唯寻国际教育

CONTENTS
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 唯寻国际教育

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UKMT

UK SENIOR MATHEMATICAL CHALLENGE

Thursday 5 November 2015
Organised by the United Kingdom Mathematics Trust
and supported by

RULES AND GUIDELINES (to be read before starting)

1. Do not open the question paper until the invigilator tells you to do so.
2. Use B or HB pencil only. Mark at most one of the options A, B, C, D, E on the
Answer Sheet for each question. Do not mark more than one option.
3. Time allowed: 90 minutes.
No answers or personal details may be entered on the Answer Sheet after the 90
minutes are over.
4. The use of rough paper is allowed.
Calculators, measuring instruments and squared paper are forbidden.
5. Candidates must be full-time students at secondary school or FE college, and must
be in Year 13 or below (England & Wales); S6 or below (Scotland); Year 14 or
below (Northern Ireland).
6. There are twenty-five questions. Each question is followed by five options marked
A, B, C, D, E. Only one of these is correct. Enter the letter A-E corresponding to
the correct answer in the corresponding box on the Answer Sheet.
7. Scoring rules: all candidates start out with 25 marks;
0 marks are awarded for each question left unanswered;
4 marks are awarded for each correct answer;
1 mark is deducted for each incorrect answer.
8. Guessing: Remember that there is a penalty for wrong answers. Note also that
later questions are deliberately intended to be harder than earlier questions. You
are thus advised to concentrate first on solving as many as possible of the first 15-
20 questions. Only then should you try later questions.

The United Kingdom Mathematics Trust is a Registered Charity.

http://www.ukmt.org.uk
© UKMT 2015

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1. What is 20152 − 2016 × 2014?
A −2015 B −1 C 0 D 1 E 2015
150
2. What is the sum of all the solutions of the equation 6x = ?
x
A 0 B 5 C 6 D 25 E 156
3. When Louise had her first car, 50 litres of petrol cost £40. When she filled up the
other day, she noticed that 40 litres of petrol cost £50.
By approximately what percentage has the cost of petrol increased over this time?
A 50% B 56% C 67% D 75% E 80%
4. In the diagram, the smaller circle touches the larger circle and also
passes through its centre. What fraction of the area of the larger
circle is outside the smaller circle?
2 3 4 5 6
A B C D E
3 4 5 6 7
5. The integer n is the mean of the three numbers 17, 23 and 2n. What is the sum of
the digits of n?
A 4 B 5 C 6 D 7 E 8
6. The numbers 5, 6, 7, 8, 9, 10 are to be placed, one in each of the circles 5
in the diagram, so that the sum of the numbers in each pair of touching
circles is a prime number. The number 5 is placed in the top circle.
Which number is placed in the shaded circle?
A 6 B 7 C 8 D 9 E 10
7. Which of the following has the largest value?

( 12 ) 1 ( )
( 12 )
3 1 ( ( ))
1
2
3

( ( ))
A B C D E
( 34 )
( )
4
( 23 ) 4 2
3
4
4

8. The diagram shows eight small squares. Six of these squares are
to be shaded so that the shaded squares form the net of a cube.
In how many different ways can this be done?
A 10 B 8 C 7 D 6 E 4
9. Four different straight lines are drawn on a flat piece of paper. The number of
points where two or more lines intersect is counted.
Which of the following could not be the number of such points?
A 1 B 2 C 3 D 4 E 5
10. The positive integer n is between 1 and 20. Milly adds up all the integers from 1 to
n inclusive. Billy adds up all the integers from n + 1 to 20 inclusive. Their totals
are the same. What is the value of n?
A 11 B 12 C 13 D 14 E 15
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 唯寻国际教育
11. Rahid has a large number of cubic building blocks. Each block has sides of length
4 cm, 6 cm or 10 cm. Rahid makes little towers built from three blocks stacked on
top of each other. How many different heights of tower can he make?
A 6 B 8 C 9 D 12 E 27
12. A circle touches the sides of triangle PQR P
at the points S, T and U as shown. Also
∠PQR = α°, ∠PRQ = β° and ∠TSU = γ°. T
Which of the following gives γ in terms
U
of α and β?
A 12 (α + β) B 180 − 12 (α + β)
C 180 − (α + β) D α + β g°
α° β°
E 13 (α + β) Q S R

