Question # 1 Alpha Bowl
FAMAT State Convention 2014
•
Let A be the smaller of the two solutions of the quadratic equation x(2x + 5) = −3 .
•
⎛ 5π ⎞
Let B equal the exact value of sec ⎜ − ⎟
⎝ 6 ⎠
•
•
Suppose f (x) = 2x − 5 and g(x) = x − 9 . Let C equal ( f  g −1 )(7)
3
1+
x =5
Let D be the largest solution for x satisfying:
1
1−
x
Use the problems above to find the value of A ⋅ B ⋅C ⋅ D .
Question # 1 Alpha Bowl
FAMAT State Convention 2014
•
Let A be the smaller of the two solutions of the quadratic equation x(2x + 5) = −3 .
•
⎛ 5π ⎞
Let B equal the exact value of sec ⎜ − ⎟
⎝ 6 ⎠
•
•
Suppose f (x) = 2x − 5 and g(x) = x − 9 . Let C equal ( f  g −1 )(7)
3
1+
x =5
Let D be the largest solution for x satisfying:
1
1−
x
Use the problems above to find the value of A ⋅ B ⋅C ⋅ D .
Question # 2 Alpha Bowl
FAMAT State Convention 2014
•
The points (2,5) , (−3, A) , and (1− A,13) are on the same line with positive slope. Find
the value of A.
•
Find the least integral value of B such that log B 2 3 ⋅ 3⋅ 5 2 ≤ 5 .
•
Let C be the amplitude of the graph f (x) =
•
Let D be the value of the determinant,
(
)
2 3 ⎛4
1⎞
− sin ⎜ π x − ⎟ .
5 5 ⎝5
5⎠
1− i 1− 5i
1+ i 2 − 2i
, where i 2 = −1 .
Use the problems above to find the value of A ⋅C + B + D .
Question # 2 Alpha Bowl
FAMAT State Convention 2014
•
The points (2,5) , (−3, A) , and (1− A,13) are on the same line with positive slope. Find
the value of A.
•
Find the least integral value of B such that log B 2 3 ⋅ 3⋅ 5 2 ≤ 5 .
•
Let C be the amplitude of the graph f (x) =
•
Let D be the value of the determinant,
(
)
2 3 ⎛4
1⎞
− sin ⎜ π x − ⎟ .
5 5 ⎝5
5⎠
1− i 1− 5i
1+ i 2 − 2i
Use the problems above to find the value of A ⋅C + B + D .
, where i 2 = −1 .
Question # 3 Alpha Bowl
FAMAT State Convention 2014
8
∑2
⎛ nπ ⎞
cos ⎜ ⎟ .
⎝ 2 ⎠
•
Let A be the value of the sum
•
Let B be the sum of the x- and y-coordinates of the focus in the first quadrant for the
curve: 9x 2 + 18x + 64y = 199 + 16y 2 .
•
Let C be the total number of distinct rearrangements of all the letters in the word
FAMAT.
•
Let D be the length of the largest side in a triangle. The two smaller side lengths of a
triangle measure 12 and 20 units and the largest angle in the triangle measures 120° .
n
n=0
Use the problems above to find the value of A − (B + C + D) .
Question # 3 Alpha Bowl
FAMAT State Convention 2014
8
∑2
⎛ nπ ⎞
cos ⎜ ⎟ .
⎝ 2 ⎠
•
Let A be the value of the sum
•
Let B be the sum of the x- and y-coordinates of the focus in the first quadrant for the
curve: 9x 2 + 18x + 64y = 199 + 16y 2 .
•
Let C be the total number of distinct rearrangements of all the letters in the word
FAMAT.
•
Let D be the length of the largest side in a triangle. The two smaller side lengths of a
triangle measure 12 and 20 units and the largest angle in the triangle measures 120° .
n=0
n
Use the problems above to find the value of A − (B + C + D) .
Question # 4 Alpha Bowl
FAMAT State Convention 2014
•
Let f (x) = rx + s for real numbers r and s and suppose f ( f ( f ( f ( f (x))))) = 32x − 93 for
all real x-values. Find the value of A = r ⋅ s .
•
3
When ( 2x − )6 is expanded, let B be the value of the constant term.
x
•
Let C be the area of the triangle two of whose sides have lengths of 4 and 9 with an
included angle measuring 150° .
