Team 1 Problem 1
Micaela has 3 shirts to wear, 4 pairs of pants to wear, and 5 different pairs of footwear she can wear.
In how many ways can she dress using exactly one of each, provided that she wears her one-of-a-kind
Apple Bottom jeans or her boots with the fur (or both)?
Team 1 Problem 2
Observe that 45x2 + 119x + 58 can be factored as (45x + 29)(x + 2). How does 45x2 + 118x + 56
factor?
Team 1 Problem 3
Let ABCD be a rectangle. Points P and Q are placed on the diagonal BD such that AP ⊥ BD
and CQ ⊥ BD. If BP = P Q = QD = 1, find the area of ABCD.
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Team 1 Problem 4
A circle and a sphere are given such that the area of the circle is numerically equal to the volume of
the sphere, and the circumference of the circle is numerically equal to the surface area of the sphere.
What is the ratio of the circle’s radius to the sphere’s radius?
Team 1 Problem 5
Find the number of positive integers n such that when the number 98 is divided by n, the result has
a non-zero remainder of 2.
Team 1 Problem 6
Two lines `1 and `2 are parallel to each other, with 20 units between them. Points A and B are
chosen on `1 that are 16 units apart. A point C is chosen between `1 and `2 so that its distance
from `2 is twice the distance from C to `1 . What is the minimum area of triangle ABC?
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Team 1 Problem 7
Find the number of positive integers n such that n2 + 216 is a perfect square.
Team 1 Problem 8
Cindi’s Imaginizer Function CIF (x, y) is defined by CIF (x, y) = x + yi, where i2 = −1. Find all
complex numbers x such that CIF (4 + 2i, x) = x2 .
Team 1 Problem 9
What is the largest power of 2 that divide 2016!?
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Team 1 Problem 10
Graham is standing in an empty square grid, completely in one 1 × 1 cell. He is facing to the right.
He then does the following routine: he walks one cell forward, turns to the left, walks two cells
forward, turns to the left, walks three cells forward, and so on until he walks ten cells forward for
the first time. We shade in every cell he has traversed (including his starting and ending cells) to
obtain a two-dimensional shape. Find the perimeter of this shape.
Team 1 Problem 11
Let Ln denote the Lucas numbers, which are defined by L0 = 2, L1 = 1 and Ln+1 = Ln + Ln−1 for
all n ≥ 1. Find the last digit of L2016
Team 1 Problem 12
How many two-digit positive integers have exactly four factors?
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Team 1 Problem 13
If you pick a random day in the year 2016, what is the probability that the date is a prime number?
For example, April 7th would be such a date, but July 10th would not.
Team 1 Problem 14
G-Dos is reciting the English alphabet as usual. We create a function ya(i) over the integers such
that for all integer 1 ≤ i ≤ 26, ya(i) is the number of vowels that G-Dos has said once he has said
the first i letters of the English alphabet. Find the average value of ya(i) as i goes from 1 to 26.
Note that we consider Y to not be a vowel.
Team 1 Problem 15
Determine the largest prime factor of
315 + 311 + 36 + 1.
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Team 1 Problem 16
Compute
F1 + F2 + F3 + 2F4 + 2F5 + 4F6 + 4F7 + 8F8 + 8F9 + 16F10 + 16F11 + 32F12 ,
where F1 = 1, F2 = 1, and for all n ≥ 3 Fn = Fn−1 + Fn−2 .
Team 1 Problem 17
How many ways are there to place 3 identical rooks on a chessboard so that no two are in the same
row or same column?
Team 1 Problem 18
Let r and s be the roots of the polynomial x2 + ax + b where a and b are integers. Find a quadratic
with leading coefficient 1 in terms of a and b, such that r + s and r2 + s2 are the roots.
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Team 1 Problem 19
M.Tip and Babbitt play a best of three series (i.e. the series is over as soon as one team wins two
games.) If the series is tied or Babbitt is ahead, M.Tip has a 60% chance of winning the next game,
but if the M.Tip are ahead, Babbitt has a 50% chance of winning the next game. What is the
probability that the M.Tip win the series in exactly three games?