13. The Knave of Hearts tells only the truth on Mondays, Tuesdays, Wednesdays and
Thursdays. He tells only lies on all the other days. The Knave of Diamonds tells
only the truth on Fridays, Saturdays, Sundays and Mondays. He tells only lies on
all the other days. On one day last week, they both said, “Yesterday I told lies.”
On which day of the week was that?
A Sunday B Monday C Tuesday D Thursday E Friday
14. The triangle shown has an area of 88 square units. What is 10 y
the value of y?
A 17.6 B 2 46 C 6 10 D 13 2 E 8 5 22
15. Two vases are cylindrical in shape. The larger vase has diameter
20 cm. The smaller vase has diameter 10 cm and height 16 cm.
The larger vase is partially filled with water. Then the empty
smaller vase, with the open end at the top, is slowly pushed down
into the water, which flows over its rim. When the smaller vase is
pushed right down, it is half full of water.
What was the original depth of the water in the larger vase?
A 10 cm B 12 cm C 14 cm D 16 cm E 18 cm
16. Fnargs are either red or blue and have 2, 3 or 4 heads. A group of six Fnargs
consisting of one of each possible form is made to line up such that no immediate
neighbours are the same colour nor have the same number of heads. How many
ways are there of lining them up from left to right?
A 12 B 24 C 60 D 120 E 720
17. The diagram shows eight circles of two different sizes. The
circles are arranged in concentric pairs so that the centres form
a square. Each larger circle touches one other larger circle and
two smaller circles. The larger circles have radius 1. What is
the radius of each smaller circle?
A 1
3 B 25 C 2−1 D 1
2 E 1
2 2

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 唯寻国际教育
18. What is the largest integer k whose square k 2 is a factor of 10!?
[ 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.]
A 6 B 256 C 360 D 720 E 5040
19. Three squares are arranged as shown so that their Q
R
bases lie on a straight line. Also, the corners P, Q and P
R lie on a straight line. The middle square has sides
that are 8 cm longer than the sides of the smallest
square. The largest square has sides of length 50 cm.
There are two possible values for the length (in cm) of the sides of the smallest
square. Which of the following are they?
A 2, 32 B 4, 42 C 4, 34 D 32, 40 E 34, 42
20. A square ink pad has sides of length 1 cm. It is covered in black ink and carefully
placed in the middle of a piece of white paper. The square pad is then rotated 180°
about one of its corners so that all of the pad remains in contact with the paper
throughout the turn. The pad is then removed from the paper. What area of paper,
in cm2, is coloured black?
A π + 2 B 2π − 1 C 4 D 2π − 2 E π + 1
21. The diagram shows a triangle XYZ. The sides Y
XY, YZ and XZ have lengths 2, 3 and 4 P T
respectively. The lines AMB, PMQ and SMT
M
are drawn parallel to the sides of triangle XYZ A B
so that AP, QS and BT are of equal length.
X S Q Z
What is the length of AP?
10 11 12 13 14
A B C D E
11 12 13 14 15
1
22. Let f (x) = x + x2 + 1 + . What is the value of f (2015)?
x − x2 + 1
A −1 B 0 C 1 D 2016 E 2015
23. Given four different non-zero digits, it is possible to form 24 different four-digit
numbers containing each of these four digits. What is the largest prime factor of the
sum of the 24 numbers?
A 23 B 93 C 97 D 101 E 113
24. Peter has 25 cards, each printed with a different integer from 1 to 25. He wishes to
place N cards in a single row so that the numbers on every adjacent pair of cards
have a prime factor in common.
What is the largest value of N for which this is possible?
A 16 B 18 C 20 D 22 E 24
25. A function, defined on the set of positive integers, is such that f (xy) = f (x) + f (y)
for all x and y. It is known that f (10) = 14 and f (40) = 20. What is the value of
f (500) ?
A 29 B 30 C 39 D 48 E 50

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 唯寻国际教育

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UKMT

UK SENIOR MATHEMATICAL CHALLENGE

Tuesday 8 November 2016
Organised by the United Kingdom Mathematics Trust
and supported by

RULES AND GUIDELINES (to be read before starting)

1. Do not open the question paper until the invigilator tells you to do so.
2. Time allowed: 90 minutes.
No answers or personal details may be entered on the Answer Sheet after the 90
minutes are over.
3. The use of rough paper is allowed.
Calculators, measuring instruments and squared paper are forbidden.
4. Candidates must be full-time students at secondary school or FE college, and must
be in Year 13 or below (England & Wales); S6 or below (Scotland); Year 14 or
below (Northern Ireland).
5. Use B or HB pencil only. Mark at most one of the options A, B, C, D, E on the
Answer Sheet for each question. Do not mark more than one option.
6. Scoring rules: all candidates start out with 25 marks;
0 marks are awarded for each question left unanswered;
4 marks are awarded for each correct answer;
1 mark is deducted for each incorrect answer.
7. Guessing: Remember that there is a penalty for incorrect answers. Note also that
later questions are deliberately intended to be harder than earlier questions. You
are thus advised to concentrate first on solving as many as possible of the first 15-
20 questions. Only then should you try later questions.

