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Individual
Competition
Part II
Questions 26 - 50
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There are 25 multiple choice
questions
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You have 2 minutes to finish each
question
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There will be no break in this round
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A trial question will now follow
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Trial Question
(2 minutes)
If 3 = k . 2r and 15 = k . 4r , then r =
(a)– log25
(b) log52
(c) log105
(d) log25
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5
(e) 2
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26
Suppose that * is an operation on the integers
defined by
a*b = a2+b.
What is the value of 3 * (2 * 1)?
(a) 12 (b) 14
(c) 54
(d) 170
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(e) 172
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27. An office building has 50 storeys, 25 of
which are painted black and the other
25 of which are painted gold. If the
number of gold storeys in the top half
of the building is added to the number
of black storeys in the bottom half of
the building, the sum is 28. How many
gold storeys are there in the top half of
the building?
A. 3
B. 14
C. 22
D. 24
E. None
of
these
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28. What is the number of distinct real
numbers r which have the property that
the median of the five numbers r,6,4,1,9
is equal to their mean?
A. 0
B. 1
C. 2
D. 3
E. 5
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29
Two perpendicular line segments divide a large rectangle
into 4 small rectangles. The areas of 3 of these 4 small
rectangles are shown. What is the area of the other small
rectangle?
6
9
8
(a) 12
(b) 13
(c) 14
(d) 15
(e) 16
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30. A can build a building in 3 hours, B can build
the building in 4 hours. Together, A and B,
and C can build the building in 1 hour. D can
build the building in half the time it takes C
to build the building. How long does it take
C and D to build the building together?
A. 36 minutes
B. 48 minutes
C. 60 minutes
D. 72 minutes
E. 84 minutes
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31. Triangle ABC has sides AB = 12, BC = 10,
and AC = 20. A circle is drawn with
radius 10 centered at C. Segment AB is
extended, intersecting the circle at point D.
Determine the length of BD.
A. 2√21
B. 10
C. 2√39
D. 13
E. None of these
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32. A block of wood in the form of a cuboid 6" x 9"
x 14" has all its six faces painted pink. If the
wooden block is cut into 756 cubes of 1" x 1" x
1", how many of these would have pink paint on
them?
A. 420
B. 560
C. 585
D. 624
E. 758
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33.
If x + y = 0 and x ≠ 0, then what is the value
of x2007
y2007 ?
(a) -2007
(b) -1 (c) 0
(d) 1
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(e) 2007
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34. The perimeter of a rectangle is P, and the area of
the rectangle is A. What is the product of the
diagonals?
A.
p2
 2A
4
D. P2 + 2A
B.
p2
 2A
4
C. P2 – 2A
E. None of these
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35.
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36.
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37.
Evaluate the sum
1- 2 + 3 – 4 + 5 – 6 +…+ 997 – 998 + 999 - 1000
A . -500
B. -1000
C. -999
D. -1001
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E. 500500
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38. Let N be the largest integer for which both
N and 7N have exactly 100 digits each.
What is the 50th digit (from the left) of N?
A. 5
D. 4
B. 1
E. 2
C. 8
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39.
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40. The number 6 is divisible by 1, 2, 3 and 6,
so 6 has 4 divisors.
How many divisors has 6718464 = (2^10) x (3^8)?
(Remark. “a^b” means “a to the power of b”.)
A. 99
B. 109
C. 98
D. 100
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E.93
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41.
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42.
A cylindrical beaker 8 cm high and 12cm in circumference
was standing on a table. On the inside of the beaker, 2 cm
from the top, is a drop of honey. Diametrically opposite
the honey and lower is a spider which is on the outside of
the beaker, 2 cm from the bottom. What is the shortest
distance the spider has to walk to reach the honey?
A. 10 cm
B. 12 cm
C. 13 cm
D. 100 cm
E. None of the above
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43.
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44. In year N, the 300th day of the year is a
Tuesday. In year N + 1, the 200th day is also a
Tuesday. On what day of the week did the 100th
day of the year N-1 occur?
A. Thursday
B. Friday
C. Saturday
D. Sunday
E. Monday
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45. A square with sides of length 1 is
divided into two congruent trapezoids
and a pentagon, which have equal areas,
by joining the centre of the square with
points on three of the sided, as shown.
Find r, the length of the longer parallel
side of each trapezoid.
r
A.
3
5
B.
2
3
C.
3
4
D.
5
6
E.
7
8
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46.
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47. In square ABCD, with sides of length 2,
segments AE, BF, CG, and DH are drawn
(figure below), bisecting the sides.
These segments form quadrilateral JKLM.
Determine the area of quadrilateral
JKLM.
D
F
A. 2/5
C
K
L
B. 4/5
E
G
C. 1
J
M
D. 2
A
1
H
1
B
E. None of these
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a , a , a ...
48. Define a sequence of real numbers
by
Then
a 1
1
a
100
and
a
 99 a n
n 1
3
3
1
for all
2
3
n  1.
equals
A. 3333
B. 3399
C. 9933
D. 9999
E. None of these
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49.The equiangular convex hexagon ABCDEF
has AB = 1 , BC = 4, CD = 2 and DE = 4.
The area of the hexagon is
A.
D.
15
3
2
39
3
4
B.
9 3
E.
43
3
4
C. 16
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50.
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