March Regional
Precalculus Condensed Team
1.
Compute the product: sin 20◦ · sin 40◦ · sin 80◦ .
2.
Let A represent the sum of the solutions to (1) and let B represent the product of the solutions to (2) where:
√
2x2 − 15x − 2 2x2 − 15x + 11
√
3x2 + 12x + 17
=
0,
(1)
=
3x2 + 12x.
(2)
What is A + B?
3.
Consider the vectors ~a < 1, 2, 3 > and ~b < 3, 2, 1 > with an acute angle of α between them. Let
A
B
C
D
= cos θ
= ~a · ~b
= ||~a|| · ||~b|| · sin α
= ||~r|| where r = ~a × ~b
Compute the product ABCD.
4.
Let
A
=
∞
X
1
2k
k=1
B
=
∞
X
1
3k
k=1
C
=
∞
X
k
4k
k=1
D
=
∞
X
2k + 1
5k
k=1
What is the value of ABCD?
5.
Let
A
=
B
=
C
=
D
=
lim
k→2
√
5 − 2x +
√
3−x
3x2 + 12x − 14
k→∞
x4 + 1
lim
x3 − 9x2 + 11
k→∞ 2x3 + 12x − 1
p
p
x2 + 12x − x2 − 4x + 7
lim
lim
k→∞
Find A + B + C + D.
6.
Find the sum of the units digits of the following 4 numbers:
9
32012 , 25625 , 20192 , 1210 .
7.
Clara and Jim play a game and to be fair, they alternate in turns (with, of course, Clara, the lady, starting
first). On whoever’s turn it is, the player draws a card from a standard deck of cards and rolls a die. If the
number shown on the die matches the number of the card drawn from the deck (count to 13 starting at 1 with
the ace) that player wins. If not, the card is replaced in the deck which is then shuffled and the other player
repeats this process. Let C represent the probability that Clara is victorious. Likewise, let J be the probability
that Jim wins. What is C − J?
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March Regional
8.
Precalculus Condensed Team
Consider the matrices:

1
 1
M =
 9
−1


2 3 0
1
4 3 2 
 and N =  −2
7 1 8 
6
2 5 −3

0 7
3 5 .
2 4
Suppose that D represents the determinant of M while S represents the sum of the elements in the inverse of
N . What is D · S?
9.
In the following 4 systems of equations find the value of x. For your answer, sum all 4 values of x (in the event
that an equation has more than one value for x, treat its part in the sum as 0).
1
x+2
=
1
2−x
1
x−1
=
1
x+2
+2
1
x−1
+
1
x−10
=
1
x−6
+
1
x−5
1
x−2
+
1
x−12
=
1
x−5
+
1
x−9
10.
In qiao-zhi-shi-jie, they use strange forms of currency. Namely, they are the yihan and the nina. Current
exchange rates between the dollar the currency of qiao-zhi-shi-jie are 31$/yihan and 29$/nina. iJustine is
traveling from America to this far off land, where, upon arrival, she wishes to buy an iPhone. Unfortunately,
in this country, you must pay exactly; i.e., you can’t pay 2 yihans for a 60 dollar item and get 2 dollars . .
. you must have exact amounts. Assume that the only two denominations of the currency that are currently
available for exchange are the aforementioned two. What is the maximum price the phone can cost, in terms
of this strange currency, that would stop iJustine from buying it? (That is, what is the largest number that
you cannot put into the form 31a + 29b for integers a and b).
11.
Suppose that
lim
2n sin
π n
is approximately a.bc (rounded to the nearest hundredth) where a, b, and c are digits. What is a + b + c?
x→∞
12.
KevJumba is on a boat. With his swim trunks and his flippie floppies, while he is flipping burgers and making
copies, he wishes to cross a river. Suppose the current of the river flows to the right at 1 m
s while the boat
is capable of moving at 2 m
.
At
what
angle
should
KevJumba
push
off
from
the
shore
to
minimize
his time
s
required to cross the river and make it to the other side (your answer should be in degrees) ?
13.
Consider a rectangle P ARK. At each of the four corners of the rectangle circles are drawn, each tangent to
two other circles, that have centers P , A, R, and K. A big circle is then drawn such that it is tangent to all 4
of the smaller circles (containing them and the rectangle completely). Suppose the radius of the large circle is
r. What is r in terms of the side lengths of the rectangle x, and y (x < y)?
14.
A triangle with sides of length 2, 4, and 5 is constructed, The the area of its incircle be I and the area of its
excircle be E, What is the difference between E and I?
15.
Left f be given by:
r q
√
x x x x···
r q
f (x) = 1 +
.
√
x x x x···
r q
1+
√
x x x x···
1+
1 + ···
Let f (3) =
√
a+ b
c .
What is abc supposing that b is not divisible by the square of any prime?
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