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Team
Problem 1.
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AwesomeMath Team Contest 2013
What is the largest whole number that is equal to the product of its digits?
Team
AwesomeMath Team Contest 2013
Problem 2.
How many of the rearrangements of the digits 123456 have the property that for each digit, no more
than two digits smaller than that digit appear to the right of that digit? For example, the rearrangement 315426 has
this property because digits 1 and 2 are the only digits smaller than 3 which follow 3; digits 2 and 4 are the only digits
smaller than 5 which follow 5, and digit 2 is the only digit smaller than 4 which follows 4.
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Team
AwesomeMath Team Contest 2013
Problem 3.
Two externally tangent circles have radius 2 and radius 3. Two lines are drawn, each tangent to both
circles, but not at the point where the circles are tangent to each other. What is the area of the quadrilateral whose
vertices are the four points of tangency between the circles and the lines?
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Team
Problem 4.
AwesomeMath Team Contest 2013
Real numbers x, y, z satisfy

2
2

x + y + xy = 1
y 2 + z 2 + yz = 4

 2
z + x2 + xz = 5
Find x + y + z.
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Team
Problem 5.
a21 + · · · + a212 ?
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Team
Problem 6.
1
Team
AwesomeMath Team Contest 2013
If a1 , · · · , a12 are twelve nonzero integers such that a61 + · · · + a612 = 450697, what is the value of
AwesomeMath Team Contest 2013
Real numbers a, b satisfy 2(sin a + cos a) sin b = 3 − cos b. Find 3 tan2 a + 4 tan2 b.
AwesomeMath Team Contest 2013
Problem 7.
In a game similar to three card monte, the dealer places three cards on the table: the queen of spades
and two red cards. The cards are placed in a row, and the queen starts in the center; the card configuration is thus
RQR. The dealer proceeds to move. With each move, the dealer randomly switches the center card with one of the
two edge cards (so the configuration after the first move is either RRQ or QRR). What is the probability that, after
2004 moves, the center card is the queen?
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Team
AwesomeMath Team Contest 2013
Problem 8.
Convex quadrilateral M AT H is given with M H/T M = 3/4, and 6 AT M = 6 M AT = 6 AHM = 60◦ .
N is the midpoint of M A and O is a point on T H such that lines M T, AH, N O are concurrent. Find the ratio
HO/OT .
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Team
Problem 9.
1
AwesomeMath Team Contest 2013
Find the number of positive integer solutions of (x2 + 2)(y 2 + 3)(z 2 + 4) = 60xyz.
Team
Problem 10.
If a recurrence an+1 =
q
an
2
+
1
2
AwesomeMath Team Contest 2013
∞
Y
has initial condition a0 = 21 , find
an .
n=0