November Problems

We hope your final exams are going great! For those who need a break, or still want to solve maths problems for fun, here are the November Problems. Also, you can find the problems below. Any queries, please comment below or send us an email (nzmosa@outlook.com).

Have fun with the problems, and don't let the exams make you stressed!

JUNIOR DIVISION

Q1: Each turn of the game, Bob shuffles a deck of cards and removes one card. If there are \(500\) cards to start with, including one and only one red card, what is the probability that the red card is removed on the \(10\)th turn?
Q2: Candle \(S\) burns for \(9\) hours before it's completely gone. Candle \(T\) burns for \(6\) hours. After \(2\) hours both candles are the same length. How many times longer than candle \(T\) is candle \(S\) ?

INTERMEDIATE DIVISION

Q3: Let \(A\) be the area of the triangle with side lengths \(5,5,6\) and let \(B\) be the area of the triangle with side lengths of \(5,5,8\). Find the value of \(A-B\).
Q4: Let \(n\) ≥ \(2\) be a whole number. Suppose that \(a,b\) are whole numbers which leave a remained of \(1\) when divided by \(n\). Prove that their product \(ab\) also leaves a remainder of \(1\) when divided by \(n\).

SENIOR DIVISION

Q5: In triangle \(ABC\), \(4 sinA + 5 cosB = 6\) and \(5 sinB + 4 cosA = 5\). Given that the ∠ \(ACB\) is acute, find its value.
Hint: Square the two equations
Q6: (a) Suppose \(A_{n}\) is a set with \(n\) distinct elements. What is the total number of subsets of \(A_{n}\)?
(b) Let \((x_{n})_{n∈ N}\) be a sequence of natural numbers. Prove that if \(x_{n+1} - x_{n} = 2^{n}\) and \(x_{0}=1\), then \(x_{n}\) represents the number of subsets of \(A_{n}\).

(Update: We are extremely sorry about the mistake in Q6; the fact that \(x_{0}=1\) is very important in the question and we forgot this. This problem is now fixed.)