GROUP ROUND INSTRUCTIONS

Transcription

1 GROUP ROUND INSTRUCTIONS Your team will have 40 minutes to answer 10 questions. Each team will have the same questions. Each question is worth 6 points. However, some questions are easier than others! You will have to decide your team s strategy for this group competition. Do you split up so that individuals work on a few questions each or do you work in pairs on a greater number of questions? Working all together on all the questions may well take too long. You decide! There is only one answer sheet per team. Five minutes before the end of the time you will be told to finalise your answers and write them on the answer sheet. This answer sheet is the only thing that will be marked.

2 Question 1 Calculate the value of

3 Question 2 The first term of a sequence is The value of the n th term of this sequence, for n > 1, is such that the sum of all the terms, from the 1 st to the n th inclusive, is half the value of the (n 1) th term. What is the lowest value of n for which the n th term is not an integer?

4 Question 3 For the year 2011 it can be noted that 2 = Find the number of positive four digit integers such that first digit = second digit + third digit + fourth digit.

5 Question If this sequence is continued, what will be the sum of the digits on the 2011 th row?

6 Question 5 Let f(x) = 3x + 4 f(f(x)) can be represented by f 2 (x) f(f(f(x))) can be represented by f 3 (x) and so on. e.g. f(5) = 19 and f 2 (5) = 61. Using the same notation, let g(x) = 2x + 5 and h(x) = 2x 5. Calculate the value of g 25 ( 2 1 ) + h 25 ( 2 1 ), giving your answer as a power of 2.

7 Question 6 A confectioner wishes to create a new spherical chocolate with a spherical chewy mint core. It is to be made with the centres of the spheres being coincident. In taste trials it was found that the most pleasing ratio of mint to chocolate was 1 : 1. Let the radius of the mint centre = r. Let the thickness of the chocolate coating = c. What is the ratio r : c? Give your answer in the form 1 : x, where x is to be given in exact form.

8 Question 7 P Q C B D A In the diagram shown, AC is the diameter of the circle and is of length 20 cm. Triangle AQB and triangle APD are congruent. Lines QB and PD both pass through the point C. Given that angle AQB is 18, calculate the length of the minor arc CD as a multiple of π.

9 Question 8 Find the smallest integer m that is one more than a multiple of each of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12.

10 Question 9 Diophantus of Alexandria was a Greek mathematician and is sometimes called the Father of Algebra. It was in a copy of his book Arithmetica that Pierre de Fermat later famously wrote that he had found a truly marvellous proof that x n + y n = z n has no integer solutions for n > 2, i.e. Fermat s Last Theorem. Although Diophantus was the first Greek mathematician who recognized fractions as numbers, Diophantine Equations are algebraic equations with integer coefficients for which integer solutions are sought is the 400 th anniversary of the birth of the English mathematician John Pell. Pell s Equations are Diophantine equations of the form x 2 ny 2 = 1 where n is a fixed, nonsquare integer and x and y are integers to be found. A trivial solution to a Pell s Equation will always be x = 1 and y = 0. Infinitely many solutions of these type of equations exist and give good rational approximations of the form y x to the square root of n. Consider the Pell s Equation x 2 2y 2 = 1. The smallest non-trivial positive values of x and y that satisfy this equation are x = 3 and y = 2. Find the next smallest. Give your answer in the form y x.

11 Question 10 The base of a right pyramid is a regular hexagon of side 2 cm. The total surface area of the pyramid is A cm 2 and the volume is V cm 3. The pyramid has proportions such that A = 3V. Find the height of the pyramid.

12 Group answer sheet Team number. Team name Value of expression = 2. Lowest value of n = 3. Number of positive four digit integers = 4. Sum of the digits = 5. Value of the function = 6. Ratio r : c = 1 : 7. CD = cm 8. m = 9. y x = 10. Height = cm Award 6 points for each correct answer. TOTAL SCORE =