Team 1 Problem 20
Suppose a and b are two distinct real roots of the quadratic ax2 + x + b = 0. Compute ab.
Team 1 Problem 21
How many ways are there to rearrange the letters in the word DIVISIBILITY such that every letter
is not adjacent to any of the letters that it is normally adjacent to in the correct spelling? For
example, the V may not be adjacent to any I’s in any such rearrangement.
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Team 1 Problem 22
How many ways are there to go from the lower-left vertex of a 20 × 16 grid (so area 320, with 20
rows of cells and 16 columns of cells) to the upper-right vertex of the grid if you may only move one
unit at a time across the grid sides, you may only go up, down, or to the right at one time, you may
never leave the grid, and you may never visit a point more than once? Express your answer as ab ,
where a and b are positive integers and a is as small as possible.
Team 1 Problem 23
Square ABCD has side length 73, and a point E is chosen in its interior uniformly and at random.
What is the expected value of the area of pentagon ABCDE? Express your answer as a common
fraction.
Team 1 Problem 24
Let n be a positive integers greater than 2016. Find the number of functions from the set {1, . . . , n}
to the set {2015, 2016} such that
f (1) + f (2) + f (3) + · · · + f (2016)
is odd.
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Team 1 Problem 25
Let f be a function with the property that
f (x2 + x + 1) = x3 + x + 1
for all real numbers x ≥ 1. What is f (x2 − x + 1) in terms of x?
Team 1 Problem 26
The function c(x) is defined on the set R − [0, 1) as
c(x) =
{x}
,
bxc
where {x} denotes the fractional part of x. A real number is chosen uniformly at random from the
interval [1, 20]. What is the probability that c(x) < 1/16?
Team 1 Problem 27
Using a segment AB, we draw a regular 18-sided polygon and a regular 20-sided polygon, both
having AB as one of their sides. The 18-sided polygon is also inside the 20-sided polygon. Let C
be the vertex of the 18-sided polygon opposite A, and let D be the vertex of the 20-sided polygon
opposite A. Find the measure of ∠CAD in degrees.
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Team 1 Problem 28
Suppose x and y are positive real numbers such that
x + y = x2 y
and
x − y = xy 2 .
Find xy.
Team 1 Problem 29
√
Find the number of positive integers m less than 10000 such that b mc | m.
Team 1 Problem 30
A hemisphere has radius 19. A square pyramid fits tightly in the hemisphere, such that the pyramid’s
base is on the base of the hemisphere, and all five vertices of the pyramid lie on the hemisphere.
What is the area of one of the triangular faces of the square pyramid?
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Team 1 Problem 31
√
Points A and B are the endpoints of a semicircle ω with radius 65. Points X and Y lie on ω such
that AX = 8 and BY = 2. Let M be the midpoint of XY . What is the area of 4AM B?
Team 1 Problem 32
Carry out the following division problem in base b2 , for b > 1:
b0b2 − 1b2
1(b + 1)b2
.
Turn in the result in terms of b, in base b2 . Note that, in the denominator, b + 1 is considered a
digit.
Team 1 Problem 33
Triangle 4ABC has AB = 37, AC = 91, and BC = 96. Point H is the orthocenter of 4ABC, and
Γ is the circumcircle of 4BHC. Suppose line AH intersects Γ at a point X 6= H. Compute AX.
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Team 1 Problem 34
Twenty-seven RAs are sitting around a table. Edgar comes by with a problem, and randomly chooses
three of these RAs to help him bring the situation under control. What is the probability that at
least two of the RAs he chooses are sitting next to each other?
Team 1 Problem 35
A circle C1 of radius 22 has its center 64 units away from a circle C2 of radius 18. There are exactly
two points on C2 that have power 2016 with respect to C1 . Compute the distance between these
two points.
