University of Mississippi
th
10
Individual Competition
Annual High School Mathematics Contest
1. When the expression (2x − 3)11 is expanded, what is the
sum of the coefficients?
(a) -1
(b) 1
(c) 4
(d) 34
(e) 146
(a)
(b)
(c)
(d)
(e)
5
6
7
8
impossible to achieve
7. What is
s
2. A function that satisfies f (f (f (x))) = f (4x) is
(a) f (x) = ex
(b) f (x) = x
(c)
f (x) = 2x
(d) f (x) = 4x
(e) none of the above
3. The sum 1 + 3 + 5 + · · · + (2n − 1) is given by:
(a) n2
(b) (n − 1)2
(c) n(n+1)
2
(d) 2n2
(e) none of the above
4. What is the value of the digit A if the five digit number
12A3B is divisible by 4 and 9, and A is different from B.
(a) 1
(b) 2
(c) 3
(d) 4
(e) 6
5. Jordan went on a 10 mile bike ride. For the first 5 miles,
his average speed was 15 miles-per-hour. Then, over the
remaining 5 miles, his average speed was 20 miles per hour.
What was his average speed over the entire bike ride? (to
the nearest decimal point)
(a) 16 mph
(b) 17.1 mph
(c) 17.5 mph
(d) 17.8 mph
(e) 18 mph
6. What number of sides must a polygon have in order to have
a sum of interior angles equal to 720◦ ?
r
2+
(a)
(b)
(c)
(d)
(e)
q
2+
2+
√
2 + ...
?
1
√
2
2
√
2 2
∞
8. What is the greatest common divisor of 15! and of 15!+132 .
(15! = 15 · 14 · . . . 3 · 2 · 1)
(a) 1
(b) 13
(c) 15
(d) 13!
(e) 15!
9. What is the largest number N of positive integers, none
of which is larger than 17, and no two of which share a
common divisor other than 1?
(a) 7
(b) 8
(c) 9
(d) 10
(e) 17
10. If an+1 =
(a)
(b)
(c)
nan
2 ,
a1 = 1, what is an ?
n!
2n
(n−1)!
2n−1
n!
2n−1
(n−1)!
2n
(d)
(e) none of the above
11. Simplify the expression log2 3 log3 4 · · · log14 15 log15 16.
(a)
(b)
(c)
(d)
(e)
1
√
2 2
e2
4
none of the above
2014 UNIVERSITY OF MISSISSIPPI INDIVIDUAL COMPETITION
2
12. What is the probability that 3 randomly chosen points on 17. In studying an extraterrestrial artifact, we come across the
a circle fall on a semi-circle?
following multiplication:
(a) 1
15 · 15 = 321.
(b) 3
4
(c) 12
(d) 78
(e) 0
√
13. In triangle ABC, sin A + cos B =
What is the measure of ∠C?
(a)
(b)
(c)
(d)
(e)
30◦
45◦
60◦
90◦
120◦
3
2
and sin B + cos A = 32 .
Knowing that aliens use a different number base than we
do, find the base for which this is correct.
(a) 4
(b) 5
(c) 6
(d) 9
(e) none of the above
18. It would take Tom 2 hours to paint the fence and Huck
would take 3 hours to paint it. They work on it together.
How long does it take them to paint the fence? (Assume
they don’t get in each other’s way.) (to the nearest decimal)
(a) 1 hour
(b) 1.2 hours
14. A band is marching in a parade at the speed of 80 meters
(c) 1.5 hours
per minute. A drummer, at the very back, has a dog who
is walking next to him. The bandleader, who is 20 meters
(d) 1.7 hours
ahead of the drummer, brings out a dog treat. The dog
(e) 2 hours
runs at 160 meters per minute to get the treat and then
runs back to the drummer. How far has the dog travelled?
19. I have 4 pairs each of 7 colors of socks. They are all jumbled
(a) 40 meters
together in my sock drawer. How many socks do I have to
(b) 100
3 meters
pull out to be sure to get a pair of the same color?
(c) 40.7 meters
(a) 7
(b)
20
(d) 160
meters
3
(c) 8
(e) none of the above
(d) 27
(e) 4
15. A committee of six mathematicians agrees to review textbooks. Each mathematician agrees to review exactly 22
textbooks and each textbook will be reviewed by exactly 3 20. A point is chosen at random from a circular board 10 inches
mathematicians. How many textbooks are there?
in diameter. What is the probability that the point is closer
(a) 22
to the center than to the boundary?
(a) 25π
(b) 44
2
1
(c) 66
(b) 4
(d) 396
π
(c) 25
(e) not enough information
1
(d) 2
(e) none of the above
16. One hundred balls are labelled with the numbers from 1 to
100 and are placed in a bag. Five balls are drawn from the
21. A 10-inch by 10-inch square is laid down on top of another
bag. What is the probability that the number on the first
square of the same size. One of the vertices of the top
ball is larger than the number on the last ball?
square is placed at the center of the bottom square (see fig(a) 0
ure). If we rotate the top square, keeping this vertex above
(b) 1/2
the center of the bottom square, what is the most area of
the bottom square we can cover?
(c) 2/3
(d) 49/50
(e) 499/500
2014 UNIVERSITY OF MISSISSIPPI INDIVIDUAL COMPETITION
(a) 20 square inches
(b) 25 square inches
(c) 27.5 square inches
(d) 50 square inches
(e) None of the above
3
22. How many zeros appear at the end of 2014! ? (recall that
2014! = 2014 · 2013 · 2012 · . . . 4 · 3 · 2 · 1)
(a) 250
(b) 251
(c) 300
(d) 501
(e) 565