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Parallelogram 29 • Level 4 • 21 Mar 2024Four brothers

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteau word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.
  • Tackle each Parallelogram in one go. Don’t get distracted.
  • Finish by midnight on Sunday if your whole class is doing parallelograms.
  • Your score & answer sheet will appear immediately after you hit SUBMIT.
  • Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1.

3 marks

1.1 Jane has 20 identical cards in the shape of an isosceles right-angled triangle. She uses the cards to make the five shapes below.

Which of the shapes has the shortest perimeter?

  • A
  • B
  • C
  • D
  • E
  • (Not answered)
Show Hint (–1 mark)
–1 mark

The isosceles triangle is also a right-angle triangle, so it has one long side, and two short sides. So one option is to measure the perimeter of each shape in terms of shorts and longs. So, shape (D) is 6 shorts.

Let the side lengths of the isosceles right-angled triangle be x, x and y as shown in the diagram.

By Pythagoras’ Theorem y2=x2+x2=2x2 and hence y=2x.

We then see that the perimeters of the shapes are as follows:

  • A: 42x
  • B: 4+22x
  • C: 4+22x
  • D: 6
  • E: 4+22x.

Now, as 1<2<1.5, we have 42<4×1.5=6=4+2<4+22. So shape A has the shortest perimeter.

[Note: that, as the value of x is irrelevant, we could have, for example, put x=1 to begin with.]

2.

3 marks

2.1 For which of the following numbers is the sum of all its factors not equal to a square number?

  • 3
  • 22
  • 40
  • 66
  • 70
  • (Not answered)

The most straightforward method here is to just list all the factors of the numbers, add them up, and see in which cases this sum is not a square number:

Since 90 is not a square, in the context of the IMC, you could stop here. However, for completeness, we also calculate the sum of the factors of the other two numbers we are given.

So, for each option given, other than 40, the sum of its factors is a square number.

3.

5 marks

3.1 The square ABCD has an area of 196.

It contains two overlapping squares; the larger of these squares has an area 4 times that of the smaller and the area of their overlap is 1.

What is the total area of the shaded region?

  • 44
  • 72
  • 80
  • 152
  • more information is needed.
  • (Not answered)
Show Hint (–1 mark)
–1 mark

We let P,Q,R,S,T and U be the points as labelled in the diagram.

Since the square ABCD has an area of 196, it has side length 14.

We suppose that the smaller square inside ABCD has side length x.

Since the larger square inside ABCD has area 4 times that of the smaller square it will have side length 2x.

We let P,Q,R,S,T and U be the points as labelled in the diagram. Since the square ABCD has area 196, it has side length 14. We suppose that the smaller square inside ABCD has side length x. Since the larger square inside ABCD has area 4 times that of the smaller square it will have side length 2x.

We note that the diagram is symmetric about the line AC. Hence the overlap of the squares is itself a square. Since this square has area 1, then it has side length 1. So QR has length 1. As UR=AP, UR has length x. Now QS has length 2x. Therefore US has length x+2x−1=3x−1. The length of US is the same as the side length of the square ABCD. Therefore 3x−1=14. Hence x=5.

So the larger square inside ABCD has side length 10 and hence area 100, and the smaller square has area 25. As their overlap has area 1, the total area they cover is 100 + 25 − 1 = 124. Therefore, the shaded area is 196 − 124 = 72.

4.

4 marks

4.1 Four brothers are discussing the order in which they were born. Two are lying and two are telling the truth. Which two are telling the truth?

Alfred: “Bernard is the youngest.”
Horatio: “Bernard is the oldest and I am the youngest.”
Inigo: “I was born last.”
Bernard: “I’m neither the youngest nor the oldest.”

  • Bernard and Inigo
  • Horatio and Bernard
  • Alfred and Horatio
  • Alfred and Bernard
  • Inigo and Horatio
  • (Not answered)
Show Hint (–1 mark)
–1 mark

Take a look at what Horatio says – does it fit with what any of the others say?

What Horatio says contradicts each of the statements made by his brothers. So he cannot be one of the two who are telling the truth.

What Alfred says contradicts both what Inigo says and what Bernard says. So the two who are telling the truth cannot include Alfred.

We deduce that the two brothers who are telling the truth are Bernard and Inigo. This is indeed possible. For example, if Alfred was born first, Horatio second, Bernard third and Inigo last, then just Bernard and Inigo are telling the truth.

5. 580-ton monster machine is building bridges across China

This machine is incredible. As you watch the video, look at when it moves the massive bridge sections forwards... what does it have to do before it can redistribute the weight?

(If you have problems watching the video, right click to open it in a new window)

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.