MT
UK
MT
UK
A1
UKMT
The equation
8(9x + 7) − 7(6x − 5) = 1
has the solution x = −k, where k is a positive integer.
Senior Team
Maths
Challenge
2016/17
Pass on the value of k.
Regional Final
Shuttle
MT
UK
MT
UK
© UKMT 2016/17
UKMT
T is the number you will receive.
Y is proportional to the reciprocal of the square of X.
Y = 20 when X = 6.
Pass on the value of Y when X = T − 1.
Senior Team
Maths
Challenge
2016/17
Regional Final
Shuttle
© UKMT 2016/17
A3
MT
UK
MT
UK
UKMT
A2
T is the number you will receive.
The expression
1
2
1
64− T + 36− 2 + 8− 3
Senior Team
Maths
Challenge
2016/17
p
can be simplified to , where p and q are positive integers with
q
no common factor greater than 1.
Pass on the value of p + q.
Regional Final
Shuttle
MT
UK
MT
UK
© UKMT 2016/17
UKMT
A4
T is the number you will receive.
In the diagram V XY and W X Z are straight lines.
The lines VW and ZY are parallel.
W X = 36 cm, XY = 30 cm, Z X = d cm and XV = T cm.
Senior Team
Maths
Challenge
2016/17
V
Z
T cm
d cm
Regional Final
X
Shuttle
36 cm
W
© UKMT 2016/17
Write down the value of d.
30 cm
Y
MT
UK
MT
UK
B1
UKMT
(10!) ÷ (6!) = n!
Pass on the value of n.
[The notation n! means the factorial of n, which is n × (n − 1) × · · · × 2 × 1.
For example, 6! means 6 × 5 × 4 × 3 × 2 × 1.]
Senior Team
Maths
Challenge
2016/17
Regional Final
Shuttle
MT
UK
MT
UK
© UKMT 2016/17
UKMT
B3
T is the number you will receive.
An equilateral triangle has its vertices
on a circle of area 15 πT cm2, as shown.
The perimeter of the triangle has
length x cm.
Senior Team
Maths
Challenge
2016/17
Regional Final
Shuttle
© UKMT 2016/17
Pass on the value of x.
[sin 30° = cos 60° =
1
2
and sin 60° = cos 30° =
√
3
2 .]
MT
UK
MT
UK
UKMT
B2
T is the number you will receive.
T 3
a
− and write your answer in the form , where
3 T
b
a and b are positive integers with no common factor greater
than 1.
Evaluate
Senior Team
Maths
Challenge
2016/17
Pass on the value of a + b − 1.
Regional Final
Shuttle
MT
UK
MT
UK
© UKMT 2016/17
UKMT
Senior Team
Maths
Challenge
2016/17
Regional Final
Shuttle
© UKMT 2016/17
B4
T is the number you will receive.
The graph of y = x 2 − 6x − T meets
the y-axis at P, and the x-axis at Q
and R, as shown.
Write down the area of the
√ triangle
PQR as a simplified surd a b, where
a and b are integers and b is not
divisible by any square greater than 1.
y
R x
Q
P
MT
UK
MT
UK
C1
UKMT
U, K, M and T are positive integers with
1 < U < K < M < T < 10
such that
Senior Team
Maths
Challenge
2016/17
U M = K × T.
Pass on the value of U + K + M + T.
Regional Final
Shuttle
MT
UK
MT
UK
© UKMT 2016/17
UKMT
T is the number you will receive.
The line y = 4x +T intersects the curve y = x 2 − (T − 14)x − 9T
at the points (x 1, y1 ) and (x 2, y2 ).
Pass on the value of x 1 + x 2.
Senior Team
Maths
Challenge
2016/17
Regional Final
Shuttle
© UKMT 2016/17
C3
MT
UK
MT
UK
UKMT
C2
T is the number you will receive.
The diagram shows parts of two regular polygons with a common
edge.
A
3(T + 1)°
Senior Team
Maths
Challenge
2016/17
Regional Final
Shuttle
B
Polygon A has five more sides than polygon B and the sum of
their exterior angles is 3(T + 1)°.
Pass on the sum of the numbers of sides of the two polygons.
MT
UK
MT
UK
© UKMT 2016/17
UKMT
T is the number you will receive.
x satisfies the equation
1
256 3 x × 2(T+1)x = 163x+T
Write down the value of x.
Senior Team
Maths
Challenge
2016/17
Regional Final
Shuttle
© UKMT 2016/17
C4
MT
UK
MT
UK
D1
UKMT
B
x°
Senior Team
Maths
Challenge
2016/17
D
20°
C
A
Regional Final
The lines C A and CB are tangents to the circle.
Shuttle
D is a point on the circle on the minor arc between A and B.
The angle BC A = 20°.
Pass on the value of x.
MT
UK
MT
UK
© UKMT 2016/17
UKMT
T is the number you will receive.
The integer k is such that the expression
√
√
√
√
√
k 2+2
3 + 3 − T − 2T − 3T
is an integer.
Senior Team
Maths
Challenge
2016/17
Regional Final
Shuttle
© UKMT 2016/17
Pass on the value of 6k.
D3
MT
UK
MT
UK
UKMT
T is the number you will receive.
D2
A rectangle is drawn with its vertices
on a circle, as shown.
The width of the rectangle is 4 cm.
√
The height of the rectangle is T cm.
Senior Team
Maths
Challenge
2016/17
The area of the circle can be written
in the form Aπ cm2.
Pass on the value of 2A − 4.
Regional Final
Shuttle
MT
UK
MT
UK
© UKMT 2016/17
UKMT
T is the number you will receive.
D4
A bag contains (T − 3) balls, each of which is red, blue or green.
There is at least one red ball. There are more blue balls than red
balls, and more green balls than blue balls.
Senior Team
Maths
Challenge
2016/17
Regional Final
Shuttle
© UKMT 2016/17
If three balls are chosen at random from the bag, without
replacement, the probability that there is one of each colour is
16
91 .
Write down the number of green balls in the bag.
Shuttle
response sheet
Team number
Senior Team Maths Challenge 2016/17
School name
A1
B1
0 1 3
A2
C1
0 1 3
A3
A total /15
3
0 1 3
C3
0 1 3
C4
B total /15
3
0 1 3
Bonus
C total /15
Circle the mark awarded for each question and cross out the others.
At the end of the round, either circle the bonus mark or cross it out.
© UKMT 2016/17
0 1 3
D4
0 1 3
Bonus
0 1 3
D3
0 1 3
B4
0 1 3
D2
0 1 3
0 1 3
0 1 3
0 1 3
C2
B3
A4
D1
0 1 3
B2
Bonus
Regional Final
3
0 1 3
Bonus
D total /15
Final score /60
3