Parallelogram 24 Year 10 26 Mar 2020Easter Challenges

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteaux 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 before the end of Easter holidays.
  • 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.

As it is the Easter holidays, you have a bit more time to complete this Parallelogram, which is why it is twice as long as a typical Parallelogram. It includes eight Intermediate Maths Challenge problems. The first four are tricky, but the second four are downright nasty, so give them some serious thought and see if you can solve them.

Because this Parallelogram is longer and tougher than a typical Parallelogram, we will be awarding TREBLE POINTS for this week's answers! So there's even more reason to give this one your best shot.

1.

3 marks

1.1 What is the value of 20032?

  • 4009
  • 400009
  • 401209
  • 4000009
  • 4012009

2000+32=20002+2×2000×3+32=4000000+12000+9=4012009.

2.

3 marks

2.1 Think of a number, double it, then add 3. Multiply your answer by 4 and from this take away 5. Now also take away the number you first thought of. No matter what your first number was, your answer will always be a multiple of:

  • 2
  • 3
  • 5
  • 7
  • 11

Let the number thought of be x. Then the final number is 42x+35x=7x+7=7x+1.

3.

3 marks

3.1 A number of years ago, a newspaper reported that Turkish football team Sarigol Municipality transferred four of its players in return for a fee of 225 sacks of cement, needed to repair their stadium. At the same rate of exchange, how many sacks of cement would the transfer fee for a full team of eleven players and one reserve?

  • 233
  • 450
  • 675
  • 900
  • 2700

The transfer fee for a total of 12 players would be three times that for 4 players.

4. Magic Hexagon

Take a look at this video by James Grime and answer the question below.

(If you have any problem seeing the video then just right-click on the video and open the video in a new window)

2 marks

4.1 How many types of magic hexagon are there?

  • 0
  • 1 (with 19 elements)
  • 2 (with 1 or 19 elements)
  • 3 (with 1, 2 or 3 elements)

5.

4 marks

5.1 Lines AB and CD are parallel and BC=BD.

Given that x is an acute angle not equal to 60°, how many other angles in this diagram are equal to x?

  • 1
  • 2
  • 3
  • 4
  • 5
Show Hint (–1 mark)
–1 mark

The answer is not 2.

  • Angle BCD = x (vertically opposite angles);
  • Angle BDC = angle BCD (base angles of isosceles triangle);
  • Angle ABC = angle BCD (alternate angles);
  • Angle EBD = angle BCD (alternate angles).

6.

4 marks

6.1 It has been estimated that the mass of insects caught by spiders in a year in the UK is equal to the mass of the human population of the UK. Assuming this population is 60 million and the average mass of a human is 70kg, what is the mass, in tonnes, of insects caught by spiders per year in the UK?

  • 4.2
  • 42
  • 4200
  • 420000
  • 4200000

60 million × 70 kg = 4200 million kg = 4.2 million tonnes, since 1 tonne = 1000 kg.

7. Liar Numbers

Here's another video by the great James Grime:

(If you have any problem seeing the video then just right-click on the video and open the video in a new window)

4 marks

7.1 Numbers that pass this prime number test, but which are not actually prime, are known as what?

  • Carmichael numbers
  • Vanmichael numbers
  • Articulatedlorrymichael numbers
  • Wagonmichael numbers
  • Busmichael numbers
4 marks

7.2 The third number that passes the prime test, but which is not a prime, is 1,729. It is known as what?

  • An uber number
  • A taxicab number
  • A minicab number
  • A chauffer number
  • A bus number

8.

4 marks

8.1 Each interior angle of a particular polygon is an obtuse angle which is a whole number of degrees. What is the greatest number of sides the polygon could have?

  • 90
  • 179
  • 180
  • 359
  • 360
Show Hint (–1 mark)
–1 mark

In order for the polygon to have the largest number of sides, the internal angle needs to be as large as possible, which is 179°, as the angle has to be a whole number and it cannot be 180° or larger.