The United Kingdom Mathematics Trust is a Registered Charity.

http://www.ukmt.org.uk

第５页
 唯寻国际教育
1. How many times does the digit 9 appear in the answer to 987654321 × 9 ?
A 0 B 1 C 5 D 8 E 9
2. On a Monday, all prices in Isla's shop are 10% more than normal. On Friday all
prices in Isla's shop are 10% less than normal. James bought a book on Monday for
£5.50. What would be the price of another copy of this book on Friday?
A £5.50 B £5.00 C £4.95 D £4.50 E £4.40
3. The diagram shows a circle with radius 1 that rolls without slipping
around the inside of a square with sides of length 5.
The circle rolls once around the square, returning to its starting point.
What distance does the centre of the circle travel?
A 16 − 2π B 12 C 6+π D 20 − 2π E 20
4. Alex draws a scalene triangle. One of the angles is 80°.
Which of the following could be the difference between the other two angles in
Alex's triangle?
A 0° B 60° C 80° D 100° E 120°
5. All the digits 2, 3, 4, 5 and 6 are placed in the grid, one in each cell,
to form two three-digit numbers that are squares.
Which digit is placed in the centre of the grid?
A 2 B 3 C 4 D 5 E 6
6. The diagram shows a square ABCD and a right-angled A D
triangle ABE. The length of BC is 3. The length of BE
is 4.
What is the area of the shaded region?
E
A 514 B 538 C 512 D 585 E 534 B C

7. Which of these has the smallest value?

A 2016−1 B 2016−1/2 C 20160 D 20161/2 E 20161
8. Points are drawn on the sides of a square, dividing each side into
n equal parts (so, in the example shown, n = 4).
The points are joined in the manner indicated, to form several
small squares (24 in the example, shown shaded) and some
triangles.
How many small squares are formed when n = 7 ?
A 56 B 84 C 140 D 840 E 5040
9. A square has vertices at (0, 0), (1, 0), (1, 1) and (0, 1). Graphs of the following
equations are drawn on the same set of axes as the square.
1
x2 + y2 = 1, y = x + 1, y = −x2 + 1, y = x, y =
x
How many of the graphs pass through exactly two of the vertices of the square?
A 1 B 2 C 3 D 4 E 5
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10. The digits from 1 to 9 are to be written in the nine cells of the 12
3 × 3 grid shown, one digit in each cell.
112
The product of the three digits in the first row is 12.
The product of the three digits in the second row is 112.
216 12
The product of the three digits in the first column is 216.
The product of the three digits in the second column is 12.
What is the product of the digits in the shaded cells?
A 24 B 30 C 36 D 48 E 140
11. In the grid below each of the blank squares and the square marked X are to be filled
by the mean of the two numbers in its adjacent squares. Which number should go
in the square marked X?
10 X 25

A 15 B 16 C 17 D 18 E 19
12. What is the smallest square that has 2016 as a factor?
A 422 B 842 C 1682 D 3362 E 20162
13. Five square tiles are put together side by side. A quarter
circle is drawn on each tile to make a continuous curve
as shown. Each of the smallest squares has side-length 1.
What is the total length of the curve?
A 6π B 6.5π C 7π D 7.5π E 8π
14. Which of the following values of the positive integer n is a counterexample to the
statement: “If n is not prime then n − 2 is not prime” ?
A 6 B 11 C 27 D 33 E 51
15. The diagram shows three rectangles and three straight
55°
lines.
What is the value of p + q + r ? q° p°

A 135 B 180 C 210

65°
D 225 E 270 60°

r°

16. For which value of k is 2016 + 56 equal to 14k ?

1 3 5 3 5
A 2 B 4 C 4 D 2 E 2
17. Aaron has to choose a three-digit code for his bike lock. The digits can be chosen
from 1 to 9. To help him remember them, Aaron chooses three different digits in
increasing order, for example 278. How many such codes can be chosen?
A 779 B 504 C 168 D 84 E 9

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18. The circumference of a circle with radius 1 is divided into four
equal arcs. Two of the arcs are ‘turned over’ as shown.
What is the area of the shaded region?
2π
1
A 1 B 2 C D 3 E 2
19. Let S be a set of five different positive integers, the largest of which is m. It is
impossible to construct a quadrilateral with non-zero area, whose side-lengths are
all distinct elements of S. What is the smallest possible value of m?
A 2 B 4 C 9 D 11 E 12
20. Michael was walking in Marrakesh when he saw a tiling X Y

formed by tessellating the square tile as shown. Z

The tile has four lines of symmetry and the length of each side
is 8 cm. The length of XY is 2 cm. The point Z is such that XZ
is a straight line and YZ is parallel to sides of the square.
What is the area of the central grey octagon?
A 6 cm2 B 7 cm2 C 8 cm2 D 9 cm2 E 10 cm2
21. The diagram shows ten equal discs that lie between two concentric
circles − an inner circle and an outer circle. Each disc touches two 1
neighbouring discs and both circles. The inner circle has radius 1.
What is the radius of the outer circle?
sin 36° 1 + sin 18° 2 9
A 2 tan 36° B C D E
1 − sin 36° 1 − sin 18° cos 18° 5
22. Three friends make the following statements.
Ben says, “Exactly one of Dan and Cam is telling the truth.”
Dan says, “Exactly one of Ben and Cam is telling the truth.”
Cam says, “Neither Ben nor Dan is telling the truth.”
Which of the three friends is lying?
A Just Ben B Just Dan C Just Cam D Each of Ben and Cam
E Each of Ben, Cam and Dan
23. A cuboid has sides of lengths 22, 2 and 10. It is contained within a sphere of the
smallest possible radius. What is the side-length of the largest cube that will fit
inside the same sphere?
A 10 B 11 C 12 D 13 E 14
24. The diagram shows a square PQRS. The arc QS is a quarter P S

circle. The point U is the midpoint of QR and the point T lies on

T
SR. The line TU is a tangent to the arc QS. What is the ratio of
the length of TR to the length of UR ?
A 3:2 B 4:3 C 5:4 D 7:6 E 9:8 Q U R

