Download BMO S1 – US Gr9 Sample Test

Bermuda S1 Mathematics Olympiad
Sunday 28 April 2013
Part A - Multiple Choice
1. The value of 2 + 0 × 1 × 3 is
(A) 6
(B) 0
(C) 2
1 2 1 5 7
, , , ,
2. The smallest number in the set
2 3 4 6 12
1
2
1
(A)
(B)
(C)
2
3
4
3. The value of
(A) 5000
5
6
(D) 5
(E) 4
is
(D)
5
6
7
12
(E)
of 15 000 is
(B) 7500
(C) 10 000
(D) 12 500
(E) 13 500
4. What number goes in the box so that 10 × 20 × 30 × 40 × 50 = 100 × 2 × 30 × 4 × ?
(A) 5000
(B) 500
(C) 50
(D) 5
(E) 0.5
5. A palindrome is a positive number which is the same when read backwards or forwards. When
the smallest four digit palindrome is subtracted from the largest four digit palindrome, the
result is:
(A) 8888
(B) 8008
(C) 7997
(D) 8998
(E) 7007
6. Lesley walks 53 of the way home in 30 minutes. If she continues to walk at the same rate, how
many minutes will it take her to walk the rest of the way home?
(A) 20
(B) 24
(C) 6
(D) 18
(E) 12
√ √
√
√
7. The expression ( 100 + 7)( 100 − 7) is equal to
(A) 93
(B) 51
(C) 9993
(D) 9951
8. In the diagram, rectangle P QRS has P S = 6 and
SR = 3. Point U is on QR with QU = 2. Point T
is on P S with ∠T U R = 90◦ . What is the length
of T R?
(E) 10 993
(A) 3
(D) 6
(B) 4
(E) 7
T
P
6
S
3
(C) 5
Q
2
U
R
2013 Bermuda S1 Mathematics Olympiad
Page 2 of 4
9. Natasha begins with 64 coins in her coin jar. Each time she reaches into the jar, she removes
half of the coins that are in the jar. How many times must she reach in and remove coins from
her jar so that exactly 1 coin remains in the jar?
(A) 5
(B) 6
(C) 7
(D) 32
(E) 63
10. The time on a cell phone is 3:52. How many minutes will pass before the phone next shows a
time using each of the digits 2, 3, and 5 exactly once?
(A) 27
(B) 59
(C) 77
(D) 91
(E) 171
11. If x = 6 and 3x + 2y = 30, what is the value of y?
(A) 12
(B) 5
(C) 8
(D) 4
(E) 6
12. The rectangle in the diagram is 4 units high by 7 units wide.
How many squares of all sizes are there in the diagram?
(A) 74
(D) 66
(B) 51
(E) 42
(C) 60
13. The surface area of a cube is 96 cm2 . The volume of the cube, in cm3 , is
(A) 81
(B) 64
(C) 16
(D) 32
(E) 4096
(C) 6
(D) 8
(E) 12
14. If 4n = 642 , then n equals
(A) 3
(B) 5
15. The square shown is divided into four congruent (identical)
rectangles. The perimeter of each of the rectangles is 25.
What is the perimeter of the square?
(A) 100
(D) 50
(B) 80
(E) 40
(C) 60
16. In the addition shown, P and Q each represent single
digits, and the sum is 1P P 7. What is P + Q?
(A) 9
(D) 15
(B) 12
(E) 13
7
6
Q
+
(C) 14
1
P
7
Q
Q
P
P
P P 7
17. An integer x is chosen so that 3x + 1 is an even integer. Which of the following must be an
odd integer?
(A) x + 3
(B) x − 3
(C) 2x
(D) 7x + 4
(E) 5x + 3
2013 Bermuda S1 Mathematics Olympiad
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18. On the 4 × 4 of unit squares shown, a Canada Goose
moves from the square labelled S to the square labelled
E. The goose can move in 3 possible ways:
A: up 2 units; B: right 1 unit; or C: up one unit and then
right one unit. How many different paths from S to E are
formed by a sequence of these moves?
(A) 9
(B) 10
(C) 11
(D) 12
(E) 13
E
S
19. The sum of the first 100 positive integers is 5050. What is the sum of the first 100 odd positive
integers?
(A) 5050
(B) 10 000
(C) 10 050
(D) 10 100
(E) 10 150
20. The integer 636 405 may be written as the product of three 2-digit positive integers. The sum
of these three integers is
(A) 259
(B) 132
(C) 74
(D) 140
(E) 192
Part B - Full Solutions
Solutions to these questions are to be written in the answer booklet on the appropriate pages.
Show the work you do to get your answers.
1. A 4 by 4 anti -magic square is an arrangement of the numbers 1 to 16 inclusive in a square,
so that the totals of each of the four rows and four columns and two main diagonals are ten
consecutive numbers in some order. The diagram shows an incomplete anti -magic square.
Complete this anti -magic square.
4
5
7
6
1 3
3
1 1
1 2
9
1 0
Explain your answer.
Continued...
1 4
2013 Bermuda S1 Mathematics Olympiad
Page 4 of 4
2. On a coordinate grid, Sharla draws a line segment of length 1 to the right from the origin,
stopping at (1, 0). She then draws a line segment of length 2 up from this point, stopping at
(1, 2). She continues to draw line segments to the right and up, increasing the length by 1 each
time.
One of the line segments ends at (100, 110).
What is the endpoint of the next line segment that Sharla draws? After this line segment is
drawn, what is the total length of the line segments that Sharla has drawn?
y
3
2
1
4
x
Explain your answer.
3. A two-digit positive integer x has the property that when 109 is divided by x,
the remainder is 4. Determine the sum of all such two-digit positive integers x.
Explain your answer.