United States of America Mathematical Talent Search - Problem 1/1/24

1/1/24. Several children were playing in the ugly tree when suddenly they all fell.
• Roger hit branches Α, Β and Γ in that order on the way down.
• Sue hit branches Δ, E, and Φ in that order on the way down.
• Gillian hit branches Κ, A, and Γ in that order on the way down.
• Marcellus hit branches B, Δ, and H in that order on the way down.
• Juan-Phillipe hit branches I, Γ, and E in that order on the way down.
Poor Mikey hit every branch on the way down. Given only this information, in how many different orders could he have hit the 9 branches on his way down?

This is one of those beautiful combinatorial problems, that require a full and analytical solution through logic. No formula could give one a full solution. I solved this problem together with Demetres some day in July through Skype. Here's our solution.

From the hypothesis we can say that:

The possible configurations including Δ are:

The possible configurations including I are:

The possible configurations including Δ and H are:

We can now multiply the possible configurations including Δ and H with the first three possible configurations including I. From this we get a total of 21 possible configurations. For the last possible configuration including I we must take two cases: I will either be before or after Δ. Working similarly, we get 12 possible configurations.

In conclusion, the total amount of possible configurations is 21+12=33.

Junior International Mathematical Olympiad 1999 Problem

I found this problem in a book that states it was set as a problem in the 3rd Junior International Mathematical Olympiad held in Hong-Kong in 1999. The hypothesis is as follows:

A furniture shop has sold 225 beds during the year 1998. At first, it sold 25 beds per month, then 16 beds per month, and finally 20 beds per month. For how many months has it been selling 25 beds per month?

I really like this problem. It is quite trivial yet may look a bit difficult at first. Here's my solution.

Based on the hypothesis, considering that:

    x is the amount of months for which the shop had been selling 25 beds per month;
    y is the amount of months for which the shop had been selling 16 beds per month;
    z is the amount of months for which the shop had been selling 20 beds per month;

we can create the following system of equations:

which can be written as:

which is equivalent to the following system:

which has (x,y,z)=(1,5,6) as a solution in the set of natural numbers.

 In conclusion, the amount of months for which the shop had been selling 25 beds per month was equal to 1.

"Balkan Mathematical Olympiad 2009 - Problem 1"

Hello guys. From now on I will be joining this blog as a moderator. My name is Demetres and I am 14 years old and I am hoping to make you love Maths. Let's start with a Number Theory problem, which uses a technique named modular method.