Write down all the 3-digit positive integers that are squares and whose individual digits are non-zero squares. Answer: Relay

Transcription

1 A Write down all the 3-digit positive integers that are squares and whose individual digits are non-zero squares.

2 A2 What is the sum of all positive fractions that are less than and have denominator 73?

3 A3 The volume of a cuboid is 60 cubic centimetres. Each of the values, in centimetres, of its length, width and height is an integer greater than one. What is the minimum possible surface area of the cuboid, in square centimetres?

4 A4 What is the sum of all the numbers that satisfy the property that the sum of the number and its square is equal to its cube?

5 A5 Triangle ABC is isosceles with CA = CB. The points D and E are the midpoints of C A and CB. The points B and E lie on the lines AG and DF so that AGFD is a parallelogram. The area of the parallelogram AGFD is seven times the area of the quadrilateral ABED. What is the value of area of the triangle DEC : area of the quadrilateral BGFE? Give your answer in the form a : b where a and b are positive integers with no common factors other than. C F E D G B A

6 A6 What is the sum of all the 2-digit primes neither of whose individual digits is prime?

7 A7 n 3 identical small cubes are stacked on a table to form a larger cube. The face of the larger cube that is on the table cannot be seen. All the other faces can be seen. For how many of the smaller cubes can exactly two faces be seen?

8 A8 What is the value of p + q so that x 4 4x 3 + 0x 2 + px + q can be expressed in the form (x 2 + ax + b) 2?

9 A9 The integers p, q, r and s are such that (i) x =, x = 2 and x = 3 are the solutions of the equation px 3 + qx 2 + r x + s = 0 and (ii) x = 4 is a solution of the equation px 3 + qx 2 + r x + s = 2. Which is the only integer that is a solution of the equation px 3 + qx 2 + r x + s = 20?

10 A0 The product of some positive integers is The largest integer is five times the smallest. What is the sum of these integers?

11 A What is ( x 6 y 6 ) ( x 6 + y 6 ) ( x 3 x 6 y 6 + y 3 ) ( x 3 + x 6 y 6 + y 3 ) in its simplest form?

12 A2 The lengths of the sides of a parallelogram are 7 and. The longer diagonal has length 4. What is the length of the shorter diagonal?

13 B a, b and c are different non-zero digits. Each of abc, acb and cba is a 3-digit square. What is the largest of these three squares?

14 B2 What is the value of ( ) ( ) ( )? Give your answer in the form a where a and b are positive b integers with no common factors other than.

15 B3 The volume of a cuboid is 90 cubic centimetres. Each of the values, in centimetres, of its length, width and depth is an integer greater than. What is the maximum possible total length, in centimetres, of the edges of the cuboid?

16 B4 What is the sum of all the numbers that satisfy the condition that three times the number equals the sum of its cube and twice its square?

17 B5 Triangle ABC is isosceles with CA = CB. The points D and E are the midpoints of C A and CB. AGFD is a parallelogram so that B and E lie on the lines AG and DF. The ratio of the area of the triangle CDE to the area of the quadrilateral BGFE is : 48. What is the value of C area of ABED : area of AGFD? Give your answer in the form a : b where a and b are positive integers with no common factors other than. F E D G B A

18 B6 A rectangle is said to be nearly square if and (i) its side lengths, in centimetres, are integers (ii) its longer sides are centimetre longer than its shorter sides. What is the area, in square centimetres, of the nearly square rectangle whose area is closest to 500 square centimetres?

19 B7 n 3 identical small cubes are stacked on a table to form a larger cube. The face of the larger cube that is on the table cannot be seen. All the other faces can be seen. For how many of the smaller cubes can exactly one face be seen? Give your answer in fully factorised form.

20 B8 The tangents from the point P to a circle have length 2. The longest distance from P to a point on the circle is 8. What is the radius of the circle?

21 B9 x and y are positive integers. How many solutions are there to the equation ( x 2 y 2) ( x 2 2y 2) = 45?

22 B0 a, b, c and d are the digits 2, 3, 4 and 5 in some order. What is the largest possible value of a b + c d?

23 B What is the sum of the 2-digit prime numbers each of whose digits is a non-zero triangular number?

24 B2 a + b = 0, b + c = 20, and c + a = 30. What is the value of abc + abc?