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2 Theoretical facts that you may find useful. Theorem 1: (Factor Theorem) Given a polynomial P, then P is divisible by x a if and only if P (a) = 0. The remainder of the division of P (x) by x a is P (a). Trigonometric formulae: sin(α ± β) = sin α cos β ± cos α sin β, cos(α ± β) = cos α cos β sin α sin β cos a cos b = sin b a sin a + b and cos a + cos b = cos a b cos a + b Double Angle Formulae: sin α = sin α cos α, cos α = cos α 1 = 1 sin α. Theorem 3: A polynomial P (x) = a 0 + a 1 x + a n x n has a rational zero, p/q (reduced form), if p divides a n and q divides a 0. Theorem 4: Quadratic formula and Viete s relations: the equation ax + bx + c = 0 has two zeros (x 1, x, real or complex) given by These zeros satisfy Viete s relations: x 1, = b ± b 4ac a. x 1 + x = b a, x 1x = c a. Theorem 5: Heron s formula: The area of a triangle is given by the formulae A = s(s a)(s b)(s c) = 1 4 (a b + b c + c a ) (a 4 + b 4 + c 4 ) where s = a+b+c and a, b and c are the sides of the triangle. Theorem 6: The Law of Sines in a triangle states that a sin A = b sin B = c sin A where A, B and C are the measurement of the interior angles of the triangle. Theorem 7: (Pythagorean Theorem:) In a right triangle with legs b, c and hypothenuse a, we have b + c = a. Theorem 8: Properties of logarithms: log a u = v u = a v log a u + log a v = log a uv, log a u v = v log a u II

3 Pre-calculus Problems 1. For two unique positive integers a and b, we have 017 = a + b 4. Also, a can be written as c 0 + c 1 b + c b + c 3 b 3 + c 4 b 4 where c 0, c 1, c, c 3 and c 4 are either 1, 0 or 1. What is the value of c 4 c 3 c c 1 c 0?. For a and b positive integers, with a not a perfect square, it is given that a b is a root of the quadratic equation x + ax b = 0 and a + b is a root of the quadratic equation x ax b = 0, what is a + b? 3. [ 1 ] Three angles α, β and γ satisfy the equalities tan α = x 1, tan β = x and tan γ = x 3, where x 1, x and x 3 are the roots of the cubic equation 5x 3 14x + x + 1 = 0. What is the value of tan(α + β + γ)? 4. It is known that 017 is the 306 th prime and it can be written as a sum of two triangular numbers (numbers of the form 1, 1 +, , etc.) in a unique way. For positive integers m, n, and k, with n > m, we have n j=m+1 j 3 = (m + 1) 3 + (m + ) n 3 = 017(017 k). Knowing that m + n is the smallest number with this property, find k. III

4 5. In the triangle ABC, AB = 51, BC = 5, AC = 53. Let D denote the midpoint of BC and let E denote the intersection of BC with the bisector of angle BAC. If the area of the triangle AED (in square units) is equal to 15x 4 what is x? (A) 1 (B) (C) 3 6. From the interior of an equilateral triangle ABC one takes a random point P. The probability that the angle AP B has measure more than 10 is equal to mπ 3 1, n 3 where the m is a positive rational number n written in reduced form. What is the value of 7m n? (A) 1 (B) (C) 3 7. [ ] The number of ordered pairs of positive real numbers (u, v) such that (u + iv) 017 = u iv is a number of three digits abc (written in base 10). What is a b? 8. [ 3 ] Positive integers a, b, and c are chosen so that a < b < c, and the system of equations x + y = 017 and y = x a + x b + x c has exactly one solution (x, y). The minimum value of c possible is a number of four digits d 1 d d 3 d 4 (written in base 10). What is d 4 + d 3 + 3d 5d 1? IV

5 9. [ 4 ] Two circles of radii R and r (R > r) are tangent to one another in such a way their common tangent lines are perpendicular (see the accompanying figure). Another circle of radius x is tangent to one of these tangent lines and tangent to both of the circles. The ratio R can be written as m + n for two x positive integers m and n. What is m + n? (A) 1 (B) (C) [ 5 ] Three real numbers x, y, and z satisfy log (x + y) = z log (x + y ) = z + 1 x + y = 6 xy > 4 What is the closest value to z?. V

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st Annual King s College Math Competition King s College welcomes you to this year s mathematics competition and to our campus. We wish you success in this competition and in your future studies. Instructions

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