Team 1 Problem 36
Compute
gcd 20163 , 20153 + 13 , 20143 + 23 , . . . , 13 + 20153 , 20163 .
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Team 1 Problem 37
Two parabolas with equations y = x2 + x + 19 and y = −2x2 − 13x + 2 are drawn in the Cartesian
coordinate plane. There are two lines that are tangent to both parabolas, but only one can have its
equation expressed as ax + by = c, for positive integers a, b, c with gcd(a, c) = gcd(b, c) = 1. What
is c?
Team 1 Problem 38
Two positive integers a and b have the same base 3 and base 5 representations, respectively. For
example, a could be 5 and b could be 7. Given that a and b are less than 1000, what is the greatest
possible value of b/a?
Team 1 Problem 39
A polynomial P has integer coefficients such that the value of P (9) is in the closed interval [4, 9]
and the value of P (22) is in the closed interval [11, 19]. We also know that P has an integer root m.
Find all possible values of m.
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Team 1 Problem 40
Compute, in terms of x,
gcd x8 − 3x + 2, x9 + 3x4 + x3 − 1 .
Team 1 Problem 41
The polynomial P (x) = x20 − x16 + 1 has 20 non-zero complex roots r1 , r2 , . . . , r20 , which are not
necessarily distinct. Compute
X ri
.
rj
i6=j
Team 1 Problem 42
An ellipse E has minor axis of length 40 and major axis of length 58. A circle Ω is drawn so that the
two foci of E are diametrically opposite points on Ω. The figures E and Ω intesect at four distinct
points. The
√ area of the quadrilateral defined by these four points can be expressed in simplest radical
form as a b/c, for square-free b. Find the sum of the distinct prime factors of a, b, and c.
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Team 1 Problem 43
Compute
sin(4◦ )
sin(4◦ )
sin(4◦ )
+
+ ··· +
.
◦
◦
◦
◦
sin(30 ) sin(34 ) sin(34 ) sin(38 )
sin(146◦ ) sin(150◦ )
Team 1 Problem 44
A regular pyramid is such that its base is a square of side length 27 and its lateral faces are equilateral
triangles. Four regular tetrahedrons are formed from these lateral faces, and they all point outward
from the original pyramid. Their top-most vertices form a square. Find that square’s area.
Team 1 Problem 45
Let ABCD be a convex cyclic quadrilateral inscribed in a circle of radius 30 satisfying DA = DC =
28 and DB = 49. If P is the intersection of lines AC and BD, compute
AB · BC − AP · P C.
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Team 1 Problem 46
Let H be the number of zeros at the end of the decimal expansion of
(1!) · (2!) · (3!) · · · (2016!).
Find the remainder when H is divided by 1000.
Team 1 Problem 47
Two infinite parallel mirrors are 20 feet apart. A ray of light originates from some point directly in
between the two mirrors, and propagates at a non-zero angle relative to the perpendicular between
the mirrors. Simultaneously, the mirrors each start moving towards the center at a speed of 1 foot
per second. Once the mirrors collide, the light is 450 feet away from its point of origin. How many
feet away would it have been if the light’s original direction of propagation had been rotated 90◦ in
either direction? Assume the speed of light is 53 feet per second.
Team 1 Problem 48
Let ω be a complex number such that ω 2016 = 1 but ω n 6= 1 for all 1 ≤ n ≤ 2015. Compute
2016
1 X ω 2016−i
.
2016 i=1 ω i + 2
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Team 1 Problem 49
How many ways are there to arrange the letters in the phrase “fourier analysis” into a different
phrase of two words (not necessarily real words) such that each word has at least five letters and is
spelled in alphabetical order? For example, the phrase ”aailnssy efiorru” is such a phrase.
Team 1 Problem 50
Consider every pair of positive integers (m, n) such that m | n and n | 1010 . What is the product of
n
, as we go over every such pair (m, n)?
the values of m
Team 1 Problem 51
There are 362880 nine-digit numbers that can be formed from a permutation of the digits 1 through
9. What fraction of these numbers are divisible by 27?
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