I live in Ireland, and I recently participated in a tough training program
to get me ready for the Irish Mathematical Olympiad, which took place on May
7, 2005. In January of 2005, I was invited to take a qualification test in late
February to determine if I was, first of all, good enough to compete. The 1,000
people lucky enough to take this qualification test came from all over Ireland.
I passed the first test, while more than 900 people were sent home for the year.

From then on, I trained with the 15 or so pupils who lived near me at a nearby
university every Saturday morning.

Now that I've taken the Irish Mathematical Olympiad, I feel that I've learned
more over the past six months, than I have in my entire life! Even to get as
far as I have is an accomplishment as far as I see it.

I have not yet received the results of my test, but I can provide you with
a few questions from the test to let you see what sort of problems you'd be
facing:

**Question 1, Paper 1**

*Prove that 2005 to the power of 2005 is a sum of two perfect squares, but
not the sum of two perfect cubes.*

**Question 3, Paper 1**

*Prove that the sum of the lengths of the medians of a triangle is at least
three quarters of the sum of the lengths of the sides.*

**Question 2, Paper 2**

*Using only the digits 1, 2, 3, 4, and 5, two players A, B compose a 2005-digit
number N by selecting one digit at a time as follows: A selects the first digit,
B the second, A the third, and so on, in that order. The last to play wins if
and only if N is divisible by 9. Who will win if both players play as well as
possible?*

Those are the only questions I can give you because, sadly, they were the
only ones I was able to complete. I wasn't even able to prove the second part
of the first question. If you have a solution to any of these questions, feel
free to Y-mail it to me, or post it in the BBS. I'll award **5,000 clams**
for each question to the citizen who can provide the correct answer. Feel free
to ask a parent or teacher for help.

By the way, if you place in the top six for the Mathematical Olympiad of your
country, you'll be sent to participate in the International Mathematical Olympiad,
a math test against the other countries.

Try to find a location near you or ask your school about the Mathematical
Olympiads. See if you can start the journey to being the best!

Good luck to anyone willing to participate. I hope you do better than I will.
I'll be sure to let everyone know if I made the team. Best of luck!

This is Deriko, signing off.