GROUP ROUND - Further Mathematicsfurthermaths.org.uk/files/group-round-final.pdf · Below is a...

GROUP ROUND

INSTRUCTIONS

Your team will have 40 minutes to answer 10 questions. Each

team will have the same questions.

Each question is worth 6 points. However, some questions are

easier than others!

You will have to decide your team’s strategy for this group

competition. Do you split up so that individuals work on a

few questions each or do you work in pairs on a greater

number of questions? Working all together on all the

questions may well take too long. You decide!

There is only one answer sheet per team. Five minutes before

the end of the time you will be told to finalise your answers

and write them on the answer sheet. This answer sheet is the

only thing that will be marked.

Senior Team Maths Challenge Regional Final 2013

Group Round Supported by Rolls-Royce plc

Question 1

1 2 3 4 …………. 100

Lucy has a long straight path in her garden that is made from

100 paving slabs. Standing on slab 1 and whilst always facing

forwards, she repeatedly takes two steps forwards and then one

step back. Each step takes her from the slab on which she is

standing to the next slab, either in front or behind. So her first

three steps are from slab 1 to slab 2, from slab 2 to slab 3 and

then from slab 3 back to slab 2.

Following this pattern, how many steps has Lucy taken when

she steps onto slab 100 for the first time?



Question 2

The recurrence relation connecting terms of a sequence 0a , 1a , 2a ,…

is 11 nnn aaa , 1n .

Also, 50 a and 010 aa .

What is the value of 1a ?



Question 3

The sum of the squares of five consecutive positive integers is

9255. What is the value of the square of the next integer?



Question 4

In triangle PQS, PQ = 6 cm, PS = 8 cm and QS = 10 cm.

An additional straight line is drawn from P to the point R on QS,

such that the area of triangle PRS is 8 cm2.

What is the length of RS ?

Not to

scale

P

6 cm 8 cm

Q S R

10 cm



Question 5

When Charlie took his car for a service last week he noticed that

his odometer, which displays the cumulative mileage of the car,

showed 54321 miles. A year ago his odometer read 43125.

Excluding 43125 and 54321, how many different permutations

of the digits 1, 2, 3, 4 and 5 (each digit appearing exactly once in

a 5-digit number) have occurred on Charlie’s odometer during

the intervening year?



Question 6

The quantity

2!– 3!

1!×

4!– 5!

3!×

6!– 7!

5!×

8!– 9!

7!

may be written in the form ba 32 .

What is the value of ?ba

n! (read as ‘n factorial’), where n is a positive integer, is the

product of all the positive integers less than or equal to n. For

example, 5! = 12345



Question 7

This is a multiplication grid with 2013 rows and 2013 columns

inside it. How many times does the number 2013 appear as a

product inside the grid?

x 1 2 3 4 5 …… 2013

1 1 2 3 4 5

2 2 4 6 8 10

3 3 6 9 12 15

4 4 8 12 16 20

5 5 10 15 20 25

…..

2013 4052169



Question 8

Two positive real numbers x and y are such that

yx , yx

xy

56

and yx

yx

11322 .

What is the value of ?yx

Robert Wallace (1796-1858) was a scientist and mathematician.

He was well ahead of his time in expounding the virtues of the use

of demonstration to explain the principles of applied mathematics.

Below is a question taken from his book ‘Elements of Algebra or

the Science of Quantity’, published in 1853 and intended for use by

self-taught students.



Question 9

The 26 pupils in a class are labelled A, B, C, …, Z. They sit in

order in a circle and, starting with pupil A, count from 1 to 156,

each pupil saying the number that is one more than the previous

pupil’s number.

As well as saying their number, pupils must follow these rules:

For each factor of 3 in a pupil’s number, they say ‘fizz’.

For each factor of 5 in a pupil’s number, they say ‘buzz’.

Each time a digit 3 appears in a pupil’s number, they say ‘dizz’.

Each time a digit 5 appears in a pupil’s number, they say ‘fuzz’.

e.g. Pupil W says ‘153 fizz, fizz, dizz, fuzz’ as 153 = 1733

Which pupil will say the number that generates the most ‘zz’s ?



Question 10

A stack of 10 identical cones is placed on a horizontal

table. The material from which the cones are made has

a constant thickness of 4

3

cm.

The vertical cross section through the vertex of the

lowest cone is the hexagon ABCDEF.

B

A F

E

C D

AF and DC rest on the table.

A line of symmetry passes through

B and E.

Angle DEF = 60º.

AC = 5 cm.

Not to

scale

What is the height of the stack of cones?



Senior Team Maths Challenge

Group answer sheet

Team number … …. Team name …………….……..

Award 6 points for each correct answer.

TOTAL SCORE = ___________

1. Number of steps =

2. Value of a1 =

3. Next square =

4. Length of RS =

5. Number of permutations = 6. Value of a + b =

7. Number of appearances of 2013

=

8. Value of x + y =

9. Pupil’s letter =

10. Height of the stack of cones

= cm

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