Problems
Which of the two following numbers is larger: \(31^{11}\) or \(17^{14}\)?
There are 2026 people at a big party. Starting one minute after midnight, the host begins sending people home in a strange way.
After 1 minute, anyone with no friends at the party leaves. After 2 minutes, anyone with exactly one friend left in the party leaves. After 3 minutes, anyone with exactly two friends left leaves, and so on.
Show that there will always be at least two people who have left at some point the party after \(2026\) minutes have passed.
Zippity the robot speaks a language of \(n\) words which can be written with \(0\)s and \(1\)s. In this language, no word appears as the first several digits of another word. For example: if “\(1001\)” is a word, then “\(100101\)” can’t be a word. Show that if \(\ell_1,\cdots, \ell_n\) are the lengths of each word (i.e: the number of digits), then \[\frac{1}{2^{\ell_1}}+\frac{1}{2^{\ell_2}}+\cdots + \frac{1}{2^{\ell_n}}\leq 1.\]
You have a \(5\)-liter bucket and a \(3\)-liter bucket, along with an unlimited supply of water. You are allowed to transfer the contents of one bucket into the other. Find two different ways to end up with exactly \(4\) liters of water.
Sometimes geometric problems are solved using a variety of plane transformations. One of such transformations is called Homothety. It requires the following data:
The center of homothety: a point on the plane, we will usually denote it as \(O\).
The koefficient of homothety: a real number, we will denote it as \(k\).
The plane transformation is described in the following way: every point \(X\) on the plane is sent to a point \(X'\) with the following properties: The points \(O, X, X'\) lie on one line, in that exact order for a coefficient \(k \geq 1\).
The distance \(OX' = kOX\).
Prove that under the homothety transformation, a circle is transferred into a circle. Consider all possible cases of \(k\): \(k<0, 0<k \leq 1, 1\leq k\).
Two circles with centres \(A\) and \(C\) are tangent at the point \(B\). The segment \(DE\) passes through the point \(B\). Prove that the tangent lines passing through the points \(D\) and \(E\) are parallel.
