19.

The diagram shows a small regular octagram (an eight-sided star)

MT
UK
surrounded by eight squares (dark grey) and eight kites (light

UK
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grey) to make a large regular octagram. Each square has area 1.
What is the area of one of the light grey kites?
21 11 UKMT
A 2 B 2+1 C D 4 2−3 E
8 4
UK SENIOR MATHEMATICAL CHALLENGE
20. Positive integers x and y satisfy the equation x − 11 = y.
x Tuesday 8 November 2011
What is the maximum possible value of ?
y Organised by the United Kingdom Mathematics Trust
A 2 B 4 C 8 D 11 E 44
and supported by
21. Each of the Four Musketeers made a statement about the four of them, as follows.
d'Artagnan: “Exactly one is lying.”
Athos: “Exactly two of us are lying.”
Porthos: “An odd number of us is lying.”
Aramis: “An even number of us is lying.”
How many of them were lying (with the others telling the truth)? RULES AND GUIDELINES (to be read before starting)
A one B one or two C two or three D three E four 1. Do not open the question paper until the invigilator tells you to do so.
22. In the diagram, ∠ABE = 10°; ∠EBC = 70°; ∠ACD = 50°; A 2. Use B or HB pencil only. Mark at most one of the options A, B, C, D, E on the
∠DCB = 20°; ∠DEF = α. E Answer Sheet for each question. Do not mark more than one option.
Which of the following is equal to tan α? 3. Time allowed: 90 minutes.
α
tan10° tan20° tan10° tan20° tan10° tan50° D F No answers or personal details may be entered on the Answer Sheet after the 90
A B C 10° 50° minutes are over.
tan50° tan70° tan70°
70° 20° 4. The use of rough paper is allowed.
tan20° tan50° tan10° tan70° B C
D E Calculators, measuring instruments and squared paper are forbidden.
tan70° tan50°
5. Candidates must be full-time students at secondary school or FE college, and must
23. What is the minimum value of x2 + y2 + 2xy + 6x + 6y + 4? be in Year 13 or below (England & Wales); S6 or below (Scotland); Year 14 or
A −7 B −5 C −4 D −1 E 4 below (Northern Ireland).
6. There are twenty-five questions. Each question is followed by five options marked
24. Three circles and the lines PQ and QR touch as shown. P A, B, C, D, E. Only one of these is correct. Enter the letter A-E corresponding to
The distance between the centres of the smallest and the correct answer in the corresponding box on the Answer Sheet.
the biggest circles is 16 times the radius of the smallest 7. Scoring rules: all candidates start out with 25 marks;
circle. What is the size of ∠PQR? Q
0 marks are awarded for each question left unanswered;
A 45° B 60° C 75° D 90° E 135°
4 marks are awarded for each correct answer;
R 1 mark is deducted for each incorrect answer.
8. Guessing: Remember that there is a penalty for wrong answers. Note also that
25. A solid sculpture consists of a 4 × 4 × 4 cube with a 3 × 3 × 3 later questions are deliberately intended to be harder than earlier questions. You
cube sticking out, as shown. Three vertices of the smaller cube lie are thus advised to concentrate first on solving as many as possible of the first 15-
on edges of the larger cube, the same distance along each. 20 questions. Only then should you try later questions.
What is the total volume of the sculpture?
The United Kingdom Mathematics Trust is a Registered Charity.
A 79 B 81 C 82 D 84 E 85
http://www.ukmt.org.uk
1. Which of the numbers below is not a whole number? 10. A triangle has two edges of length 5. What length should be chosen for the third
2011 + 0 2011 + 1 2011 + 2 2011 + 3 2011 + 4 side of the triangle so as to maximise the area within the triangle?
A B C D E
1 2 3 4 5 A 5 B 6 C 5 2 D 8 E 5 3
2. Jack and Jill went up the hill to fetch a pail of water. Having filled the pail to the 11. PQRSTU is a regular hexagon and V is the midpoint of PQ. P V Q
full, Jack fell down, spilling 23 of the water, before Jill caught the pail. She then What fraction of the area of PQRSTU is the area of triangle STV ?
tumbled down the hill, spilling 25 of the remainder. 1 2 1 2 5 U R
What fraction of the pail does the remaining water fill? A B C D E
4 15 3 5 12
11 1 4 1 1 T S
A B C D E
15 3 15 5 15
12. The primorial of a number is the product of all of the prime numbers less than or
3. The robot Lumber9 moves along the number line. Lumber9 starts at 0, takes 1 step
equal to that number. For example, the primorial of 6 is 2 × 3 × 5 = 30. How
forward (to 1), then 2 steps backward (to −1), then 3 steps forward, 4 steps
many different whole numbers have a primorial of 210?
backward, and so on, moving alternately forwards and backwards, one more step
each time. At what number is Lumber9 after 2011 steps? A 1 B 2 C 3 D 4 E 5
A 1006 B 27 C 11 D 0 E −18 13. The diagram represents a maze. Given that you can only move
2011 horizontally and vertically and are not allowed to revisit a
4. What is the last digit of 3 ? square, how many different routes are there through the maze?
A 1 B 3 C 5 D 7 E 9 A 16 B 12 C 10 D 8 E 6
5. The diagram shows a regular hexagon inside a rectangle.
What is the sum of the four marked angles?
A 90° B 120° C 150° D 180° E 210° 14. An equilateral triangle of side length 4 cm is divided into smaller equilateral
triangles, all of which have side length equal to a whole number of centimetres.
Which of the following cannot be the number of smaller triangles obtained?
A 4 B 8 C 12 D 13 E 16
6. Granny and her granddaughter Gill both had their birthday yesterday. Today,
Granny's age in years is an even number and 15 times that of Gill. In 4 years' time 15. The equation x2 + ax + b = 0, where a and b are different, has solutions x = a and
Granny's age in years will be the square of Gill's age in years. How many years x = b. How many such equations are there?
older than Gill is Granny today? A 0 B 1 C 3 D 4 E an infinity
A 42 B 49 C 56 D 60 E 64
16. PQRS is a rectangle. The area of triangle QRT is 15 of the Q R
7. Two sides of a triangle have lengths 4 cm and 5 cm. The third side has length x cm, area of PQRS, and the area of triangle TSU is 18 of the T
where x is a positive integer. How many different values can x have? area of PQRS. What fraction of the area of rectangle
A 4 B 5 C 6 D 7 E 8 PQRS is the area of triangle QTU ?
27 21 1 19 23 P
U S
8. A 2 × 3 grid of squares can be divided into A B C D E
1 × 2 rectangles in three different ways. 40 40 2 40 60
17. Jamie conducted a survey on the food preferences of pupils at a school and
How many ways are there of dividing the bottom shape into
discovered that 70% of the pupils like pears, 75% like oranges, 80% like bananas
1 × 2 rectangles?
and 85% like apples. What is the smallest possible percentage of pupils who like
A 1 B 4 C 6 D 7 E 8 all four of these fruits?
A at least 10% B at least 15% C at least 20%
9. Sam has a large collection of 1 × 1 × 1 cubes, each of which is either red or
yellow. Sam makes a 3 × 3 × 3 block from twenty-seven cubes, so that no cubes D at least 25% E at least 70%
of the same colour meet face-to-face. 1 1 1
What is the difference between the largest number of red cubes that Sam can use 18. Two numbers x and y are such that x + y = 20 and + = . What is the
x y 2
and the smallest number? value of x2y + xy2?
A 0 B 1 C 2 D 3 E 4 A 80 B 200 C 400 D 640 E 800