Olympiad Pls Help

Question

5A

Charlotte buys two biscuits and a carton of milk for $5.00.

Oliver buys three biscuits and a carton of milk for $6.50.

How much more does a carton of milk cost than a single biscuit?

5B

27 ÷ N has a remainder of 3.

How different positive integers could N be?

5C

Some identical cubes are arranged in a pyramid-like form with 5 layers. Starting from the bottom layer, the number of cubes in the layers are 5 x 5, then 4 x 4, then 3 x 3, then 2 x 2, then 1 x 1. The whole exterior of the pyramid is painted including the bottom. How many of the cubes will have none of their faces painted?

5D

Chloe notices that 91 = 100 - 9 = (10 + 3) x (10 - 3) = 13 x 7.

Since 7 and 13 are primes, 91 has two prime factors.

Mia notices that 391 = 400 - 9 = (20 + 3) x (20 - 3) = 23 x 17.

Since 17 and 23 are primes, 391 has two prime factors.

Using the same method, Ruby finds prime factors of 1591.

Find the sum of Ruby's prime factors.

5E

There are 5 different coloured cups in a row.

1. The green cup is to the left of the red cup (but not necessarily next to it).
2. The yellow cup is to the left of the purple cup (but not necessarily next to it).
3. The blue cups is on one end of the row of cups.

For all these 3 statements to be true, how many different ways could these 5 cups be arranged?

Answer 1

5A) B=price of 1 biscuit, M=price of carton of milk:

2B+M=5, 3B+M=6.50, so B=6.50-5=$1.50, therefore M=5-3=$2. M-B=$0.50 or 50c more than a biscuit.

5B) 27/N has a remainder of 3 is another way of saying qN+3=27, where q is an integer, qN=24. The factors of 24 bigger than 3 but less than 24 are 4,6,8,12. So there are 4 possible integers for N.

5C) We need only consider the third and fourth layers because, in the other three layers the cubes will be exposed and so painted on at least one face. In the third layer, the central cube is completely hidden, and in the fourth layer 4 cubes are hidden. Therefore 5 cubes will have no painted faces.

5D) 1591=1600-9=(40+3)(40-3)=43×37, the sum of these prime factors is 43+37=80.

5E) The cups can be represented by letters (initial letters of the colours) which represent numbers in relationships. "To the left" means "less than", and "to the right" means "greater than".

So G<R, Y<P, and there are two conditions for B: either (1) B<G, P, R, Y; or (2) B>G, P, R, Y.

Let's look at (1):

B<G<R and B<Y<P, so we know B is the smallest. There are 24 arrangements for the remaining letters but only 6 of these satisfying the other conditions: BGRYP, BGYPR, BGYRP, BYGPR, BYGRP, BYPGR.

For (2) G<R<B and Y<P<B, we therefore have GRYPB, GYPRB, GYRPB, YGPRB, YGRPB, YPGRB, where B is the largest.

So in all there are 12 arrangements.

Answer 2

For question 5A the increase of one biscuit is equal to $1.50 - therefore one biscuit is $1.50...

The questions asks the difference in the price of the milk and the biscuit. We can simply use the first statement to solve this question. 2B + 1M = $5.00. We multiply the biscuits by 2 and then whatever remaining price is will be the milk carton price ($2.00). This means the answer must be $0.50.