Test Report: Math for Olympiad from MUMS 1
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These collection of math questions were collected from Melbourne
University Mathematics & Statistics Society.
http://ms.unimelb.au.edu/~mums/
1.
"Baker's Dozen" doughnuts are sold only in boxes of 7, 13 or 25.
To buy 14 doughnuts you must order two boxes of 7, but you
cannot buy exactly 15 since no combination of boxes contains 15
doughnuts. What is the largest number of doughnuts that cannot
be ordered using combinations of these boxes?
A.
44
B.
45
C.
46
D.
47
E.
48
2.
A circular table has exactly 60 chairs around it. There are N
people seated at this table in such a way that the next person
to be seated must sit next to someone else. What is the smallest
possible value of N?
A.
18
B.
19
C.
20
D.
21
E.
22
3.
A farm has cows and chickens. Together they have 35 heads and 94
legs. How many more chickens than cows?
A. 8
B. 9
C.
10
D.
11
E.
12
4.
A rectangular tank with base 4 meters by 4 meters and height 3
meters contains water to a depth of 1 meter. A lead cube of edge
length two meters is placed in the tank. By how much (in
centimeters) does the depth of water increase?
A.
35
B.
45
C.
50
D.
75
E.
80
5.
A sequence of numbers k1, k2,
k3, … satisfies k1 = 0 and for
all .
Find k1999.
A.
699
B.
666
C.
777
D.
888
E.
999
6.
A set of consecutive numbers beginning with 1 is written on a
blackboard. One number is erased. The average (arithmetic mean)
of the remaining numbers is 35 1/4. What
number was erased?
A.
21
B.
32
C.
16
D.
19
E.
18
7.
A square is cut along two lines parallel to a side to form three
identical rectangles . If the perimeter of each rectangle is 24,
then what is the area of the original square?
A.
79
B.
80
C.
81
D.
89
E.
90
8.
A square island 2km on a side is centered in a circular lake 4km
in diameter. Find the shortest distance, in kilometers, from the
coast of the island to the mainland.
9.
An urn is filled with coins and beads, all of which are either
silver or gold. Twenty percent of the objects in the urn are
beads. Forty percent of the coins in the urn are silver. What
percentage of objects in the urn are gold coins?
A.
80%
B.
48%
C.
60%
D.
40%
E.
68%
10.
Bobbi and Chris were walking up the stairs of a tower. Bobbi was
constantly 52 steps
ahead of Chris. When Bobbi was halfway up the stairs, she said
to Chris, "When I've reached the top, you'll be three times as
far as you are now." What is the number of stairs
in
the tower?
A.
108
B.
208
C.
218
D.
280
E.
288
1. A
Note
that we can buy any number of doughnuts from 45 through to 51 as
follows:
45 =
1 x 7 + 1 x 13 + 1 x 25
46 =
3 x 7 + 0 x 13 + 1 x 25
47 =
3 x 7 + 2 x 13 + 0 x 25
48 =
5 x 7 + 1 x 13 + 0 x 25
49 =
7 x 7 + 0 x 13 + 0 x 25
50 =
0 x 7 + 0 x 13 + 2 x 25
51 =
0 x 7 + 2 x 13 + 1 x 25
Since these are seven consecutive numbers, we can buy any number
of doughnuts which is 45 or greater, by buying extra boxes of 7.
However, it is not possible to buy 44 doughnuts.
2. C
If
there is a gap of 3 anywhere, then we can let the next person
sit in the middle of that
gap,
so every 3 seats must have at least one person, giving 20 as a
lower bound. By putting 20 people evenly around the table with a
space of 2 seats in between each pair, it is easy to see that it
works. Thus the minimum value of N is 20.
3. D
4. C
5. E
6. E
7. C
Let
the side of the square be 3x. Then the rectangles will be
x by 3x. Since they have perimeter 24, x =
3. Then the area of the square is (3x)2 = 81.
8. E
9. B
100%−20% = 80% are coins, and 100%−40% = 60% of coins are gold.
Multiplying gives 48% gold coins.
10.
B
Let
x be the total number of stairs, and let y be the
number that Chris has climbed when
Bobbi is halfway. Then and .
Eliminating y gives x = 208.
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