# September Problems

Finally, it's spring!!! Start a great season with awesome September Problems.

You can download the pdf here. The problems are also repeated below. If you have any questions or if you want clarification on the problems, feel free to email us, message us on Facebook, or leave a comment below.

Good luck, have fun, and send your solutions in to nzmosa@outlook.com! Remember to include your name, school and school year in your email!

(P.S: If you've just joined us, you can check out the rules about the competition.)

(P.P.S: We'll still accept August problem solutions! Submitting will put you in a pretty good position for a prize. We will announce winners as soon as we have enough people; June and July Problems prize winners will be released soon!)

## JUNIOR DIVISION

Q1: What is the smallest 5-digit number divisible by 15?

Q2: The All Whites are preparing for the penalty shootout. There are 11 players on the
pitch, and among them, the coach has to choose 1st, 2nd, 3rd, 4th and 5th kickers.
How many combinations are there?

## INTERMEDIATE DIVISION

Q3: 10 years ago, my sister's age was twice the square of my age then. 6 years ago, my
sister's age was twice my age then. How old am I, and how old is my sister now?

Q4:  (a) Prove that $$PC^{2}=PA\times PB$$
(b) Prove that $$PA\times PD=PB \times PC$$

## SENIOR DIVISION

Q5: (a) What is the maximum number of intersections when $$n$$ lines are drawn on a plane?
(b) What is the maximum number of areas when $$n$$ lines are drawn on a plane? (For example, the following diagram has 7 areas)
Q6:  Prove that $$(a^2 + b^2)(x^2 + y^2) \geq (ax + by)^2$$ for real numbers $$a, b, x$$ and $$y$$.