A1 What is the value of 1 2 + 2 3 + 3 4 + 4 5 5 2?
A2 What is the value of k? 11 2 22 2 33 2 = 66 2 k.
A3 The four-digit integers 5634 and 6435 share the following two properties: (i) they consist of four consecutive digits in some order; (ii) they are divisible by 3. How many such four-digit integers are there?
A4 Shona, Jo and Bev run round a 400 m athletic track in the same direction. Shona runs at a constant speed of 3 m/s, Jo runs at a constant speed of 4 m/s whilst Bev runs at a constant speed of 5 m/s. They start together. How many metres in total do the three of them run before they are together again?
A5 The operation A B is defined by A B = 20A B 7AB. What is the value of 2 ( 0 (1 7) )?
A6 For how many integers n, is it true that 3 47 < 1 n < 4 37?
A7 In the triangle ABC, AB has length 7, BC has length 8 and C A has length 9. D is the midpoint of BC. What is the length of AD?
A8 For a > 0, the triangle with vertices at the points (0, a), (2a, 10) and (10, 0) has area 4a 2. What is the value of a?
A9 A circle with centre A and radius 12 has diameter BC. A second circle with diameter AC is drawn. The tangent shown from B to this circle meets circle centre A at the point D. What is the length of BD? Give your answer in fully simplified form. B A D C
A10 The positive integers x and y are such that 20x + 17y = 2017. What is the maximum value of x + y?
A11 abc is a three-digit prime. Its individual digits a, b and c are different primes. Write down all such three-digit primes abc.
A12 For each value of x, F(x) is defined to be the minimum value of x + 5, 2x 4 and 5x 7. What is the maximum value of F(x)?
B1 What is the value of 1 1 + 2 2 + 3 3 + 4 4 + 5 5 7 2?
B2 What is the value of k? 88 4 22 2 44 2 = 2k.
B3 For how many integer values of x is 24 x 6 an integer?
B4 A match takes place between two teams, each of 11 players. There are three referees. Before the match, each player exchanges handshakes with each of their opponents and each of the referees. What was the total number of handshakes?
B5 What is the sum of all digits that cannot be the units digit of the square of an integer?
B6 For which integer values of x is x 4 40x 2 + 144 negative?
B7 In the triangle ABC, AB has length 20, BC has length 11 and CA has length 13. The line through A perpendicular to BC meets the line through B and C at D. What is the length of AD?
B8 The diagram shows three congruent rectangles of height 5 and breadth 13. As shown, two pairs of rectangles share a vertex. Two rectangles are placed so that a vertex of each lies on the long side of the third rectangle. What is the area of the region where all three rectangles overlap?
B9 A circle centre A has diameter BC. A second circle with diameter AC is drawn. The tangent shown from B to this circle meets circle centre A at the point D. BD has length 16. What is the length of the radius of the circle centre A? Give your answer in fully simplified form. B A D C
B10 The sum of all two-digit primes, in which all the individual digits are prime, is A. The sum of all single-digit primes is B. The integer A + B is the product of two primes p and q. What are p and q?
B11 a, b, c, d and e are nonzero digits. The five-digit integers abcde and edcba satisfy the multiplication sum shown alongside. a e b d c c d e 4 b a What is the digit c?
B12 For every value of x, f (x) satisfies the equation f (x) + 3 f (1 x) = 2x. What is the value of f (25)?