•
Let D be the value of ⎡⎢ log 5 2013⎤⎥ + ⎢⎣ log 2013 5 ⎥⎦ .
Use the problems above to find the value of
B
+D.
A ⋅C
Question # 4 Alpha Bowl
FAMAT State Convention 2014
•
Let f (x) = rx + s for real numbers r and s and suppose f ( f ( f ( f ( f (x))))) = 32x − 93 for
all real x-values. Find the value of A = r ⋅ s .
•
3
When ( 2x − )6 is expanded, let B be the value of the constant term.
x
•
Let C be the area of the triangle two of whose sides have lengths of 4 and 9 with an
included angle measuring 150° .
•
Let D be the value of ⎡⎢ log 5 2013⎤⎥ + ⎢⎣ log 2013 5 ⎥⎦ .
Use the problems above to find the value of
B
+D.
A ⋅C
Question # 5 Alpha Bowl
FAMAT State Convention 2014
•
On the interval [0,2π ) , let A equal the largest solution to
sec 2 θ − tan 2 θ − cos 2 θ
=3
csc 2 θ − cot 2 θ − sin 2 θ
•
⎧(x − 1)2 + (y + 4)2 ≤ 100
Let B be the area of the region enclosed by ⎨
.
⎩ y ≥ 20x − 24
Let C be the greatest number of sides in a convex polygon with at most 40 diagonals.
•
Let D be the smallest root of the polynomial p(x) = 6x 3 + 7x 2 − 1 .
•
Use the problems above to find the value of
B⋅D
.
A ⋅C
Question # 5 Alpha Bowl
FAMAT State Convention 2014
•
On the interval [0,2π ) , let A equal the largest solution to
sec 2 θ − tan 2 θ − cos 2 θ
=3
csc 2 θ − cot 2 θ − sin 2 θ
•
⎧(x − 1)2 + (y + 4)2 ≤ 100
Let B be the area of the region enclosed by ⎨
.
y
≥
20x
−
24
⎩
Let C be the greatest number of sides in a convex polygon with at most 40 diagonals.
•
Let D be the smallest root of the polynomial p(x) = 6x 3 + 7x 2 − 1 .
•
Use the problems above to find the value of
B⋅D
.
A ⋅C
Question # 6 Alpha Bowl
FAMAT State Convention 2014
•
When the sum i1 + i 2 + i 3 + ...+ i 2014 is simplified to a single complex number, let A denote
the modulus of the complex number. Note that i = −1 .
•
Let B be the minimum value of f (x) = 4 cos 3x + 4 sin 3x .
•
⎡ log 2 5
3
1
4
⎢
log 5 7
−1
5
⎢ 0
Let C be the value of the determinant ⎢
0
0
log 7 11
9
⎢
⎢ 0
0
0
log11 16
⎣
•
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Let D be the number of distinct intersection points of the polar curves r = 2 and
r = 1− 2sin θ .
Use the problems above to find the value of A ⋅ B + C ⋅ D .
Question # 6 Alpha Bowl
FAMAT State Convention 2014
•
When the sum i1 + i 2 + i 3 + ...+ i 2014 is simplified to a single complex number, let A denote
the modulus of the complex number. Note that i = −1 .
•
Let B be the minimum value of f (x) = 4 cos 3x + 4 sin 3x .
•
⎡ log 2 5
3
1
4
⎢
log 5 7
−1
5
⎢ 0
Let C be the value of the determinant ⎢
0
0
log 7 11
9
⎢
⎢ 0
0
0
log11 16
⎣
•
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Let D be the number of distinct intersection points of the polar curves r = 2 and
r = 1− 2sin θ .
Use the problems above to find the value of A ⋅ B + C ⋅ D .
Question # 7 Alpha Bowl
FAMAT State Convention 2014
•
Polynomials p(x) and q(x) have degrees of 7 and 12, respectively. Let A be the maximum
number of intersection points between their graphs.
•
Let B equal the maximum value of n so that n − 2, 8 − 3n, 2n forms a geometric
progression when written in that order.
•
3
Let C be the simplified value of tan(2 cos −1 ) .
5
•
⎛ D ⎞
⎛ n⎞
Find D so that ⎜
= 35 . Note that this ⎜ ⎟ notation is a combination.
