Parallelogram 17 • Level 5 • 28 Dec 2023Chrismaths - Part 4

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

a portmanteau word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.

It’s the end of Christmaths, so here is your last instalment of your Christmaths maths challenge paper!

Simon.

PS: I want to say thank you to the UK Mathematics Trust, who own the copyright to these questions.

1.

3 marks

The sum of two numbers is 2. The difference between them is 4. What is their product?

-8
-3
0
3
8
(Not answered)

Let the numbers be x and y. Then x+y=2; x−y=4. Adding these equations gives 2x=6, so x=3. Hence y=2−3=−1 so the numbers are 3 and -1.

2.

4 marks

What is the mean of 12, 13, 14 and 16?

15
115
512
724
516
(Not answered)

The four fractions total 54, so their mean is 54÷4=516.

3.

4 marks

Only one of these triangles can actually be made.

Which is it?

A
B
C
D
E
(Not answered)

As triangle A has two equal sides, it should have two equal angles, but its angles are 25°, 110° and 45° so it is impossible to make.

In a triangle, the smallest angle lies opposite the shortest side which makes B impossible since 20° is the smallest angle, but 5 cm is not the shortest side.

Triangle C is impossible as 42+72≠82, so it does not obey Pythagoras' Theorem.

The longest side of a triangle must be shorter than the sum of the other two sides, but this is not the case in triangle E, so it cannot be made either.

Triangle D, however, is obtained by cutting an equilateral triangle of side 6cm in half along an axis of symmetry and so can certainly be made.

4.

4 marks

What is the largest power of 2 that divides 1272−1?

21
27
28
263
2127
(Not answered)

Using the formula for the difference of two squares:

1272−1=1272−12=127+1127−1=128×126=27×2×63=28×32×7.

5.

5 marks

The diagram shows a square with two lines from a corner to the middle of an opposite side.

The rectangle fits exactly inside these two lines and the square itself.

What fraction of the square is occupied by the shaded rectangle?

A) 13
B) 25
C) 310
D) 12
E) 38
(Not answered)

Triangles ABG, GFE and EFA are similar, so AF:FE=EF:FG=GB:BA=1:2.

Thus if AF=a, then FE=2a, FG=4a, and the shaded area is 8a2.

By Pythagoras' Theorem, AE=5a, so AD=25a and the area of the square is 20a2.

Thus the required fraction is 820, which is 25.

Alternatively, as we have shown that FG=4AF, we can divide parallelogram AGCE into 10 congruent triangles, 8 of which make up rectangle EFGH.

So the area of the rectangle is 45 of the area of the parallelogram, which in turn is half the area of square ABCD.

Before you hit the SUBMIT button, here are some quick reminders:

You will receive your score immediately, and collect your reward points.
You might earn a new badge... if not, then maybe next week.
Make sure you go through the solution sheet – it is massively important.
A score of less than 50% is ok – it means you can learn lots from your mistakes.
The next Parallelogram is next week, at 3pm on Thursday.
Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.

Cheerio, Simon.