Download Department of Mathematics and Statistics

Department of Mathematics and Statistics
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AUS MathFest 2016
Competition for Teachers
School: ______________________________________________
Teacher Names: ______________________________________________
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Instructions:
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Duration: 75 minutes
This exam consists of 5 long questions and 5 multiple choice questions
For the multiple choice part:
Each question has one correct answer
Circle the right answer
3 points for a correct answer
−1 point for an incorrect answer
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Calculators are not allowed
No cell phones
No Questions allowed
It is very unlikely you will be able to solve all the questions in the given
time
The results announced by the judges are final
Good Luck!
AUS MathFest
Competition for teachers
Part I. Long questions.
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1. (1 point) The largest known prime number is 23021377 −1. Is it true?
Explain your answer.
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AUS MathFest
Competition for teachers
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2. (2 points) 4 football players are placed on the football field. There are
six distances between players: between player 1 and player 2, between
player 1 and player 3, between player 1 and player 4, between player 2
and player 3 and so on. Is it possible to place these 4 players on the
football field in a way that the distances between them would be 1, 2,
3, 4, 5 and 6 meters?
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AUS MathFest
Competition for teachers
3. (3 points) John, Jack and Steve are sitting in the office and sending
email to each other. One email can reach only one person. Also email
can be placed in the spam folder and the receiver will not read it. After
some time they stopped sending emails. The known facts are:
(a) Jack sent the first email to John or Steve.
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(b) After Jack got and read an email, he sent 4 emails randomly to
John or Steve (not necessarily to one man, for example, it can be
2 emails to John and 2 emails to Steve).
(c) After Steve got and read an email, he sent 5 emails randomly to
Jack and John.
(d) After John got and read an email, he sent 6 emails randomly to
Jack and Steve.
(e) The total number of emails in the spam folders is 13.
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How many emails did John, Jack and Steve read? You should specify
3 numbers in your answer.
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AUS MathFest
Competition for teachers
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4. (3 points) One man throws a coin 10 times. Another man throws a
coin 11 times. What is the probability that the second man (the one
who throws 11 times) has more heads than the first one?
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AUS MathFest
Competition for teachers
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5. (2 points) Let ABCD be a rhombus with AC = 16 and BD = 30. Let
N be a point on AB, and let P and Q be the feet of the perpendiculars
from N to AC and BD respectively. What is the minimum possible
value of P Q as N moves from A to B?
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AUS MathFest
Competition for teachers
Part 2. Multiple choice questions.
(a)
(b)
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(c) 15
(d)
39
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(e) 24
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1. Let S be the set of ordered triples (x, y, z) of real numbers for which
log10 (x + y) = z and log10 (x2 + y 2 ) = z + 1. There are real numbers a
and b such that for all ordered triples (x, y, z) in S we have x3 + y 3 =
a · 103z + b · 102z . What is the value of a + b?
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AUS MathFest
Competition for teachers
2. Let A0 = (0, 0) . Distinct points A1 , A2 , ... lie on the
√ x-axis, and
distinct points B1 , B2 , ... lie on the graph of y = x. For every
positive integer n, An−1 Bn An is an equilateral triangle. What is the
least n for which the length A0 An > 100?
(b) 15
(c) 17
(d) 19
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(e) 21
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(a) 13
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AUS MathFest
Competition for teachers
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with ra3. Consider two solid spherical balls, one centered at 0, 0,
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dius 6, and the other centered at (0, 0, 1) with radius . How many
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points with only integer coordinates (lattice points) are there in the
intersection of the balls?
(a) 7
(b) 9
(c) 11
(d) 13
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(e) 15
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AUS MathFest
Competition for teachers
4. How many positive two-digits integers are factors of 224 − 1?
(a) 4
(b) 8
(d) 12
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(e) 14
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(c) 10
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AUS MathFest
Competition for teachers
5. You provide your students with 10 questions to study from for the
exam. One of your students can solve six of the questions. For the
exam, you select 5 questions at random from the list of 10. What is
the probability that the student will score 100% on the exam?
(b) 0.6
(c) 0.83
(d) 0.024
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(e) 0.221
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(a) 0.5
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