25. Let n be the smallest integer for which 7n has 2016 digits.
What is the units digit of n?
A 0 B 1 C 4 D 6 E 8
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UKMT

UK SENIOR MATHEMATICAL CHALLENGE

Tuesday 7 November 2017
Organised by the United Kingdom Mathematics Trust
and supported by

RULES AND GUIDELINES (to be read before starting)

1. Do not open the question paper until the invigilator tells you to do so.
2. Time allowed: 90 minutes.
No answers or personal details may be entered on the Answer Sheet after the 90
minutes are over.
3. The use of rough paper is allowed.
Calculators, measuring instruments and squared paper are forbidden.
4. Candidates must be full-time students at secondary school or FE college, and must
be in Year 13 or below (England & Wales); S6 or below (Scotland); Year 14 or
below (Northern Ireland).
5. Use B or HB pencil only. Mark at most one of the options A, B, C, D, E on the
Answer Sheet for each question. Do not mark more than one option.
6. Scoring rules: all candidates start out with 25 marks;
0 marks are awarded for each question left unanswered;
4 marks are awarded for each correct answer;
1 mark is deducted for each incorrect answer.
7. Guessing: Remember that there is a penalty for incorrect answers. Note also that
later questions are deliberately intended to be harder than earlier questions. You
are thus advised to concentrate first on solving as many as possible of the first 15-
20 questions. Only then should you try later questions.

The United Kingdom Mathematics Trust is a Registered Charity.

http://www.ukmt.org.uk
© UKMT 2017

第９页
 唯寻国际教育

1. One of the following numbers is prime. Which is it?

A 2017 − 2 B 2017 − 1 C 2017 D 2017 + 1 E 2017 + 2
2. Last year, an earthworm from Wigan named Dave wriggled into the record books
as the largest found in the UK. Dave was 40 cm long and had a mass of 26 g.
What was Dave's mass per unit length?
A 0.6 g/cm B 0.65 g/cm C 0.75 g/cm D 1.6 g/cm E 1.75 g/cm
3. The five integers 2, 5, 6, 9, 14 are arranged into a different order. In the new
arrangement, the sum of the first three integers is equal to the sum of the last three
integers. What is the middle number in the new arrangement?
A 2 B 5 C 6 D 9 E 14
1
4. Which of the following is equal to 2017 − ?
2017
20172 2016 2018 4059 2018 × 2016
A B C D E
2016 2017 2017 2017 2017
12
5. One light-year is nearly 6 × 10 miles. In 2016, the Hubble Space Telescope set a
new cosmic record, observing a galaxy 13.4 thousand million light-years away.
Roughly how many miles is that?
A 8 × 1020 B 8 × 1021 C 8 × 1022 D 8 × 1023 E 8 × 1024
6. The circles in the diagram are to be coloured so that any two circles
connected by a line segment have different colours.
What is the smallest number of colours required?
A 2 B 3 C 4 D 5 E 6
7. The positive integer k satisfies the equation 2 + 8 + 18 = k . What is the
value of k ?
A 28 B 36 C 72 D 128 E 288
8. When evaluated, which of the following is not an integer?
1 2 3
A 1−1 B 4− 2 C 60 D 83 E 16 4
9. The diagram shows an n × (n + 1) rectangle tiled with k × (k + 1)
rectangles, where n and k are integers and k takes each value from
1 to 8 inclusive.
What is the value of n?
A 16 B 15 C 14 D 13 E 12
10. A rectangle is divided into three smaller congruent rectangles as
shown.
Each smaller rectangle is similar to the large rectangle.
In each of these four rectangles, what is the ratio of the length of
a longer side to that of a shorter side?
A 2 3 : 1 B 3 : 1 C 2 : 1 D 3 : 1 E 2 : 1

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 唯寻国际教育

11. The teenagers Sam and Jo notice the following facts about their ages:
The difference between the squares of their ages is four times the sum of their ages.
The sum of their ages is eight times the difference between their ages.
What is the age of the older of the two?
A 15 B 16 C 17 D 18 E 19
12. The diagram shows a square and a
regular decagon that share an edge. One
x°
side of the square is extended to meet an
extended side of the decagon.
What is the value of x?
A 15 B 18 C 21 D 24 E 27
13. Isobel: “Josh is innocent” Genotan: “Tegan is guilty”
Josh: “Genotan is guilty” Tegan: “Isobel is innocent”
Only the guilty person is lying; all the others are telling the truth. Who is guilty?
A Isobel B Josh C Genotan D Tegan E More information required
14. In the diagram, all the angles marked • are equal in size
to the angle marked x°.
What is the value of x?
A 100 B 105 C 110 D 115 E 120 x°

