Shuttle. Senior Team Maths. Challenge 2017/18. National Final UKMT 2017/18 UKMT UKMT UKMT. United Kingdom Mathematics Trust. Pass on the value of a b.

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1 The line x + 2y = T 3 meets the curve x2 + 2xy = T at the point (a, b). In the diagram, three vertices of the parallelogram ABCD lie on a circle so that BC and DC are tangents to the circle. A1 Pass on the value of a b. the value of k. on Pass 10k. is ABC of size The A3

2 Total surface area of a cone = πr(r + h 2 + r 2 ). Total surface area of a cylinder = 2πrh + 2πr 2. Note: Total surface area of a sphere = 4πr cm, and the total surface area of a cylinder of radius 10 cm and height 8 cm. The surface area of a sphere of radius r cm is T times the sum of the total surface area of a cone of radius 5 cm and height The shaded area enclosed between the positive x-axis, the positive y-axis and the lines 2x + 2y = T and x + 3y = T is b. A2 Write down the value of r. Pass on the value of b. A4

3 B1 A circle is drawn through all six vertices of a regular hexagon. The number area of hexagon area of circle may be written in the form 3 3 aπ. Pass on the value of a. B3 The mean, median, mode and range of four numbers are all T. Pass on the product of the four numbers.

4 B2 The straight lines y = x + T and y = x T are both tangents to the same circle. The area of the circle is πa. Pass on the value of a. Calculate the value of B4 T ( ). Write down your answer in the form a + b c, where a, b and c are positive integers, b and c have no common factors other than 1, and b < c.

5 Pass on the value of C C3 a, b, c are three primes such that a b c, and a+b+c = 1 2 T +3. N is the number of different ways in which three such primes may be chosen. Pass on the value of N.

6 Kingdom United than 1. Write down the value of b a. b where a and b are positive integers with no common factor other The probability that this team comprises two boys and two girls is a A group of children comprises T boys and three girls. A team of four is selected at random from the group. The perimeter of the rectangle is P cm, where P is an integer. Pass on the value of P. The lengths of the sides of the rectangle are whole numbers of centimetres. A circle is drawn through the vertices of a rectangle. The area of the circle is Tπ cm 2. C4 C2

7 The integer k is given by D1 k = ( 8 ) ( 81 ) ( 25 ) ( ) 2 3 ( ) Pass on the value of k. D3 In a certain arithmetic sequence the ratio of the third term of the sequence to the first term is equal to the ratio of the seventh term of the sequence to the third term. All the terms of the sequence are different. The first term of the sequence is T. Pass on the value of the third term. Note that an arithmetic sequence is a sequence in which the difference between consecutive terms is always the same.

8 D2 All the members of the Camford Puzzle Club were asked whether they liked or Physics. Twice as many people said they only liked as only liked Physics. Twice as many people said they only liked Physics as liked both subjects. T people said they liked neither subject. Three-fifths of the members liked. Pass on the number of members of the Camford Puzzle Club. D4 The volume of a jug is 480 cm 3 and its total surface area is 2T cm 2. The volume of a similar but smaller jug is 3 5 T cm3. Write down the total surface area, in cm 2, of the smaller jug.

9 response sheet STMC Team number School name A1 B1 C1 D1 A2 B2 C2 D2 A3 B3 C3 D3 A4 B4 C4 D4 Bonus 3 Bonus 3 Bonus 3 Bonus 3 A total /15 B total /15 C total /15 D 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. Final score /60