1. Consider the sequence a<sub>0</sub>, a<sub>1</sub>, a<sub>2</sub>, ... which is an increasing sequence of real numbers (ie a<sub>n</sub> < a<sub>n+1</sub> for all non-negative integers n), defined by a<sub>n+1</sub> = 2<sup>n</sup> - 3a<sub>n</sub>, for n = 0, 1, 2, ...
(a) If the first term of the sequence is a<sub>0</sub> > 0, find expressions for a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub>, in terms of a<sub>0</sub>, and show that
a<sub>4</sub> = 2<sup>3</sup> - 3 * 2<sup>2</sup> + 3<sup>2</sup> * 2 - 3<sup>3</sup> + 3<sup>4</sup>a<sub>0</sub>
(b) By examining the pattern, show that a<sub>n</sub> = 2<sup>n</sup> / 5 + (-1)<sup>n</sup>(a<sub>0</sub> - 1 / 5)3<sup>n</sup>
(c) Show that a<sub>0</sub> = 1 / 5 produces such a sequence.
(d) By considering large values of n, with A = a<sub>0</sub> - 1 / 5 <> 0, show that a<sub>0</sub> = 1 / 5 is the only value of a<sub>0</sub> for which this sequence increases.
2. Consider the arc of the parabola y = x<sup>2</sup> for 0 <= x <= b, where b is a positive constant. Let the points A and P be, respectively, (0, a) and (t, t<sup>2</sup>), where a => 0 and 0 <= t <= b.
(a) Find AP<sup>2</sup> in terms of t.
(b) We seek to find the position of P such that the distance AP is minimised. Discuss the value of AP<sub>min</sub>, and the value of t for which it occurs, and how they vary depending on the values of a and b. (Hint: there are three cases to consider).
[I can make this question more directed if people need more help.]
3. Following an unfortunate accident involving a football in a Chemistry Laboratory at a certain co-educational school, each boy in year 12 was fined 30 cents, and each girl 2 cents. There were a total of 120 students in year 12, and the boys outnumber the girls. The money raised by the fines was enough to buy 12 new test tubes (costing 7 cents each), and some number of beakers (costing 90 cents each), with no money left over. We seek to find out how many boys there are in year 12, and how many new beakers were purchased.
---Note: For those who want a real challenge, try and solve this problem without looking at the parts that follow---
(a) If b, g and x are, respectively, the number of boys, girls and beakers, show that 1758 = 14g + 45x, g < 60.
(b) Explain why the number of new beakers purchased must have been an even number.
(c) If X = x / 2, and G = g / 3, discuss the range of possible values of X and G.
(d) Show that X - 6 is divisible by 7.
(e) Find b, g and x.
4. The circumcircle of triangle ABC is the circle that passes through all of its vertices.
(a) Without using the sine rule, show that the radius R of the circumcircle is given by
R = a / (2sin A) = b / (2sin B) = c / (2sin C), where a, b and c are the lengths of BC, CA and AB, respectively.
(b) Prove that sin<sup>2</sup>A + sin<sup>2</sup>B + sin<sup>2</sup>C <= 9 / 4
(c) Show that if the circumcircle of a triangle ABC has a radius of at most 1 cm, then the sum of the squares of the sides of the triangle does not exceed 9 cm<sup>2</sup>
5. If a, b, c, d, e and f are six positive real numbers, such that
a<sup>2</sup> + b<sup>2</sup> + c<sup>2</sup> = 16
d<sup>2</sup> + e<sup>2</sup> + f<sup>2</sup> = 49
ad + be + cf = 28
show that (a + b + c) / (d + e + f) = 4 / 7
[Hint available]
(a) If the first term of the sequence is a<sub>0</sub> > 0, find expressions for a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub>, in terms of a<sub>0</sub>, and show that
a<sub>4</sub> = 2<sup>3</sup> - 3 * 2<sup>2</sup> + 3<sup>2</sup> * 2 - 3<sup>3</sup> + 3<sup>4</sup>a<sub>0</sub>
(b) By examining the pattern, show that a<sub>n</sub> = 2<sup>n</sup> / 5 + (-1)<sup>n</sup>(a<sub>0</sub> - 1 / 5)3<sup>n</sup>
(c) Show that a<sub>0</sub> = 1 / 5 produces such a sequence.
(d) By considering large values of n, with A = a<sub>0</sub> - 1 / 5 <> 0, show that a<sub>0</sub> = 1 / 5 is the only value of a<sub>0</sub> for which this sequence increases.
2. Consider the arc of the parabola y = x<sup>2</sup> for 0 <= x <= b, where b is a positive constant. Let the points A and P be, respectively, (0, a) and (t, t<sup>2</sup>), where a => 0 and 0 <= t <= b.
(a) Find AP<sup>2</sup> in terms of t.
(b) We seek to find the position of P such that the distance AP is minimised. Discuss the value of AP<sub>min</sub>, and the value of t for which it occurs, and how they vary depending on the values of a and b. (Hint: there are three cases to consider).
[I can make this question more directed if people need more help.]
3. Following an unfortunate accident involving a football in a Chemistry Laboratory at a certain co-educational school, each boy in year 12 was fined 30 cents, and each girl 2 cents. There were a total of 120 students in year 12, and the boys outnumber the girls. The money raised by the fines was enough to buy 12 new test tubes (costing 7 cents each), and some number of beakers (costing 90 cents each), with no money left over. We seek to find out how many boys there are in year 12, and how many new beakers were purchased.
---Note: For those who want a real challenge, try and solve this problem without looking at the parts that follow---
(a) If b, g and x are, respectively, the number of boys, girls and beakers, show that 1758 = 14g + 45x, g < 60.
(b) Explain why the number of new beakers purchased must have been an even number.
(c) If X = x / 2, and G = g / 3, discuss the range of possible values of X and G.
(d) Show that X - 6 is divisible by 7.
(e) Find b, g and x.
4. The circumcircle of triangle ABC is the circle that passes through all of its vertices.
(a) Without using the sine rule, show that the radius R of the circumcircle is given by
R = a / (2sin A) = b / (2sin B) = c / (2sin C), where a, b and c are the lengths of BC, CA and AB, respectively.
(b) Prove that sin<sup>2</sup>A + sin<sup>2</sup>B + sin<sup>2</sup>C <= 9 / 4
(c) Show that if the circumcircle of a triangle ABC has a radius of at most 1 cm, then the sum of the squares of the sides of the triangle does not exceed 9 cm<sup>2</sup>
5. If a, b, c, d, e and f are six positive real numbers, such that
a<sup>2</sup> + b<sup>2</sup> + c<sup>2</sup> = 16
d<sup>2</sup> + e<sup>2</sup> + f<sup>2</sup> = 49
ad + be + cf = 28
show that (a + b + c) / (d + e + f) = 4 / 7
[Hint available]
(sorry to offtopic your thread)
