| 1. (1) given x = (1 + 1/n)n, y = (1 + 1/n)n+1, show that xy = yx. (2) Show that 12 - 22 + 32 - 42 + ... + (-1)n+1n2 = (-1)n+1(1 + 2 + ... + n). |
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| 2. Given the points A (0, 1), B (0, 0), C (1, 0), D (2, 0), E (3, 0), F (3, 1). Show that angle FBE + angle FCE = angle FDE. |
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| 3. All coefficients of the polynomial p(x) are non-negative and none exceed p(0). If p(x) has degree n, show that the coefficient of xn+1 in p(x)2 is at most p(1)2/2. |
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| 4. What is the maximum possible value for the sum of the absolute values of the differences between each pair of n non-negative real numbers which do not exceed 1? |
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| 5. AB is a diameter of a circle. X is a point on the circle other than the midpoint of the arc AB. BX meets the tangent at A at P, and AX meets the tangent at B at Q. Show that the line PQ, the tangent at X and the line AB are concurrent. |
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| 6. What is the largest integer n which cannot be represented as 8a + 15b with a and b non-negative integers? |
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| 7. Bus A leaves the terminus every 20 minutes, it travels a distance 1 mile to a circular road of length 10 miles and goes clockwise around the road, and then back along the same road to to the terminus (a total distance of 12 miles). The journey takes 20 minutes and the bus travels at constant speed. Having reached the terminus it immediately repeats the journey. Bus B does the same except that it leaves the terminus 10 minutes after Bus A and travels the opposite way round the circular road. The time taken to pick up or set down passengers is negligible. A man wants to catch a bus a distance 0 < x < 12 miles from the terminus (along the route of Bus A). Let f(x) the maximum time his journey can take (waiting time plus journey time to the terminus). Find f(2) and f(4). Find the value of x for which f(x) is a maximum. Sketch f(x). |
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To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
15 June 2002
Last corrected/updated 1 Mar 03