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EF Exam
Texas A&M Math Contest
23 October, 2010
(NOTE: If units are appropriate, please include them in your answer.)
1.
How many 3-digit positive integers have digits whose product equals 24?
2.
△ABC has integer side lengths, and BD bisects ∠ABC. If AD = 3 and DC = 8,
what is the smallest possible value of the perimeter?
3.
The polynomial x3 −ax2 +bx−2010 = 0 has three positive integer solutions. What
is the smallest possible value of a?
4.
1
For nonzero numbers x and y define the operation ♠(x, y) = x − . Given that
y
B = ♠(a, ♠(a, a)) is defined, but ♠(a, B) is undefined, list all possible values of a.
x − 2y − 3z = 2
5. Given the system of equations x − 4y − 13z = 14 has an infinite number of soax + by + cz = 0
b+c
lutions, what is
?
a
6.
Given rectangle ABCD and
√ a point P inside the rectangle such that
P A = 4, P B = 3, P C = 10. Find P D.
7.
Given △ABC with cos(2A − B) + sin(A + B) = 2 and AB = 4, what is BC?
8.
Compute
9.
A frog makes 3 jumps, each 1 meter in length. The directions of the jumps are
chosen independently and at random. What is the probability that the frog’s final
position is at most 1 meter from its starting position?
tan 75◦ − 1
. Answer must be a single fraction with a rationalized detan 75◦ + 1
nominator.
1
For problems 10-11, given a set S, a permutation p on S is a function p : S → S
such that p is one-to-one and onto. Permutations are often written in cyclic
notation. For example, if S = {1, 2, 3, 4, 5, 6} and p = (1 3 6)(2 5), then p(1) =
3, p(3) = 6, p(6) = 1, p(2) = 5, p(5) = 2, and p(4) = 4.
10.
For permutations p and q on the set S above, define the operator p ⊗ q to mean
“apply p, then apply q. Compute (1 4 2 5 3) ⊗ (2 4 6)(3 5). Write your answer in
cyclic notation, starting with 1.
11.
Since p = (1 4 2 5 3) is one-to-one, it has an inverse (which is also a permutation).
Write p−1 in cyclic notation, starting with 1.
12.
You are given P and Q are monic quadratic polynomials, P (Q(x)) has zeros at
x = −23, x = −21, x = −17, and x = −15, and Q(P (x)) has zeros at
x = −59, x = −57, x = −51, and x = −49. Find the sum of the minimum values
of P and Q.
13.
Find the number of positive integers x ≤ 2010 such that 2x − x2 is not divisible
by 7.
14.
Given f (x) =
15.
x50
, find f (50) (0) (the fiftieth derivative of f evaluated at x = 0).
1−x
x2
y2
+
= 1 at the point (a, b). Find
9
4
positive values y1 and y2 such that, if b ∈ (y1 , y2), the y-intercept of N lies outside
the ellipse.
Let N be the line normal to the ellipse
16.
An ellipsewith
semi-axes 3 and 4 rolls without slipping along the curve
x
(c > 0). Given that the ellipse completes one revolution when it
y = c sin
3
traverses one period of the curve, find c.
17.
For any positive integer n, define hni to be the closest integer to
∞
X
2hni + 2−hni
.
n
2
n=1
2
√
n. Evaluate