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.
Question 1
Amal wants to write the number 1000 as the sum of different
powers of 3. How many powers does he require?
Group Round
STMC National Final 2014
Question 2
What is the value of x that makes all of these fractions equal?
5  x x  13
,
15 x  18
and
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STMC National Final 2014
4 .
9 x
Question 3
Starting with the number 1, consecutive integers are used to fill the
grid as shown.
….
21 ….
11 20 ….
10 12 19 ….
4
9 13 18 ….
3
5
8 14 17 ….
1
2
6
7 15 16 ….
The number 18 is at position (4 , 3).
What is the position of the number 2014?
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STMC National Final 2014
Question 4
ACEG is a square which is white on the front and black on the
reverse.
The point B divides the edge AC in the ratio 1 : m
The point D divides the edge CE in the ratio 1 : m
The point F divides the edge EG in the ratio 1 : m
The point H divides the edge GA in the ratio 1 : m
The square is folded towards you, along each of BD, DF, FH and
HB so that the resulting shape continues to have rotational
symmetry of order 4.
The visible white area is now
1
of the area of ACEG.
3
Given that m  1, what is the exact value of m?
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STMC National Final 2014
Question 5
The first term of a sequence is a positive three digit integer.
Subsequent terms are created as follows:
If the previous term is even, halve it.
If the previous term is odd, add 1.
The sequence ends at the first occurrence of the number 1.
What is the value of the first term which gives rise to the sequence
with the greatest number of terms?
Group Round
STMC National Final 2014
Question 6
A regular octahedron has surface area 24 3 cm2.
Jo draws the shortest, continuous, non-intersecting path on the
surface of the octahedron which passes through the centre of each
face and returns to its starting point.
What is the length of her path in cm?
Group Round
STMC National Final 2014
Question 7
The radius of the circle at the top of a frustum of a right circular
cone is half the radius of the circle at its base. This frustum has the
same volume as a hemisphere whose radius is equal to that of the
circle at the frustum’s base.
What is the ratio of the height of the frustum to the radius of its
base?
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STMC National Final 2014
Question 8
A
An isosceles triangle ABC
is inscribed in a circle of
radius R.
AB = AC = 10 cm
BC = 12 cm
C
B
What is the value of R?
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STMC National Final 2014
Question 9
Given that x  1  1  1  1  1  .... , write down an
expression for x2, and hence find the exact value for x.
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STMC National Final 2014
Question 10
An octagon ABCDEFGH has the following properties:
AB = 1 cm, BC = 2 cm, CD = 3 cm, DE = 4 cm, EF = 5 cm,
FG = 6 cm, GH = 7 cm, HA = 8 cm.
Also, any two adjoining edges meet at 90º.
What is the area of the octagon in cm2 ?
Group Round
STMC National Final 2014
Senior Team Maths Challenge
Group answer sheet
Team number … ….
Team name …………….……..
1. Number of powers =
3. Position of 2014 = ( ,
2. Value of x =
)
4. Value of m =
5. First term =
6. Length of path =
cm
7. Height of frustum : radius of base
8. Value of R =
cm
=
9. Value of x =
10. Area of octagon ABCDEFGH
=
cm2
Award 6 points for each correct answer.
TOTAL SCORE = ___________
Group Round
STMC National Final 2014