1.
Consider the polynomial
f (x) = x4 − 6x2 + 12x3 − 4x + 7,
whose roots, when f (x) = 0, are α, β, γ, and δ. If
A
=
α + β + γ + δ,
B
=
αβγδ,
C
=
1
α
D
=
αβ + βγ + γα + βδ + γδ + αδ,
+
1
β
+
1
γ
+ 1δ , and
then compute A + B + C + D.
2.
For Question 2, examine the vectors v1 = h3, 1, −2i and v2 = h−1, 5, 4i. Let θ be the angle between them.
Find the value of the
3.
B
D
+
A
=
v1 · v2 (where · represents dot–product)
B
=
||v1 × v2 ||
C
=
cos θ (θ <
D
= ||v1 || · ||v2 || · sin θ
π
2)
A
C.
Consider the following sums:
(17)
(−6)
(24)
(−22)
1
11
1
1
31 + 32
∞
k
X
k=1
∞
X
2
33
+
3
34
+ ··· +
Fn
3n
+ · · · , where Fn is the nth fibonacci number with F1 = F2 = 1.
x
, where x > 0.
k!
k 2010
.
2k
k=1
1
1
1 + 2
∞
X
k=1
+
+
1
k
k−2
1
3
+
1
4
+ ···.
.
Each of the sums has a value listed next to it in parenthesis. Let P represent the sum of the parenthesis whose
expressions converge (that is, the ones that are summable). Let Q represent the sum of the ones that don’t
converge. What is P Q?
4.
Suppose that
∞
X
k=2
1
=
k3 − k
m
n
where m and n are relatively positive prime integers. Compute m + n.
5.
On the interval [0, 2012π], cos 3x − cos 5x = 0 has N solutions. Let n be the sum of the digits of N . What is
n?
6.
Suppose that a + b = 3, and a3 + b3 = 27 and a < b. Let
What is the value of A + B + C + D ?
A
= a−b
B
= a2 + b2
C
= ab
D
= a3 − b3
7.
Refer to the matrix:

1
M = 0
−5
What is
A
B
+
C
D

−3
4 .
7
2
−1
6
if
A
B
C
D
= the determinant of M,
= The sum of the eigenvalues of M ,
= The product of the eigenvalues of M , and
= The trace of M .
8.
Peter is a swimmer, and a fast one at that. Paddling downstream in the rapids with a current of 15mph for 50
miles, and then returning upstream (okay, yea, he is sort of a superhero at that speed) to the same point from
which he left, takes him the same amount of time that it would take him to travel 200 miles in still water.
Find how much time it would take him to travel 20 miles downstream in a calmer river with current of 3mph
(an un-rationalized value is acceptable).
9.
Consider
f (a) = a +
1
2a +
1
1
2a+ 2a+···
.
Let
A
B
C
D
= f (2012)
= f (2011)
= f (2010)
= f (2009).
Compute A2 − B 2 + C 2 − D2 .
10.
Jingyi, Michelle, Leo, and Jeremy all have awful eyesight. So, take it from me when I say this, it is extremely
interesting to watch them play a game of darts. Anyways, each, in turn, takes a stab at trying to hit the bulls
eye; but, because each person forgot his or her glasses, there is only a 14 probability that he or she will hit the
target. The first person to hit the bulls eye wins the game. Suppose
A
B
Let
11.
A
B
=
=
=
The probability that Jingyi wins if she goes first, Michelle second, Jeremy third, and Leo last.
The probability that Jingyi wins if Jeremy goes first, Michelle second, Jingyi third, and Leo last.
m
n
where m and n are relatively prime positive integers. What is m + n?
Consider the general ellipse
(x − h)2
(y − k)2
+
= 1.
a2
b2
Inscribe a rectangle in the ellipse such that the rectangle has maximum area. In the following calculations,
assume a = 12 and b = 8.
A
B
C
D
What is AB − CD?
= The area of
= The area of
= The area of
= The area of
the
the
the
the
rectangle
rectangle
rectangle
rectangle
when
when
when
when
h = 7 and k
h = 1 and k
h=√
1 and k
h = 5 and
= 9.
= 12.
= 1. √
k = 7.
12.
Consider a table with n urns and an eager game player who wants to toss k balls into the n urns (an urn
doesn’t have to receive any balls; the player, could, for example, toss the k balls into just one urn — believe
me, they won’t fill up). Let
A
B
C
D
= The number of ways
= The number of ways
= The number of ways
= The number of ways
to
to
to
to
distribute
distribute
distribute
distribute
them
them
them
them
when
when
when
when
n = 2012 and k = 1.
n = 10 and k = 3.
n = 3 and k = 10.
n = 4 and k = 7.
Compute A + B + C + D.
13.
What is the sum
π
π
π
π
sin
+ sin
cos ?
10
10
10
10
(Leave your answer in its simplest possible form.)
cos
14.
In chemistry, the subject of balancing equations is extremely important. Here’s why: different compounds
and molecules have different atomic weights (though they are extremely small, they still have weight). When a
chemical reaction takes place between certain elements/compounds it is obvious that the weight can’t change.
That is to say, if we could measure the mass of the stuff we put it, it must equal the mass of the stuff we get
out (even though it is in a different form). Consider the reaction:
a · CH4 + b · O2 =⇒ c · CO2 + d · H2 O,
where a, b, c, and d are the coefficients of the products/reactants we are working with. Find the sum of a, b, c,
and d (all 4 are relatively prime).
For clarification, suppose a = b = c = d = 1. If you take the coefficients and multiply them by the subscripts
on each molecules, the number of C’s, O’s, and H’s won’t equate on both sides. Hence, these aren’t the
coefficients that we are looking for.
15.
An elementary school class wishes to lineup in the lunch line to eat. Interestingly enough, all the gentleman in
the class got into an argument on the way to lunch, and hence the teacher won’t let them stand next to one
another while in line.. Suppose S is the number of ways for the children to line up if there are 4 girls and 3
boys. Likewise, let T be the number of ways for the kids to line up if there are 7 girls and 4 boys. What is the
difference T − S ? (No one child is a clone of another; e.g. they are distinguishable).