \(n\) such that \(3^{n-1}-2^{n-1}\) is divisible by \(n\).
Show that there are infinitely many numbers \(n\) such that \(2^n+1\) is divisible by \(n\). Find all prime numbers, that satisfy this property.
If \(k>1\), show that \(k\) does not divide \(2^{k-1}+1\). Find all prime numbers \(p,q\) such that \(2^p+2^q\) is divisible by \(pq\).
Find all positive integers \((x,n)\) such that \(x^n + 2^n + 1\) is a divisor \(x^{n+1} + 2^{n+1} + 1\).
Show that if \(n\) is an integer, greater than \(1\), then \(n\) does not divide \(2^n-1\).
Find all the integers \(n\) such that \(1^n + 2^n + ... + (n-1)^n\) is divisible by \(n\).