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High School Math Contest University of South Carolina January 28, 2012

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High School Math Contest University of South Carolina January 28, 2012
High School Math Contest
University of South Carolina
January 28, 2012
I NSTRUCTIONS
1. Do not open the booklet until you are told to do so.
2. This is a thirty (30) question multiple-choice test with questions on six (6) pages. Each question is followed by
answers marked (a), (b), (c), (d), and (e). Exactly one of these answers is correct for each question.
3. The test will be scored as follows: Five (5) points for each correct answer, one (1) point for each answer that is
left blank, and zero (0) points for each incorrect answer.
4. Your teacher should have provided you with a SCANTRON sheet. Fill in your 5-digit registration number from
your name tag. For cross-reference, also print your name and school on the blank lines labeled NAME and TEST,
respectively.
5. Using a #2 pencil, record your answers on the SCANTRON sheet. Do not make any further marks on the
SCANTRON sheet except for your answers. Use only the first 30 lines of the side of the form where you filled
in your registration information. Record your answers with heavy marks and, if you make corrections, erase
thoroughly. Use a blank sheet of paper to cover your answers as you work through the test.
6. No calculators or reference materials are allowed.1 Test proctors and judges have been instructed not to elaborate
on any specific test questions. All electronic devices must be turned off and put out of sight. Any electronic
device seen during the exam will be confiscated and returned at the end of the exam.
7. Drawings on the test are not necessarily drawn to scale.
8. When you are given the signal, begin working the problems. You have 90 minutes working time for the test.
9. If you booklet has a defective, missing, or illegible page, please notify the proctor in your room as soon as
possible.
10. You must stay in the room for the entire exam period. If you need to use the restroom before the end of the exam
period, quietly notify a proctor. Be sure to completely cover all exam materials before getting up from your desk.
Return directly to your testing room.
11. Work quietly until the 90 minute time period has elapsed. Return the SCANTRON sheet and pencil to the
proctor. You may keep this booklet. (Solutions will be posted at http://www.math.sc.edu/contest/ .) Unless
instructed otherwise by your teachers, take your belongings and meet your teachers in the first floor lobby.
N OTATION
• When n is a positive integer, n! denotes 1 · 2 · 3 · · · · · n. For example, 3! = 1 · 2 · 3 = 6.
• We denote by {1, 2, 3, . . . , 99, 100} the set of positive integers from 1 to 100.
• We denote by AB the line seqment with endpoints A and B, and we denote by AB the length of line segment
AB.
G OOD L UCK !
1
Unless specifically approved in advance by Dr. Meade.
High School Math Contest
University of South Carolina
January 28, 2012
1. How many two digit prime numbers are there in which both digits are prime numbers? (For
example, 23 is one of these numbers but 31 is not, since 1 is not a prime number.)
(a) 3
(b) 4
(c) 5
(d) 8
(e) 15
2. You own thirteen pairs of socks, all different, and all of the socks are individually jumbled in
a drawer. One morning you rummage through the drawer and continue to pull out socks until
you have a matching pair. How many socks must you pull out to guarantee having a matching
pair?
(a) 3
(b) 12
(c) 13
(d) 14
(e) 25
3. A jeweler has a 20 gram ring that is 60% gold and 40% silver. He wants to melt it down and
add enough gold to make it 80% gold. How many grams of gold should be added?
(a) 4 grams
(b) 8 grams
(c) 12 grams
(d) 16 grams
(e) 20 grams
4. Consider the following game. A referee has cards labeled A, B, C, and D, and places them
face down in some order. You point to each card in turn, and guess what letter is written on
the bottom. You guess each of A, B, C, and D exactly once (otherwise there is no chance of
getting them all right!).
You play this game once, and then the referee tells you that you guessed exactly n of the letters
correctly. Which value of n is not a possible value of n?
(a) 0
(b) 1
5. What is the value of
(a) 1
(c) 2
(d) 3
(e) 4
p √
8 6
√
(e) 8 6
p
p
√
√
10 + 4 6 − 10 − 4 6?
(b) 4
√
(c) 2 6
1
(d)
6. The triangle 4ABC has sides of the following lengths: AB = 24, BC = 7, and AC = 25.
Let M be the midpoint of AB. What is the length of CM ? (The figure below is not drawn to
scale.)
(a) 1
(b)
√
139
(c) 12
(d)
√
193
(e) 16
7. What is the value of (log2 3)(log3 4)(log4 5) · · · (log63 64)?
(a)
1
6
(b) 2
(c)
5
2
(d) 6
(e) 32
8. On a test the passing students had an average of 83, while the failing students had an average
of 55. If the overall class average was 76, what percent of the class passed?
(a) 44%
(b) 66%
(c) 68%
(d) 72%
(e) 75%
9. Jack and Lee walk around a circular track. It takes Jack and Lee respectively 6 and 10 minutes
to finish each lap. They start at the same time, at the same point on the track, and walk in the
same direction around the track. After how many minutes will they be at the same spot again
(not necessarily at the starting point) for the first time after they start walking?
(a) 15
(b) 16
(c) 30
(d) 32
(e) 60
10. If sin(x) + cos(x) = 21 , what is the value of sin3 (x) + cos3 (x)?
(a)
1
8
(b)
5
16
(c)
2
3
8
(d)
5
8
(e)
11
16
11. The two roots of the quadratic equation x2 − 85x + c = 0 are prime numbers. What is the
value of c?