15. The diagram shows a square PQRS. Points T , U , V S V R

and W lie on the edges of the square as shown, such
that PT = 1, QU = 2, RV = 3 and SW = 4.
The area of TU VW is half that of PQRS. What is
the length of PQ? U
W
A 5 B 6 C 7 D 8 E 9
P T Q

16. The diagram shows two right-angled triangles inside a

square. The perpendicular edges of the larger triangle
20
have lengths 15 and 20.
What is the area of the shaded quadrilateral?
A 142 B 146 C 150 D 154 E 158 15

17. Amy, Beth and Claire each has some sweets. Amy gives one third of her sweets to
Beth. Beth gives one third of all the sweets she now has to Claire. Then Claire
gives one third of all the sweets she now has to Amy. All the girls end up having
the same number of sweets.
Claire begins with 40 sweets. How many sweets does Beth have originally?
A 20 B 30 C 40 D 50 E 60

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18. The arithmetic mean, A, of any two positive numbers x and y is defined to be
A = 12 (x + y) and their geometric mean, G, is defined to be G = xy. For two
particular values x and y, with x > y, the ratio A : G = 5 : 4. For these values of x
and y, what is the ratio x : y?
A 5 : 4 B 2 : 1 C 5 : 2 D 7 : 2 E 4 : 1
19. The diagram shows a circle of radius 1 touching three
sides of a 2 × 4 rectangle. A diagonal of the rectangle Q
intersects the circle at P and Q, as shown. What is the P
length of the chord PQ?
4 2 5 5
A 5 B C 5 − D E 2
5 5 6
20. The diagram shows a square PQRS with edges of length 1, U
and four arcs, each of which is a quarter of a circle. Arc
TRU has centre P; arc VPW has centre R; arc U V has V
S
R
centre S; and arc WT has centre Q.
What is the length of the perimeter of the shaded region?
Q
A 6 B (2 2 − 1) π C ( 2 − 1
2 )π P T

D 2π E (3 2 − 2) π W

21. How many pairs (x, y) of positive integers satisfy the equation 4x = y2 + 15?
A 0 B 1 C 2 D 4 E an infinite number
22. The diagram shows a regular octagon and a square formed by
drawing four diagonals of the octagon. The edges of the square
have length 1.
What is the area of the octagon?
6 4 7 3
A B C D 2 E
2 3 5 2
2
23. The parabola with equation y = x is reflected in the line with equation y = x + 2.
Which of the following is the equation of the reflected parabola?
A x = y2 + 4y + 2 B x = y2 + 4y − 2 C x = y2 − 4y + 2
D x = y2 − 4y − 2 E x = y2 + 2
24. There is a set of straight lines in a plane such that each line intersects exactly ten
others. Which of the following could not be the number of lines in that set?
A 11 B 12 C 15 D 16 E 20
25. The diagram shows a regular nonagon N . Moving clockwise
around N , at each vertex a line segment is drawn perpendicular to
the preceding edge. This produces a smaller nonagon S, shown S
shaded.
What fraction of the area of N is the area of S? N
1 − cos 40° cos 40° sin 40° 1 − sin 40° 1
A B C D E
1 + cos 40° 1 + cos 40° 1 + sin 40° 1 + sin 40° 9

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第１３页
 唯寻国际教育
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第１４页
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第１５页
 唯寻国际教育
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第１６页
 唯寻国际教育

MT
UK
MT

UK
UKMT

United Kingdom
Mathematics Trust

Senior Mathematical Challenge

Thursday 7 November 2019
Organised by the United Kingdom Mathematics Trust

supported by

Candidates must be full-time students at secondary school or FE college.

England & Wales: Year 13 or below
Scotland: S6 or below
Northern Ireland: Year 14 or below

Instructions
1. Do not open the paper until the invigilator tells you to do so.
2. Time allowed: 90 minutes.
No answers, or personal details, may be entered after the allowed time is over.
3. The use of blank or lined paper for rough working is allowed; squared paper, calculators
and measuring instruments are forbidden.
4. Use a B or an HB non-propelling pencil. Mark at most one of the options A, B, C, D, E
on the Answer Sheet for each question. Do not mark more than one option.
5. Do not expect to finish the whole paper in the time allowed. The questions in this paper
have been arranged in approximate order of difficulty with the harder questions towards the
end. You are not expected to complete all the questions during the time. You should bear
this in mind when deciding which questions to tackle.
6. Scoring rules:
All candidates start with 25 marks;
0 marks are awarded for each question left unanswered;
4 marks are awarded for each correct answer;
1 mark is deducted for each incorrect answer (to discourage guessing).
7. Your Answer Sheet will be read by a machine. Do not write or doodle on the sheet except
to mark your chosen options. The machine will read all black pencil markings even if
they are in the wrong places. If you mark the sheet in the wrong place, or leave bits of eraser
stuck to the page, the machine will interpret the mark in its own way.
8. The questions on this paper are designed to challenge you to think, not to guess. You
will gain more marks, and more satisfaction, by doing one question carefully than by guessing
lots of answers. This paper is about solving interesting problems, not about lucky guessing.

