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2013 WMI Competition
Grade 6 Part 2 Applications Test
Problems 1–15: 10 points each for a total of 150 points.
1. Compute: (1 −
1
1
1
1
1
) + (1 − × 2) + (1 − × 3) + ... + (1 − × 12) + (1 − × 13) .
39
39
39
39
39
2. A, B, C, and D are all good friends of each other. When D has no money, A gives
1
8
1
1
of his money to D, and C gives
of his money to D.
4
6
If they have given the same amount to D, then the amount D has now is what
percentage of A, B, and C’s total?
of his money to D, B gives
3. If the first box of fruits is 28 kilograms heavier than the second box of fruits and
3
5
the weight of the first box is the same as
the weight of the second, what is the
4
9
sum of the weights of both boxes of fruits?
4. A sequence of numbers is listed according to the following pattern:
1, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 10,……
(1) What is the 200th number?
(2) Where does the number 79 last appeared?
5. As shown in the figure on the right, ABCD is a square with
lengths as indicated and the arc BD is part of the circle with
center at C and radius BC . Find the area of the shaded region.
6. Let S = the set of two consecutive positive integers a and b such that their product
plus 17 is composite. What is the smallest possible a+b?
7. Consider integers from 3007 to 6004, what is the total number of single digit, 2–digit,
and 3–digit numbers each with distinct digits?
8. Suppose there are two prime numbers such that their sum is: (1) odd, (2) larger than
20, (3) smaller than 200, and (4) a multiple of 19. What is the difference of these two
prime numbers?
9. There are 9 people in the park. It is given that their average age is 49 and none of
these 9 people are older than 54. Also, there are, at the most, 3 people have the same
age. What is the least possible age of the youngest of these 9 people?
10. Let P, Q, and R be 2–digit positive integers and P＞Q＞R. If their sum is an even
number and their product is 5544, what is the value for P＋R－Q?
11. As shown in the figure on the right, ABCD is a
parallelogram where all the line segments in it are
parallel to AB , AD , or AE . Using these line
segments, how many parallelograms can be
obtained that will contain 「＊」?
12. Use numbers 5, 6, or 7 (2 of each) to form a 6–digit number that is divisible by 312.
What is this 6–digit number?
13. Find all 4–digit numbers abcd that are multiples of 9 with no digits being 0 and
satisfying the conditions that c – b = d and bc is a perfect square.
14. People start to line up outside a concert hall certain number of minutes before the
concert officials admitting them into the hall. Suppose the same number of people
joining the lineup every minute and that people would constantly arriving to join the
line at the same rate while the gates are admitting people. If three gates were open,
then it takes 40 minutes to admit all the people. If 4 gates were open, then it would
take 25 minutes. Assuming the admission rate is constant at each gate during the
entire process. In order to admit all the people in 10 minutes, how many gates must
be open?
15. The figure on the right represents a 400 meters track where
each of the two straight tracks is 140 meters long and each
of the two curved tracks is 60 meters long. Suppose A and
B start at point A and start running at the same time. Let the
speeds of A and B on the straight track be 5 meters per second
and 4 meters per second, respectively, and let their speeds on
the curved tracks be 4 meters per second and 3 meters per
second, respectively. In how many minutes and seconds will
A be able to catch up to B the second time?