Euclid Contest 2008 - CEMC - University of Waterloo

(b) Determine all values of k, with k ≠ 0, for which the parabola
has its vertex on the x-axis.
y = kx 2 + (5k + 3)x + (6k + 5)
6. (a) The function f(x) satisfies the equation f(x) = f(x − 1) + f(x + 1) for all values
of x. If f(1) = 1 and f(2) = 3, what is the value of f(2008)?
(b) The numbers a, b, c, in that order, form a three term arithmetic sequence (see
below) and a + b + c = 60.
The numbers a − 2, b, c + 3, in that order, form a three term geometric sequence.
Determine all possible values of a, b and c.
(An arithmetic sequence is a sequence in which each term after the first is
obtained from the previous term by adding a constant. For example, 3, 5, 7 is
an arithmetic sequence with three terms.
A geometric sequence is a sequence in which each term after the first is obtained
from the previous term by multiplying it by a constant. For example, 3, 6, 12 is
a geometric sequence with three terms.)
7. (a) The average of three consecutive multiples of 3 is a.
The average of four consecutive multiples of 4 is a + 27.
The average of the smallest and largest of these seven integers is 42.
Determine the value of a.
(b) Billy and Crystal each have a bag of 9 balls. The balls in each bag are numbered
from 1 to 9. Billy and Crystal each remove one ball from their own bag. Let b be
the sum of the numbers on the balls remaining in Billy’s bag. Let c be the sum of
the numbers on the balls remaining in Crystal’s bag. Determine the probability
that b and c differ by a multiple of 4.
8. (a) Points A, B, C, and D are arranged, as
shown, with AB parallel to DC and P the
point of intersection of AC and BD. Also,
∠ACB = 90 ◦ , AC = CB, AB = BD = 2.
Determine the measure of ∠DBC.
A
D
P
C
2
B
(b) In the diagram, ABC is a right-angled
triangle with P and R on AB. Also, Q is on
AC, and P Q is parallel to BC. If RP = 2,
BR = 3, BC = 4, and the area of △QRC
is 5, determine the length of AP .
C
4
Q
B
3 R 2
P
A