Enquiries about the Senior Mathematical Challenge should be sent to:

UK Mathematics Trust, School of Mathematics, University of Leeds, Leeds LS2 9JT
T 0113 343 2339 enquiry@ukmt.org.uk
第１７页 www.ukmt.org.uk
Senior Mathematical Challenge Thursday 7 November 2019
唯寻国际教育
1. What is the value of 1232 − 232 ?
A 10 000 B 10 409 C 12 323 D 14 600 E 15 658

2. What is the value of (2019 − (2000 − (10 − 9))) − (2000 − (10 − (9 − 2019)))?
A 4040 B 40 C −400 D −4002 E −4020

3. Used in measuring the width of a wire, one mil is equal to one thousandth of an inch. An inch is
about 2.5 cm.
Which of these is approximately equal to one mil?
1 1 1
A mm B mm C mm D 25 mm E 40 mm
40 25 4

4. For how many positive integer values of n is n2 + 2n prime?

A 0 B 1 C 2 D 3 E more than 3

5. Olive Green wishes to colour all the circles in the diagram so that, for each circle,
there is exactly one circle of the same colour joined to it.
What is the smallest number of colours that Olive needs to complete this task?
A 1 B 2 C 3 D 4 E 5

6. Each of the factors of 100 is to be placed in a 3 by 3 grid, one per cell, in such a
x 1 50
way that the products of the three numbers in each row, column and diagonal are all
equal. The positions of the numbers 1, 2, 50 and x are shown in the diagram.
What is the value of x? 2
A 4 B 5 C 10 D 20 E 25

7. Lucy is asked to choose p, q, r and s to be the numbers 1, 2, 3 and 4, in some order, so as to make the
p r
value of + as small as possible.
q s
What is the smallest value Lucy can achieve in this way?
7 2 3 5 11
A B C D E
12 3 4 6 12
x
8. The number x is the solution to the equation 3(3 ) = 333.
Which of the following is true?
A 0<x<1 B 1<x<2 C 2<x<3 D 3<x<4 E 4<x<5

9. A square of paper is folded in half four times to obtain a smaller square. Then a corner
is removed as shown.
Which of the following could be the paper after it is unfolded?
A B C D E

© UK Mathematics Trust 2019 第１８页

www.ukmt.org.uk
Senior Mathematical Challenge Thursday 7 November 2019
唯寻国际教育
10. Which of the following five values of n is a counterexample to the statement in the box below?

For a positive integer n, at least one of 6n − 1 and 6n + 1 is prime.

A 10 B 19 C 20 D 21 E 30
p √
11. For how many integer values of k is 200 − k also an integer?
A 11 B 13 C 15 D 17 E 20

12. A circle with radius 1 touches the sides of a rhombus, as shown. Each of
the smaller angles between the sides of the rhombus is 60°.
What is the area of the rhombus?
√
√ √ 8 3 60◦
A 6 B 4 C 2 3 D 3 3 E
3

13. Anish has a number of small congruent square tiles to use in a mosaic. When he forms the tiles into a
square of side n, he has 64 tiles left over. When he tries to form the tiles into a square of side n + 1, he
has 25 too few.
How many tiles does Anish have?
A 89 B 1935 C 1980 D 2000 E 2019

14. One of the following is the largest square that is a factor of 10!. Which one?
Note that, n! = 1 × 2 × 3 × · · · × (n − 1) × n.
A (4!)2 B (5!)2 C (6!)2 D (7!)2 E (8!)2

15. The highest common factors of all the pairs chosen from the positive integers Q, R and S are three
different primes.
What is the smallest possible value of Q + R + S?
A 41 B 31 C 30 D 21 E 10

16. The numbers x, y and z satisfy the equations 9x + 3y − 5z = −4 and 5x + 2y − 2z = 13.

What is the mean of x, y and z?
A 10 B 11 C 12 D 13 E 14

17. Jeroen writes a list of 2019 consecutive integers. The sum of his integers is 2019.
What is the product of all the integers in Jeroen’s list?
2019 × 2020
A 20192 B C 22019 D 2019 E 0
2
18. Alison folds a square piece of paper in half along the dashed line shown in the
diagram. After opening the paper out again, she then folds one of the corners
onto the dashed line. α◦

What is the value of α?

A 45 B 60 C 65 D 70 E 75

© UK Mathematics Trust 2019 第１９页

www.ukmt.org.uk
Senior Mathematical Challenge Thursday 7 November 2019
唯寻国际教育
19. Which of the following could be the graph of y 2 = sin x 2 ?

