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2014 MATHEMATICS TEAM CHALLENGE SENIOR SECONDARY RELAY CONTEST Time: 1 hour Calculators may be used Each question is worth 5 points Total of 100 points R1 (5 points) (95 points remaining) The last four digits of my telephone number add to six. If none of the digits are zero, what is the probability that you can correctly guess the last four digits? R2 (5 points) (90 points remaining) Tom bought 100 plants for exactly $100. Roses cost $1.15 and ferns cost $0.95. How many roses did Tom buy? R3 (5 points) (85 points remaining) What is the smallest 4-digit number which has all even digits and is also a square number? R4 (5 points) (80 points remaining) A bowls club has 160 members. All members play in a tournament and when a member loses a set, he or she is out of the tournament. If there are no ties, determine the minimum number of sets that must be played to determine the overall winner. ` R5 (5 points) (75 points remaining) John has a one-in-four chance of hitting the bull’s eye when he tosses a dart. If he tosses four darts, what is the probability that he will hit the bull’s eye at least once? R6 (5 points) (70 points remaining) A palindrome is a number that reads the same forwards and backwards. For example 1331 is a palindrome. What is a four-digit palindrome that is also the same when looked at upside down and its reflection in a mirror is also the same? R7 (5 points) (65 points remaining) In a triangle, the lengths of the sides are, 10 cm, 13 cm and 13 cm, respectively. Find the length of the altitude to the shortest side. R8 (5 points) (60 points remaining) Sixty-four unit cubes are joined to form a 4 × 4 × 4 cube. Two stripes each two units wide are painted entirely around the cube in two directions as shown. How many of the sixty-four cubes received no paint? R9 (5 points) (55 points remaining) How many times do the hands of a clock cross in 12 hours? R10 (5 points) (50 points remaining) For what value of “n” (greater than 1) is the expression a perfect square? 1 + 2 + 3 + 4 + 5 + ……+ n R11 (5 points) (45 points remaining) A circle is inscribed in a regular hexagon. Next, a second circle is circumscribed about the hexagon. What is the ratio of the area of the outer circle to the area of the inner circle? R12 (5 points) (40 points remaining) How many four-digit numbers xyzw exist such that x is even, y is divisible by 5, z is prime, and w is odd? R13 (5 points) (35 points remaining) Find two numbers such that their difference, sum and product are in the ratio 1: 9 : 40, respectively. R14 (5 points) (30 points remaining) Ten people sitting in a circle shook hands with everyone except the people on either side of them. How many handshakes took place? R15 (5 points) (25 points remaining) 150 seats are arranged in a row. What is the least number of seats that must be occupied so the next person to be seated has to sit next to someone? R16 (5 points) (20 points remaining) If one cubic centimetre of sand contains 10,000 grains of sand. How many grains of sand will fit in rectangular room with dimensions length = 9 metres, width = 4.5 metres and height = 2.4 metres? R17 (5 points) (15 points remaining) A golfer hits a ball 250 metres but 2 degrees left of his intended line. How many metres from his intended spot did the golf-ball land? R18 (5 points) (10 points remaining) What is the only 4-digit perfect square that is a pair of two-digit square numbers written side by side? R19 (5 points) What is the value of x + y, given the following? (5 points remaining) 3 4 5 6 𝑥 √ × × × × ……… = 5 2 3 4 5 𝑦 R20 (5 points) (0 points remaining) Two numbers are amicable if each is the sum of the proper divisors of the other; 284 and 220 are amicable. [220 has proper divisors 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 and 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284] What number is amicable to 1184? SENIOR SECONDARY MATHS TEAM CHALLENGE (2014) Relay Answer Sheet Problem Answer 1 R 1 (5 points) 10 R 2 (5 points) R 3 (5 points) R 4 (5 points) Attempts √ or × 5 4 3 2 1 Score Progressive Score or 0.1 25 4624 159 Change R 5 (5 points) 175 or 0.68359. 1881, 8118, 8888, 8008 or 1111 will do. 12 cm 24 256 R 6 (5 points) R 7 (5 points) R 8 (5 points) Change R 9 (5 points) R 10 (5 points) R 11 (5 points) R 12 (5 points) Change R 13 (5 points) R 14 (5 points) R 15 (5 points) R 16 (5 points) Change R 17 (5 points) R 18 (5 points) R 19 (5 points) R 20 (5 points) Change Change Change Change Change Change 11 times n=8 4: 3 160 8 & 10 35 50 9.72 × 1011 8.73 metres 1681 99 1210 Total School:___________________________________ Team 1: Team 2: