2010 Sun Life Financial Canadian Open Mathematics Challenge – PART B
5:00 PM
Muhammad Yusuf
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1. In each part of this problem,
each of the variables in the grid is to be replaced with an integer. The sum of
the integers in a row is given to the right of the row. The sum of the integers
in a column is given at the bottom of the column. For example, from the grid to
the right we can conclude that X+13 = 30, Y +11 = 23, X+Y = 29, and 13 + 11 = 24.
(a) Determine the value of C.
(b) Determine the value of n, the sum of the integers in the second column.
(c) Determine the value of P + Q.
2. The parabola with equation y = x2 - 4x + 12
intersects the line with equation y = -2x + 20 at points A and B.
(a) Determine the coordinates of
the points A and B.
(b) Determine the coordinates of
the midpoint, M, of the segment AB.
(c) A line parallel to the line
with equation y = -2x+20 intersects the parabola at distinct points P(p, p2 - 4p+12) and Q(q, q2 - 4q +12). Prove that p+q = 2.
(d) Point N is the midpoint of PQ.
Explain why line segment MN is vertical.
3. In the diagram, the circle has
centre O, diameter AC, and radius 1. A chord is drawn from A to an arbitrary point B (di_erent from A) on the circle and
extended to the point P with BP = 1. Thus P can take many positions.
Let S be the set of points P.
(a) Let U be a point in S for which UO is perpendicular to AC.
Determine the length of UO.
(b) Let V be a point in S for which V C is perpendicular to AC. Determine the length of V C.
(c) Determine whether or not
there is a circle on which all points of S lie.
4. Let ⎿x⏌ denote the greatest integer less
than or equal to x. For example, ⎿3:1⏌ = 3 and ⎿-1,4⏌ = -2. For x > 0, define
For Example,
(a) Determine all x > 0 so that f(x) = x.
(b) Suppose that x = a/(a + 1) for some positive integer a > 1.
Prove that x ≠ f(x), but that f(x) = f(f(x)).
(c) Prove that there are
in_nitely many rational numbers u so that
- 0 < u < 1,
- u, f(u) and f(f(u)) are all distinct, and
- f(f(u)) = f(f(f(u))).
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