Canadian Open Mathematics Challenge

The Canadian Mathematical Society in collaboration with The Center for Education in Mathematics and Computing The Second Canadian Open Mathematics Challenge Wednesday, November 26, 1997 Examination Paper c Canadian Mathematical Society 1997 Part A Note: All questions in part A will be graded out of 5 points. 1. In triangle ABC , \A equals 120 degrees. A point D is inside the triangle such that \DBC = 2
\ABD and \DCB = 2
\ACD. DeterB mine the measure, in degrees, of \BDC . 2. Solve the following system of equations: A D C 3 xy2 = 108; xy = 1010 : 3. Determine all points on the straight line which joins (,4; 11) to (16; ,1) and whose coordinates are positive integers. 4. Given three distinct digits a; b and c, it is possible, by choosing two digits at a time, to form six two-digit numbers. Determine all possible sets fa; b; cg for which the sum of the six two-digits numbers is 484. 5. Two cubes have their faces painted either red or blue. The rst cube has ve red faces and one blue face. When the two cubes are rolled simultaneously, the probability that the two top faces show the same colour is 21 . How many red faces are there on the second cube? 6. The triangle ABC has sides AB = 137; AC = 241, and BC = 200. There is a point D, on BC , such that both incircles of triangles ABD and ACD touch AD at the same point E . Determine the length of CD. A E B D C 7. Determine the minimum value of f (x) where f (x) = (3 sin x , 4 cos x , 10)(3 sin x + 4 cos x , 10): 8. An hourglass is formed from two identical cones. Initially, the upper cone is lled with sand and the lower one is empty. The sand ows at a constant rate from the upper to the lower cone. It takes exactly one hour to empty the upper cone. How long does it take for the depth of sand in the lower cone to be half the depth of sand in the upper cone? (Assume that the sand stays level in both cones at all times.) Part B Note: All questions in part B will be graded out of 10 points. 1. The straight line l1 with equation x , 2y +10 = 0 meets the circle with equation x2 + y2 = 100 at B in the rst quadrant. A line through B , perpendicular to l1 cuts the y-axis at P (0; t). Determine the value of t. 2. Consider the ten numbers ar; ar2; ar3;


; ar10. If their sum is 18 and the sum of their reciprocals is 6, determine their product. B 3. In an isosceles right-angled triangle AOB , points P; Q and S are chosen on sides OB; OA and AB respectively such that a S square PQRS is formed as shown. If the P lengths of OP and OQ are a and b respectively, and the area of PQRS is 52 that of a triangle AOB , determine a : b. R Ob Q 4. Find all real values of x; y and z such that pyz = 42 x,p y , pxz = 6 z , xy = ,30: A