Problems

On the Problemland Space Station, there are \(1000\) tonnes of air, of which \(99\%\) is oxygen. After an unfortunate asteroid impact, some of the air is vented into space. The hull is quickly repaired, and no further loss occurs.

Afterward, measurements reveal that oxygen now makes up only \(98\%\) of the remaining air, and that only oxygen was lost during the incident.

How many tonnes of oxygen remain on the space station?

Find all solutions of the following puzzle. Each animal represents a different number.

What is the smallest positive whole number whose digits add up to \(2026\)?

The thirteen dwarves sat down around a bonfire in a circle, and their leader Thorin proposed a challenge to pass the time:

Each dwarf would choose an integer so that for every group of three neighboring dwarves, the sum of their numbers must be exactly \(13\).

Could Thorin’s challenge be solved?

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 \[\frac{1}{2^{\ell_1}}+\frac{1}{2^{\ell_2}}+\cdots + \frac{1}{2^{\ell_n}}\leq 1.\]