As each interior angle of the polygon is a whole number of degrees, the same must apply to each exterior angle. The sum of the exterior angles of a polygon is 360° and so the greatest number of sides will be that of a 360-sided polygon in which each interior angle is 179°, thus making each exterior angle 1°.

9.

4 marks

9.1 After a year's training, Minnie Midriffe increased her average speed in the London Marathon by 25%.

By what percentage did her time decrease?

  • 50
  • 30
  • 20
  • 10
  • 5
Show Hint (–1 mark)
–1 mark

It helps to think about this problem in terms of fractions, rather than percentages. If the new speed is 5/4 of the original speed, then what is the new time?

Minnie's average speed this year is 54 of last year's value. Hence her time this year will be 45, ie 80% of last year's time.

10.

4 marks

10.1 The diagram shows three semicircles, each of radius one.

What is the size of the total shaded area?

  • π+2
  • 3
  • 32π+1
  • 4
  • 2π1
Show Hint (–2 mark)
–2 mark

Redraw the diagram as shown below and the problem becomes much easier.

The shaded area may be divided into a 2 × 1 rectangle plus a semicircle and two quarter circles, all of radius 1.

Hence the total area is that of the rectangle plus that of a circle of radius 1, ie 2 + π.

11.

4 marks

11.1 When dates are written using eight digits, eg 06 02 2020, the 20th February 2002 is a palindromic date, since 20 02 2002 has the ame digits int he same order when read in reverse.

The previous palindromic date and the next few all occur in the month of February.

What will be the next month other than February to have a palindromic date in it?

  • March
  • April
  • October
  • November
  • December

All palindromic dates this century will fall in February as they will bee of the form ab 02 20ba. The palindromic dates next century will fall in December as they will be of the form cd 12 21dc and the first of these will be the 10th December 2101, ie 10 12 2101.

12. Classic Countdown

Take a look at this classic number game from Countdown.

(If you have any problem seeing the video then just right-click on the video and open the video in a new window)

Embded https://www.youtube.com/watch?v=

Here is an explanation of what happened by Junaid Mubeen, Director of Education, at Whizz Education, who was also a Countdown octochamp.

(If you have any problem seeing the video then just right-click on the video and open the video in a new window)

If you want to practice some Countdown problems, then you can visit this interactive website.

13.

5 marks

13.1 What is the area of the pentagon shown?

  • 12abc
  • 12ba+c
  • 12ab+c
  • 12bca
  • 12ca+b

The area of the pentagon is that of a rectangle of length b and breadth a plus that of a triangle of base b and height ca, ie ab+12bca=12b2a+ca=12ba+c.

The area may also be considered to be the sum of the areas of two trapezia which have parallel sides a and c and whose heights have total length b.

14.

5 marks

14.1 Suppose that e, i, n and t represent different positive whole numbers, n+i+n+e=9, t+e+n=10 and i=1.

What is t?

  • 2
  • 3
  • 4
  • 5
  • 6

We are given that 2n+e=8and t+e+n=10. Subtracting the first equation from the second gives tn=2. As n cannot equal 1, the minimum value of t is 4 but this gives n=2, e=4 which is impossible. If t=5 then n=3 and e=2, which is allowed. If t>5 then n>3 and e is not a positive whole number, so 5 is the only possible value of t.

15.

5 marks

15.1 The width:height ratio of television screens is changing form the traditional 4:3 to the widescreen 16:9. If a traditional screen and a widescreen have the same area, then what is the ratio widescreen width:traditional width?

Assume that television screens are rectangles.

  • 2:3
  • 3:2
  • 3:2
  • 4:3
  • none of these
Show Hint (–2 mark)
–2 mark

Let the widescreen width and traditional width be w and W respectively. Then the respective heights are 9w16 and 3W4.

As the areas are equal:

w×9w16=W×3W4, ie w2=43W2. Hence w : W = 2:3

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 after the Easter holidays.
  • 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.