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2013 MATHEMATICS TEAM CHALLENGE SENIOR SECONDARY TEAMS CONTEST Time: 45 minutes Calculators may be used Each question is worth 10 points Total of 100 points T1 (10 points) The picture shows 25 dots evenly spaced in a five by five grid arrangement. How many squares can be drawn using the dots as vertices? One example is drawn below. T2 (10 points) The coefficients (a, b and c) for the quadratic expression below are obtained by rolling a standard six sided die. ax2+ bx + c Some of the quadratic expressions can be factorised. For example 1x2 + 5x + 6 = (x + 2)(x + 3) What is the probability of obtaining a quadratic expression which can be factorised? An example of a factorised quadratic is: 4x2 + 6x + 4 = 2(2x2 + 3x + 2) Note: Putting a 1 outside the brackets is not a factorised quadratic. eg 1(x2 + x + 1) T3 (10 points) Binary uses only 0’s and 1’s. Powers of 2 Binary decimal 25 1 24 0 23 1 22 0 21 0 20 1 2-1 0 2-2 1 2-3 0 2-4 0 The binary decimal (101001.0100)2 = 1 × 25 + 1 × 23 + 1 × 20 + 1 × 2-2 = 41.25 What is the binary representation of 3.1415? This is π to four decimal places. T4 (10 points) Permutable primes remain prime when their digits are jumbled. For example 13 is a permutable prime. Find one three-digit permutable prime. T5 (10 points) A packaging firm makes cardboard boxes according to the following specifications. Length + width + height = 24 cm and each box has integral side lengths. How many different box sizes can the packaging firm provide for customers? The size in this case refers to the volume of the box. [Note: a box with length 5 cm, width 9 cm and height 10 cm has the same volume as a box with length 10 cm, width 5 cm and height 9 cm] T6 (10 points) The equation 91 = 12 + 22 + 32 + 42 + 52 + 62 shows that 91 is the sum of the first six squares, so 91 is a square pyramidal number. It is also a triangular number since. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 = 91 Fifty-five is another two-digit number that is both a square pyramidal number and a triangular number. To date, there is one other number that is both a square pyramidal number and a triangular number. This number has six digits. What is this number? T7 (10 points) Five two metre diameter spheres are stacked as shown below. What is the distance between the centres of the top and bottom spheres? T8 (10 points) The crescent moon is made of two circles. A is the centre of the bigger circle. The width of the crescent between points B and C is 18 cm. The width between points D and E is 10 cm. What is the radius of the bigger circle? T9 (10 points) A certain type of screw comes in packs of 6, 9 or 20. Determine the largest number of screws that you cannot buy when combining various packets. T10 (10 points) For what value of “n” is the following equation correct? 1 1+√ + 3 1 √3+√ + 5 1 √5+√7 + 1 √7+√ + ⋯…+ 9 1 √2𝑛−1 + √2𝑛+1 = 1,000 School Name:________________________ Team 1: Team 2: 2013 MATHS TEAM CHALLENGE SENIOR SECONDARY TEAM EVENT ANSWER SHEET Question T1. (10 points) Answers Points T2. (10 points) T3. (10 points) T4. (10 points) T5. (10 points) T6. (10 points) T7. (10 points) T8. (10 points) T9. (10points) T10. (10 points) Total /100 2013 MATHS TEAM CHALLENGE SENIOR SECONDARY TEAM EVENT ANSWER SHEET Question T1. (10 points) T2. (10 points) Answers 52 13 , 216 54 T3. (10 points) T4. (10 points) 50 or 0.24074 11.001001000011 113, 199, or 337 T5. (10 points) 48 T6. (10 points) 208,335 T7. (10 points) 4√ 2 3 cm or 3.266 m T8. (10 points) 50 cm T9. (10 points) 43 T10. (10 points) n = 2,002,000 Points