D - E - F - G (1967 Jr.) Given that then find the number of real solutions ( ) of this equation.
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1 D - E - F - G (1975 Jr.) Given and. Two circles, with centres and, touch each other and also the sides of the rectangle at and. If the radius of the smaller circle is 2, then find the radius of the larger circle. 2. (1967 Jr.) Given that then find the number of real solutions ( ) of this equation. 3. (1968 Jr.) A square is inscribed in an equilateral triangle with one side along the base of the triangle. Find the ratio of the area of the square to the area of the triangle. 4. (1969 Jr.) 105 quarters are lying on a flat surface with their edges in contact. They just fit in an equilateral triangle with the inside perimeter of the frame of 84 cm. Find the radius of a quarter. 5. (1956 Sen.) is formed by three tangents to circle and. Find. (Generalize this proof) 6. (1958 Sen.) If and then: (a) has no min or max value (b) max value of is 1 (c) min value of is 1 (d) max value of is 4 (e) min value of is 4 7. (1958 Sen.) The area of a circle inscribed in a regular hexagon is. Find the area of the hexagon. A X D P M B 40 S A E R T O Y F B C
2 8. (1979 Sen.) If, ( ) value of ( ). ( ) ( ) ( ) ( ) then find the minimum 9. (1960 Sen.) Let be any two odd numbers with. Find the largest integer which divides all numbers of form. 10. (1996 Sen.) If the sum is a perfect square and if is less than, then find possible values of.
3 F - G (1986 AIME) Find the sum of the roots for. 2. (1986 AIME) If and then find ( ). 3. (1970 DesCartes) An ellipse has its centre at the origin, its focus on the axis, and its major axis three times as long as its minor axis. Given that the ellipse passes through the point ( ), find its equation. 4. (1972 DesCartes) Lines and meet at so that, and. A circle is drawn to pass through. Find the radius of the circle. 5. (1972 DesCartes) Simplify ( ). 6. (1972 DesCartes) Determine the coordinates of a point, other than the origin, on the curve, where the tangent at which passes through ( ). 7. (1971 DesCartes) Evaluate the sum of the series ( ) ( ) ( ) ( ) ( ) 8. (1971 DesCartes) (a) Evaluate ( ) where and. (b) Deduce the value of. 9. (1973 DesCartes) Determine the equations of all the common tangents to and. 10. (1971 DesCartes) Prove that the equation has no solution in integers except.
4 D - E - G F (1970 Sen. Math) Simplify. 2. (1984 Euclid) Find the equation of the circle passing through the three points ( ) ( ) and ( ). 3. (1955 Sen. Math) Find the intersection of and. 4. (1955 Sen. Math) Find when. 5. (1955 Sen. Math) A 60 cm and 180 cm diameter poles are touching. Find the length of the shortest wire that will go around them. 6. (1955 Sen. Math) In circle with centre is produced so equals the radius of the circle. is drawn and extended to as shown. is drawn. Find the connection between and. A B y C D x O 7. (19 Sen. Math) Find the largest value of if. 8. (19 Sen. Math) What solutions are possible if a man spends a $10000 on steers at $250 and cows at $260? 9. (19 Sen. Math) Find the sum of the first 50 terms of sequence 10. (19 Sen. Math) Find the sum of the infinite series
5 F - G (1973 DesCartes) A geometric series, whose common ratio is real, has the sum of the third and fourth term equal to and the sum to infinity equal to. Find. 2. (1968 DesCartes) (a) Prove ( ) ( ). (b) Evaluate ( ) ( ) ( ) ( ). 3. (1969 DesCartes) The polynomial ( ) ( ) has factors and. What are and and what are the other factors? 4. (1968 DesCartes) A multiplicative group is made up of distinct items where is the identity element. If it is given that for prove that. Write a multiplication table for this group for. 5. (1969 DesCartes) In an isosceles triangle, lies in and lies in so that. Find. 6. (1973 DesCartes) Five points with integral abscissae and ordinates ( coordinates) are given in Cartesian plane, no three collinear. (a) How many line segments are determined by 5 points? (b) Prove that the midpoint of at least one of the line segments has integral coordinates. 7. (1972 DesCartes) Evaluate ( ) ( ) ( ) ( ) ( ). 8. (1974 DesCartes) Use induction to show is divisible by 9. (1975 DesCartes) Given the equation with rational coefficients has root, find and. 10. (1975 DesCartes) Given are sides of a triangle such that form an arithmetic sequence, show that form a geometric sequence.
6 D - E - F - G - 20 Math Contest Problems 1. Find an unending arithmetic sequence all of whose terms are relatively prime to 7, 11 and Express as a rational number. 3. If and then find and. 4. There are three pumps which may be run in either direction. If all run together in the same direction it takes hours to fill a given tank. If the first and third are filling the tank but the second is emptying it, the time to fill is hours. If first and second are emptying and third filling, the full tank empties in tank. hours. Find the times it would take each pump to fill the 5. [ ] denotes the greatest integer. If [ ] [ ] [ ] [ ], find. 6. Some unit cubes are assembled to form a larger cube then some of the faces of this larger cube are painted. When disassembled into unit cubes, 45 of these have no paint on any faces. How many faces of the larger cube were painted? 7. If, find the values of and that satisfy. (Lattice points) 8. The sum of 101 consecutive odd numbers is Find the smallest one. 9. If then find in terms of. 10. Find the sum of the numerical coefficients in the complete expansion of.
7 F - G (Math Contest Problems) Find if. 2. (Math Contest Problems) What is the probability that in drawing three cards from a deck without replacement one will obtain a spade, a queen, and a diamond in that order? 3. (Math Contest Problems) Given that find. 4. (Math Contest Problems) Determine all positive integer pairs of values of and for which. 5. (1968 DesCartes) Prove ( ). 6. (1968 DesCartes) Deduce from # 5 a formula for sum of ( ) ( ) ( ) ( ) ( ) 7. (1976 DesCartes) is a polynomial of degree 5 such that is divisible by and is divisible by. Find. 8. (1976 DesCartes) The first three terms of a binomial expansion are. Find the fourth term. 9. (1977 DesCartes) Identify. 10. (1977 DesCartes) Show that point lies on # (1977 DesCartes) Find the equation of tangent to # 9 at # 10.
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