Bermuda M3 Mathematics Olympiad
Sunday 28 April 2013
Part A - Multiple Choice
1. The value of 2 × 0 − 1 + 3 is
(B) −1
(A) 2
(C) 4
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) 6000
2
3
(D) 5
(E) −2
is
(D)
5
6
(E)
7
12
of 15 000 is
(B) 10000
(C) 9000
(D) 3000
(E) 4000
4. If the length of each line segment in the figure is
2 cm, the perimeter of the figure, in cm, will be
(A) 24
(B) 16
(D) 30
(E) 36
(C) 20
5. What number goes in the box so that 10 × 20 × 30 × 40 × 50 = 100 × 2 × 300 × 4 × ?
(A) 0.5
(B) 5
(C) 50
(D) 500
(E) 5000
6. 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
7. At Laurel Park, an adult’s season pass costs $45 and a child’s season pass costa $30. A season
pass for a family costs $110. For a family of one adult and three children, how much does the
family season pass save as compared to buying individual passes?
(A) $45
(B) $35
(C) $25
(D) $15
(E) $5
8. The product of the digits in the year 2013 is 0; that is, 2 × 0 × 1 × 3 = 0. The number of years
between 1900 and 2100 whose digits have a product of 0 is
(A) 121
(B) 120
(C) 41
(D) 40
(E) 21
2013 Bermuda M3 Mathematics Olympiad
Page 2 of 4
9. The point (−2, −3) is reflected in the x-axis.
What are the coordinates of its image after the
reflection?
y
4
(A) (2, −1)
(B) (3, −2)
(D) (−3, −2)
(E) (−2, 3)
2
(C) (2, 3)
- 4
- 2
( - 2 ,- 3 )
- 2
2
4
x
- 4
10. 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
11. If x = 4 and 3x + 2y = 30, what is the value of y?
(A) 18
(B) 6
(C) 3
(D) 4
(E) 9
12. The mean (average) of five consecutive even numbers is 12. The mean of the smallest and
largest of these numbers is
(A) 12
(B) 10
(C) 14
(D) 8
(E) 16
13. 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
14. The surface area of a cube is 54 cm2 . The volume of the cube, in cm3 , is
(A) 81
(B) 343
(C) 18
(D) 27
(E) 729
(C) 4
(D) 10
(E) 12
15. The value of (23 )2 − 43 is
(A) 0
(B) −8
16. 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
2013 Bermuda M3 Mathematics Olympiad
Page 3 of 4
17. P QRS is a rectangle with diagonals P R and QS, as shown.
The value of y is
(A) 30
(D) 50
(B) 40
(E) 60
P
Q
( 5 x )
(C) 45
( 4 x )
y
S
R
18. One day Clarise begins on page 1 and reads 15 of a 300 page book. The next day, she reads
of the remaining pages. On the third day, on what page does she start to read?
(A) 141
(B) 125
(C) 101
(D) 81
(E) 65
19. 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
(C) 14
20. 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
4
15
7
6
Q
+
1
P
7
Q
Q
P
P
P P 7
E
S
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. Each of the numbers from 1 to 9 is placed, one per circle, into the pattern shown. The sums
along each of the four sides are equal. Which of the numbers could go in the circle in the
middle?
Explain your answer.
2013 Bermuda M3 Mathematics Olympiad
Page 4 of 4
2. 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.
Note that the top row has a sum of 30 and the diagonal from the left bottom to the top right
has sum 39.
4
5
7
6
1 3
3
1 1
1 2
9
1 4
1 0
Explain your answer.
3. 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?
y
3
2
1
Explain your answer.
4
x