⎟
⎝ D − 3⎠
⎝r⎠
Use the problems above to find the value of A ⋅ B + C ⋅ D .
Question # 7 Alpha Bowl
FAMAT State Convention 2014
•
Polynomials p(x) and q(x) have degrees of 7 and 12, respectively. Let A be the maximum
number of intersection points between their graphs.
•
Let B equal the maximum value of n so that n − 2, 8 − 3n, 2n forms a geometric
progression when written in that order.
•
3
Let C be the simplified value of tan(2 cos −1 ) .
5
•
⎛ D ⎞
⎛ n⎞
Find D so that ⎜
= 35 . Note that this ⎜ ⎟ notation is a combination.
⎟
⎝ D − 3⎠
⎝r⎠
Use the problems above to find the value of A ⋅ B + C ⋅ D .
Question # 8 Alpha Bowl
FAMAT State Convention 2014
•
•
•
Let A be the value of the arithmetic series (−23 − 6π ) + (−20 − 5π ) + ...+ (25 + 10π ) .
Let B be the sum of the coefficients of the quadratic equation whose graph passes through
the points (2, -3), (-1, -6), and (6, -13).
Let C be the circumference of the circumcircle of a triangle whose side lengths are 8, 15,
and 17.
Use the problems above to find the value of A − B − 2C.
Question # 8 Alpha Bowl
FAMAT State Convention 2014
•
•
•
Let A be the value of the arithmetic series (−23 − 6π ) + (−20 − 5π ) + ...+ (25 + 10π ) .
Let B be the sum of the coefficients of the quadratic equation whose graph passes through
the points (2, -3), (-1, -6), and (6, -13).
Let C be the circumference of the circumcircle of a triangle whose side lengths are 8, 15,
and 17.
Use the problems above to find the value of A − B − 2C.
Question # 9 Alpha Bowl
FAMAT State Convention 2014
•
Let A be the cosine of the angle between the vectors 3,−6,6 and −1,−2,−2 .
2
• An unfair coin has the probability of
to show heads when tossed. Let B be the
3
probability that when tossed 3 times, the coin shows heads exactly once.
∞
j
• Let C be the value of the infinite series ∑ j .
j=1 10
C
Use the problems above to find the value of
.
A⋅B
Question # 9 Alpha Bowl
FAMAT State Convention 2014
•
Let A be the cosine of the angle between the vectors 3,−6,6 and −1,−2,−2 .
2
• An unfair coin has the probability of
to show heads when tossed. Let B be the
3
probability that when tossed 3 times, the coin shows heads exactly once.
∞
j
• Let C be the value of the infinite series ∑ j .
j=1 10
C
Use the problems above to find the value of
.
A⋅B
Question # 10 Alpha Bowl
FAMAT State Convention 2014
•
Let A be the maximum value of the function f (x) = sin ( x ) ⋅ tan ( x ) + cos ( 2x ) ⋅sec ( x ) .
•
Let B be the number of factors of x17 − x when written as a product of irreducible
polynomials of degree at least 1 with integer coefficients.
•
The interior angle measures of a convex nonagon are consecutive integers in degrees.
Let C denote the smallest of these angle measures.
•
Let D be the value of 999 ⊗ (68 ⊗ (60 ⊗ (45 ⊗15))) where ⊗ is defined as follows:
⎧⎪ the largest odd divisor of a + b,
a⊗b = ⎨
the maximum of a or b,
⎩⎪
if a + b is even
if a + b is odd
Use the problems above to find the value of A + C − B − D .
Question # 10 Alpha Bowl
FAMAT State Convention 2014
•
Let A be the maximum value of the function f (x) = sin ( x ) ⋅ tan ( x ) + cos ( 2x ) ⋅sec ( x ) .
•
Let B be the number of factors of x17 − x when written as a product of irreducible
polynomials of degree at least 1 with integer coefficients.
•
The interior angle measures of a convex nonagon are consecutive integers in degrees.
Let C denote the smallest of these angle measures.
•
Let D be the value of 999 ⊗ (68 ⊗ (60 ⊗ (45 ⊗15))) where ⊗ is defined as follows:
⎧⎪ the largest odd divisor of a + b,
a⊗b = ⎨
the maximum of a or b,
⎪⎩
if a + b is even
if a + b is odd
Use the problems above to find the value of A + C − B − D .