(a) 84
(b) 166
(c) 332
(d) 664
(e) 1328
12. How many pairs (x, y) of integers satisfy x4 − y 4 = 16?
(a) 0
(b) 1
(c) 2
(d) 4
(e) infinitely many
13. A circle passes through two adjacent vertices of a square and is tangent to one side of the
square. If the side length of the square is 2, what is the radius of the circle?
(a)
3
2
(b)
4
3
(c)
5
4
(d)
6
5
(e) None of these
14. If x and y are positive real numbers, neither of which is equal to 1, what is the smallest nonnegative value of logx (y) + logy (x)?
(a) 0
(b)
√
15. What is the value of sin
(a) 1
2
2π
5
(b) −1
(c)
+ sin
4π
5
+ sin
√
π
6π
5
(d) 2
+ sin
8π
5
?
1
(d) √
5
(c) 0
(e) 10
1
(e) − √
5
n2 − 38
16. What is the largest integer n such that
is an integer?
n+1
(a) 36
(b) 38
(c) 72
3
(d) 76
(e) None of these
17. For a positive integer n, define S(n) to be the sum of the positive divisors of n. Which of the
following is the smallest?
(a) S(2010)
(b) S(2011)
(c) S(2012)
(d) S(2013)
(e) S(2014)
18. A class has three girls and three boys. These students line up at random, one after another.
What is the probability that no boy is right next to another boy, and no girl is right next to
another girl?
(a)
1
20
(b)
1
12
(c)
1
10
(d)
3
10
(e)
1
2
19. Suppose f (x) = ax + b and a and b are real numbers. We define
f1 (x) = f (x)
and
fn+1 (x) = f (fn (x))
for all positive integers n. If f7 (x) = 128x + 381, what is the value of a + b?
(a) 1
(b) 2
(c) 5
(d) 7
(e) 8
20. A bag contains 11 candy bars: three cost 50 cents each, four cost $1 each and four cost $2
each. How many ways can 3 candy bars be selected from the 11 candy bars so that the total
cost is more than $4?
(a) 8
(b) 28
(c) 46
(d) 66
(e) 70
21. Consider the following game, in which a referee picks a random integer between 1 and 100.
One after the other, each of three players tries to guess the number the referee picked. Each
player announces his or her guess before the next player guesses. Each guess has to be different from the previous guesses. The winner is the player who comes closest to the referee’s
number without exceeding it. (It is possible for none of the players to win.)
Suppose that Player 1 guesses 24, and that Player 3 will guess a number that gives her/him the
best chance of winning. What number should Player 2 guess to maximize his/her chances of
winning?
(a) 1
(b) 25
(c) 62
4
(d) 63
(e) 64
22. The Sierpiński Triangle involves a sequence of geometric figures. The first figure in the sequence is an equilateral triangle. The second has an inverted (shaded) equilateral triangle
inscribed inside an equilateral triangle as shown. Each subsequent figure in this sequence is
obtained by inserting an inverted (shaded) triangle inside each non-inverted (white) triangle
of the previous figure, as shown below. How many regions (both shaded and white together)
are in the ninth figure in this sequence?
For example, the first three figures in the sequence have 1 region, 4 regions, and 13 regions
respectively.
(a) 4021
(b) 4022
(c) 4023
(d) 9841
(e) 9842
23. How many positive integers n have the property that when 1,000,063 is divided by n, the
remainder is 63?
(a) 29
(b) 37
(c) 39
(d) 49
(e) 79
24. I have twenty 3¢ stamps and twenty 5¢ stamps. Using one or more of these stamps, how many
different amounts of postage can I make?
(a) 150
(b) 152
(c) 154
(d) 396
(e) 400
25. All of the positive integers are written in a triangular pattern, beginning with the following
four lines and continuing in the same way:
10
5
11
1
3
7
13
2
6
12
4
8
14
9
15
16
Which number appears directly below 2012?
(a) 2100
(b) 2102
(c) 2104
5
(d) 2106
(e) 2108
26. Two spies agreed to meet at a gas station between noon and 1pm, but they have both forgotten
the arranged time. Each arrives at a random time between noon and 1pm and stays for 6
minutes unless the other is there before the 6 minutes are up. Assuming all random times are
equally likely, what is the probability that they will meet within the hour (noon to 1pm)?
(a) 0.12
(b) 0.15
(c) 0.17
(d) 0.19
(e) 0.25
27. A farmer has 12 plots of land, arranged in a row. To ensure viability of the soil, the farmer
never uses two adjacent plots at the same time. This season, the farmer wishes to plant one
plot of each of the following: corn, wheat, soybeans, and rice. Each crop is assigned its own
plot of land. How many ways can the farmer allocate plots of land for these crops?
(a) 1680
(b) 3024
(c) 5040
(d) 7920
(e) 11880
28. How many triples (x, y, z) of rational numbers satisfy the following system of equations?
x+y+z = 0
xyz + z = 0
xy + yz + xz + y = 0
(a) 1
(b) 2
(c) 3
(d) 4
(e) 5
29. A coin has a probability of 1/3 for coming up heads and 2/3 for coming up tails. On average,
how many flips of this coin are needed to guarantee both heads and tails appear at least once?
(a) 2.25
(b) 2.5
(c) 3
(d) 3.5
(e) 5
30. Suppose a, b, and c are three successive terms in a geometric progression, and are also the
lengths of the three sides opposite the angles A, B, and C, respectively, of 4ABC.
Which of the following intervals is the set of possible values of
√
(a) (0, +∞) (b)
0,
5+1
2
!
(c)
!
√
√
5−1 5+1
,
(d)
2
2
6
√
sin A cot C + cos A
?
sin B cot C + cos B
!
5−1
, +∞ (e)
2
√
!
5+1
, +∞
2
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