By Titu Andreescu

ISBN-10: 0883858177

ISBN-13: 9780883858172

ISBN-10: 0883858193

ISBN-13: 9780883858196

Publication by means of Andreescu, Titu, Feng, Zuming, Loh, Po-Shen

**Read or Download USA and International Mathematical Olympiads, 2003 PDF**

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**Extra info for USA and International Mathematical Olympiads, 2003**

**Sample text**

Rearranging equation ᭺ 2: Substituting for x in equation ᭺ (7 – 2y)2 + y 2 = 10 Multiplying out the (7 – 2y) # (7 – 2y) gives 49 – 14y – 14y + 4y 2 = 49 – 28y + 4y 2, so the equation is 49 – 28y + 4y 2 + y 2 = 10. 30 This is rearranged to give 0 0 0 0 A quadratic in y which you can now solve using factorisation or the formula. 6 or x = 1, y = 3. Always substitute into the linear equation. Substituting in the quadratic will give you extra answers which are not correct. For the routine questions (numbers 1 and 5), solve the pairs of simultaneous equations, and if possible use a calculator or a computer algebra system to check your answers.

This fact allows you to find the equation of a straight line from first principles. 3 Find the equation of the straight line with gradient 2 through the point (0, –5). 11. The gradient of the line joining (0, –5) to (x, y) is given by gradient = y - ]- 5g y + 5 = . x-0 x Since we are told that the gradient of the line is 2, this gives y+5 =2 x & y = 2x - 5. Since (x, y) is a general point on the line, this holds for any point on the line and is therefore the equation of the line. The example above can easily be generalised (see pages 46 to 47) to give the result that the equation of the line with gradient m cutting the y axis at the point (0, c) is y = mx + c.

Find the area of the quadrilateral. 8 A sum of £5 000 is invested and simple interest is paid at a rate of 8%. (i) (ii) Adding the interest to the sum invested gives the amount of the investment after a period of time. (iii) 9 C1 2 Exercise 2C Calculate the interest received after 1, 2, and 3 years, and hence sketch the graph of interest against time, and find its equation. Use the equation to find the length of time for which the money must be invested for the total interest to reach £2500. Sketch the graph of amount against time and find its equation.