TEAM MATHEMATICS COMPETITION 11
Team……………………………………………………….…
GROUP ROUND: SOLUTIONS
Answers can be left as surds where appropriate
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
TEAM MATHEMATICS COMPETITION 11
GROUP ROUND: QUESTION 1
Chemists at FizzPop Inc have mixed
chemicals incorrectly – into the bucket for
the North production line they put 30ml of
Ingredient X with 100ml water, but into
the bucket for the South production line
they put 20ml of Ingredient X with 150ml
water. Fortunately there is one more
50ml tin of Ingredient X.
How much
Ingredient X should be added to the South
production line bucket so that the two
mixtures are of equal strength and all the
Ingredient X is used up?
TEAM MATHEMATICS COMPETITION 11
GROUP ROUND: QUESTION 2
Arrows, if they are thin cylinders, can be
packed in hexagonal bundles. If there are 36
arrows on the outside of the bundle, how
many arrows in total are in the quiver?
TEAM MATHEMATICS COMPETITION 11
GROUP ROUND: QUESTION 3
Find the size of x.
TEAM MATHEMATICS COMPETITION 11
GROUP ROUND: QUESTION 4
Regular eight-sided polygons and regular foursided polygons are arranged on a flat surface
such that:
 Each eight-sided polygon is in contact with
4 others and with 4 four-sided polygons
 Each four -sided polygon is in contact with
4 eight-sided polygons only
If the pattern is extended to cover an infinite
plane, what proportion is covered with foursided polygons?
(simplify the fraction and leave in surd form).
TEAM MATHEMATICS COMPETITION 11
GROUP ROUND: QUESTION 5
An upside-down number is an
integer where the nth digit from
the left plus the nth digit from the
right is always equal to 10. For
example 13579 is an upsidedown number since 1+9 = 10,
3+7 = 10 and (since 5 is both
the 3rd digit from the left and
from the right) 5+5=10. Find the
100th upside down number.
TEAM MATHEMATICS COMPETITION 11
GROUP ROUND: QUESTION 6
A column of people, 1km long,
marches at a constant speed in a
straight line. A motorcyclist starts at
the rear of the column, moves at a
constant speed until the front of the
column is reached, and immediately
turns around and moves to the back of
the column at the same, constant,
speed. By this time, the column has
moved forward so that the last person
in the column is in the position the
first
person
was
before
the
motorcyclist started.
How far did the motorcyclist travel?
TEAM MATHEMATICS COMPETITION 11
GROUP ROUND: QUESTION 7
How many different (non-congruent)
triangles are there whose vertices are
on the lattice points of a 3 x 3 grid?
(i.e. the triangles could be made with a rubber band on a
3 x 3 pin board)
TEAM MATHEMATICS COMPETITION 11
GROUP ROUND: QUESTION 8
Diophantus
of
Alexandria
wrote
the
Arithmetica, a book on equations, in the third
century AD.
A friend wrote this obituary: ‘Diophantus was
a child for one sixth of his life and he grew a
beard after a twelfth more. He married after
another seventh of his life and his wife bore a
son five years later. The son lived to half his
father’s age and Diophantus died four years
after his son.
How old was Diophantus when he died?’
TEAM MATHEMATICS COMPETITION 11
GROUP ROUND: QUESTION 9
Encountering a man on the porch of
his house, a census taker asked “What
are the ages of the persons living
here?” the man replied, “All our ages
are square integers. My age is the sum
of the ages of my wife, son and
daughter. My father’s age is the sum
of my age and the ages of my wife
and daughter. Although he has passed
the prime of life, his age is a prime
number.”
What is the sum of the ages the
census taker recorded?
TEAM MATHEMATICS COMPETITION 11
GROUP ROUND: QUESTION 10
Points (-2, 1), (5,0) and (6, 7) form a
triangle whose points lie on the edge
of a circle.
Find the area inside the circle and
outside the triangle.
Give your answer in terms of π.
TEAM MATHEMATICS COMPETITION 11
Team……………………………………………………….…
BONUS QUESTION 1
The solution to this question can be submitted at any time during
the competition.
The curve y=x2 - 4 is rotated 180
about the point (-1,1).
What is the equation of the new
curve?
Solution
TEAM MATHEMATICS COMPETITION 11
Team……………………………………………………….…
BONUS QUESTION 2
The solution to this question can be submitted at any time during
the competition.
What proportion of the area of the
square is the shaded area?
The arcs are cicular arcs, centred at
corners of the square.
Solution
TEAM MATHEMATICS COMPETITION 11
GROUP ROUND: SOLUTIONS
1.
2.
40
3.
127
4.
15cm
5.
3 - 2√2
6.
9911
7.
1 + √2 km
8.
8
9.
84
10.
187
Bonus 1.
y =6 -(x+2)2
Or
2
y =-x - 4x + 2
25(π-1)
Bonus 2.