Problems

In the month of January of a certain year, there are \(4\) Mondays and \(4\) Fridays. What day of the week is the \(20^{\text{th}}\) day of this month?

We wish to place the numbers \(1\) to \(10\) in the circles of the following picture, so that each circle contains exactly one number, in such a way that each line of three circles sums to the same number, can we do this?

We have a \(17\) digit number, and we form a new number by reading the original number from right to left. We add this new number to the original number. Show that this resulting sum will have at least one even digit.

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.

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.