A y B y C y

x x x

D y E y

x x

20. The "heart" shown in the diagram is formed from an equilateral triangle X
ABC and two congruent semicircles on AB. The two semicircles meet
at the point P. The point O is the centre of one of the semicircles. On A
O
B
P
the semicircle with centre O, lies a point X. The lines XO and X P are
Z
extended to meet AC at Y and Z respectively. The lines XY and X Z are
of equal length. Y
What is ∠Z XY ?
C
A 20° B 25° C 30° D 40° E 45°

21. In a square garden PQRT of side 10 m, a ladybird sets off from Q and moves along edge QR at 30 cm
per minute. At the same time, a spider sets off from R and moves along edge RT at 40 cm per minute.
What will be the shortest distance between them, in metres?
√
A 5 B 6 C 5 2 D 8 E 10
22. A function f satisfies the equation (n − 2019) f (n) − f (2019 − n) = 2019 for every integer n.
What is the value of f (2019)?
A 0 B 1 C 2018 × 2019 D 20192 E 2019 × 2020
23. The edge-length of the solid cube shown is 2. A single plane cut goes through Y
the points Y , T, V and W which are midpoints of the edges of the cube, as shown.
What is the area of the cross-section? T
√ √ √
A 3 B 3 3 C 6 D 6 2 E 8 W

V
p √ p √ p √ p √
24. The p
numbers x,py and z are given by x = 12 − 3 7 − 12 + 3 7, y = 7 − 4 3 − 7 + 4 3 and
√ √
z = 2 + 3 − 2 − 3.
What is the value of x yz ?
A 1 B –6 C –8 D 18 E 12
25. Two circles of radius 1 are such that the centre of each circle lies on the
other circle. A square is inscribed in the space between the circles.
What is the area of the square?
√ √ √
7 7 √ 5
A 2− B 2+ C 4− 5 D 1 E
2 2 5

© UK Mathematics Trust 2019 第２０页

www.ukmt.org.uk
 唯寻国际教育

MT
UK
MT

UK
UKMT

United Kingdom
Mathematics Trust

Senior Mathematical Challenge

2 – 5 November 2020
Organised by the United Kingdom Mathematics Trust

supported by

Candidates must be full-time students at secondary school or FE college.

England & Wales: Year 13 or below
Scotland: S6 or below
Northern Ireland: Year 14 or below

Instructions
1. Do not open the paper until the invigilator tells you to do so.
2. Time allowed: 90 minutes.
No answers, or personal details, may be entered after the allowed time is over.
3. The use of blank or lined paper for rough working is allowed; squared paper, calculators
and measuring instruments are forbidden.
4. Use a B or an HB non-propelling pencil. Mark at most one of the options A, B, C, D, E
on the Answer Sheet for each question. Do not mark more than one option.
5. Do not expect to finish the whole paper in the time allowed. The questions in this paper
have been arranged in approximate order of difficulty with the harder questions towards the
end. You are not expected to complete all the questions during the time. You should bear
this in mind when deciding which questions to tackle.
6. Scoring rules:
All candidates start with 25 marks;
0 marks are awarded for each question left unanswered;
4 marks are awarded for each correct answer;
1 mark is deducted for each incorrect answer (to discourage guessing).
7. Your Answer Sheet will be read by a machine. Do not write or doodle on the sheet except
to mark your chosen options. The machine will read all black pencil markings even if
they are in the wrong places. If you mark the sheet in the wrong place, or leave bits of eraser
stuck to the page, the machine will interpret the mark in its own way.
8. The questions on this paper are designed to challenge you to think, not to guess. You
will gain more marks, and more satisfaction, by doing one question carefully than by guessing
lots of answers. This paper is about solving interesting problems, not about lucky guessing.

Enquiries about the Senior Mathematical Challenge should be sent to:

UK Mathematics Trust, School of Mathematics, University of Leeds, Leeds LS2 9JT
T 0113 365 1121 enquiry@ukmt.org.uk
第２１页 www.ukmt.org.uk
Senior Mathematical Challenge 2 – 5 November 2020
唯寻国际教育
2020
1. What is the value of ?
20 × 20
A 10.1 B 5.5 C 5.1 D 5.05 E 0.55

2. What is the remainder when 1234 × 5678 is divided by 5?

A 0 B 1 C 2 D 3 E 4

3. A shape is made from five unit cubes, as shown.

What is the surface area of the shape?
A 22 B 24 C 26 D 28 E 30

4. The numbers p, q, r and s satisfy the equations p = 2, p × q = 20, p × q × r = 202 and

p × q × r × s = 2020.
What is the value of p + q + r + s?
A 32 B 32.1 C 33 D 33.1 E 34
√
5. What is 123454321?
A 1111111 B 111111 C 11111 D 1111 E 111

6. There are fewer than 30 students in the A-level mathematics class. One half of them play the piano,
one quarter play hockey and one seventh are in the school play.
How many of the students play hockey?
A 3 B 4 C 5 D 6 E 7

7. Official UK accident statistics showed that there were 225 accidents involving teapots in one year.
However, in the following year there were 47 such accidents.
What was the approximate percentage reduction in recorded accidents involving teapots from the first
year to the second?
A 50% B 60% C 70% D 80% E 90%