Question # 11 Alpha Bowl
FAMAT State Convention 2014
•
•
2x 2 − 8x + 6
:
x2 − x
Let A be the sum of the x- and y-coordinates for the hole in the graph.
Let B equal lim f (x) .
•
•
Let x = C be the equation of the vertical asymptote.
Let D be the sum of x-values where f (x) = 0 .
For the rational function f (x) =
x→−∞
Use the problems above to find the value of A B + C D .
Question # 11 Alpha Bowl
FAMAT State Convention 2014
2x 2 − 8x + 6
For the rational function f (x) =
:
x2 − x
• Let A be the sum of the x- and y-coordinates for the hole in the graph.
• Let B equal lim f (x) .
x→−∞
•
•
Let x = C be the equation of the vertical asymptote.
Let D be the sum of x-values where f (x) = 0 .
Use the problems above to find the value of A B + C D .
Question # 12 Alpha Bowl
FAMAT State Convention 2014
•
Let A be the largest solution to 4 x + 2 x+2 = 32 .
•
⎛π⎞
⎛π⎞
Let B be the value of sin ⎜ ⎟ + cos ⎜ ⎟ .
⎝ 12 ⎠
⎝ 12 ⎠
•
A rectangular prism with a square base has a volume of 54 cubic units and a total surface
area of 90 square units. Let C be the largest distance between any two vertices of the
prism.
Use the problems above to find the value of A ⋅ B ⋅C .
Question # 12 Alpha Bowl
FAMAT State Convention 2014
•
Let A be the largest solution to 4 x + 2 x+2 = 32 .
•
⎛π⎞
⎛π⎞
Let B be the value of sin ⎜ ⎟ + cos ⎜ ⎟ .
⎝ 12 ⎠
⎝ 12 ⎠
•
A rectangular prism with a square base has a volume of 54 cubic units and a total surface
area of 90 square units. Let C be the largest distance between any two vertices of the
prism.
Use the problems above to find the value of A ⋅ B ⋅C .
Question # 13 Alpha Bowl
FAMAT State Convention 2014
•
Let A be the units digit of 5 2014 + 12 2014 + 132014 .
•
In base B, 154 B + 435 B = 622 B . Find the value of B.
•
Given xy = 1 and x + y = 1 , let C be the value of x 3 + x 2 + x + y 3 + y 2 + y + 1
•
Let D be the number of non-congruent triangles with integer side lengths that have a
perimeter of 16.
Use the problems above to find the value of A + B + C + D .
Question # 13 Alpha Bowl
FAMAT State Convention 2014
•
Let A be the units digit of 5 2014 + 12 2014 + 132014 .
•
In base B, 154 B + 435 B = 622 B . Find the value of B.
•
Given xy = 1 and x + y = 1 , let C be the value of x 3 + x 2 + x + y 3 + y 2 + y + 1
•
Let D be the number of non-congruent triangles with integer side lengths that have a
perimeter of 16.
Use the problems above to find the value of A + B + C + D .
Question # 14 Alpha Bowl
FAMAT State Convention 2014
•
Let A be the total number of ordered triples of non-negative integers (x, y, z) such that
x + y + z = 10 .
•
A lattice point (x, y) is a point in a 2-dimensional coordinate plane where both x and y
are integers. Let B be the number of lattic points that satisfy (x − 1)2 + (y + 3)2 ≤ 9 .
•
Let C be the smallest positive integer that has exactly 7 positive integer factors.
•
The equation x 2 + 8x + 9 = 0 has complex solutions of x = r and x = s. Let D be the
value of r 2 + s 2 .
Use the problems above to find the value of ( A − B ) − ( C − D ) .
Question # 14 Alpha Bowl
FAMAT State Convention 2014
•
Let A be the total number of ordered triples of non-negative integers (x, y, z) such that
x + y + z = 10 .
•
A lattice point (x, y) is a point in a 2-dimensional coordinate plane where both x and y
are integers. Let B be the number of lattic points that satisfy (x − 1)2 + (y + 3)2 ≤ 9 .
•
Let C be the smallest positive integer that has exactly 7 positive integer factors.
•
The equation x 2 + 8x + 9 = 0 has complex solutions of x = r and x = s. Let D be the
value of r 2 + s 2 .
Use the problems above to find the value of ( A − B ) − ( C − D ) .