8. What is the largest prime factor of 1062 − 152 ?

A 3 B 7 C 11 D 13 E 17

9. In 2018, a racing driver was allowed to use the Drag Reduction System provided that the car was
within 1 second of the car ahead. Suppose that two cars were 1 second apart, each travelling at 180
km/h (in the same direction!).
How many metres apart were they?
A 100 B 50 C 10 D 5 E 1

10. Six friends Pat, Qasim, Roman, Sam, Tara and Uma, stand in a line for a photograph. There are three
people standing between Pat and Qasim, two between Qasim and Roman and one between Roman and
Sam. Sam is not at either end of the line.
How many people are standing between Tara and Uma?
A 4 B 3 C 2 D 1 E 0

© UK Mathematics Trust 2020 第２２页

www.ukmt.org.uk
Senior Mathematical Challenge 2 – 5 November 2020
唯寻国际教育
11. Two congruent pentagons are each formed by removing a right-angled isosceles
triangle from a square of side-length 1. The two pentagons are then fitted
together as shown.
What is the length of the perimeter of the octagon formed?
√ √
A 4 B 4+2 2 C 5 D 6−2 2 E 6

12. A three-piece suit consists of a jacket, a pair of trousers and a waistcoat. Two jackets and three pairs
of trousers cost £380. A pair of trousers costs the same as two waistcoats.
What is the cost of a three-piece suit?
A £150 B £190 C £200 D £228
E more information needed

13. The number 16! ÷ 2k is an odd integer. Note that n! = 1 × 2 × 3 × · · · × (n − 1) × n.

What is the value of k?
A 9 B 11 C 13 D 15 E 17

14. Diane has five identical blue disks, two identical red disks and one yellow disk.
She wants to place them on the grid opposite so that each cell contains exactly
one disk. The two red disks are not to be placed in cells that share a common
edge.
How many different-looking completed grids can she produce?
A 96 B 108 C 144 D 180 E 216

15. The shaded area shown in the diagram consists of the interior of a circle of radius
3 together with the area between the circle and two tangents to the circle. The
60◦
angle between the tangents at the point where they meet is 60°.
What is the shaded area?
√ √ √ √
9 3
A 6π + 9 3 B 15 3 C 9π D 9π + 4 3 E 6π + 4

16. Which diagram represents the set of all points (x, y) satisfying y 2 − 2y = x 2 + 2x?
A 𝑦 B 𝑦 C 𝑦 D 𝑦 E 𝑦

𝑥 𝑥 𝑥 𝑥 𝑥

3
17. The positive integers m, n and p satisfy the equation 3m + 1
= 17.
n+ p
What is the value of p?
A 2 B 3 C 4 D 6 E 9

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Senior Mathematical Challenge 2 – 5 November 2020
唯寻国际教育
18. Two circles C1 and C2 have their centres at the point (3,4) and touch a third circle, C3 . The centre of
C3 is at the point (0,0) and its radius is 2.
What is the sum of the radii of the two circles C1 and C2 ?
A 6 B 7 C 8 D 9 E 10

19. The letters p, q, r, s and t represent different positive single-digit numbers such that p − q = r and
r − s = t.
How many different values could t have?
A 6 B 5 C 4 D 3 E 2

1 √
20. The real numbers x and y satisfy the equations 4 y = √ and 9 x × 3 y = 3 3.
8( 2) x+2
What is the value of 5 x+y ?
√ √ 1
A 5 5 B 5 C 5 D E √1
5 5

21. When written out in full, the number (102020 + 2020)2 has 4041 digits.
What is the sum of the digits of this 4041-digit number?
A 9 B 17 C 25 D 2048 E 4041

22. A square with perimeter 4 cm can be cut into two

congruent right-angled triangles and two congruent
trapezia as shown in the first diagram in such a way
that the four pieces can be rearranged to form the
rectangle shown in the second diagram.

What is the perimeter, in centimetres, of this rectangle?

√ √ √ √
A 2 5 B 4 2 C 5 D 4 3 E 3 7

f (20) − f (2)
23. A function f satisfies y 3 f (x) = x 3 f (y) and f (3) , 0. What is the value of ?
f (3)
A 6 B 20 C 216 D 296 E 2023

24. In the diagram shown, M is the mid-point of PQ. The line PS 𝑅

bisects ∠RPQ and intersects RQ at S. The line ST is parallel to
PR and intersects PQ at T. The length of PQ is 12 and the length
𝑆
of MT is 1. The angle SQT is 120°.
𝑃 120◦
What is the length of SQ? 𝑀𝑇 𝑄
A 2 B 3 C 3.5 D 4 E 5

25. A regular m-gon, a regular n-gon and a regular p-gon share a vertex and
pairwise share edges, as shown in the diagram.
𝑚-gon 𝑛-gon
What is the largest possible value of p? 𝑝-gon
A 6 B 20 C 42 D 50 E 100

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