University of Mississippi
High School Mathematics Contest – Individual Competition
March 24, 2006
1. How many numbers are in the set {747, 748, 749, . . . 1048, 1049}?
(a) 301
(b) 302
(c) 303
(d) 1049
2. Which of the following is equal to log6 2 + log6 3?
(a) 1
(b) √
log6 2 √
log6 3
(c) 6 + 3
(d) log6 5
3. The area of a right triangle is 30, and the difference between the lengths of the two
legs is 7. What is the sum of the lengths of the two legs?
(a) 17
(b) 18
(c) 19
(d) 20
4. One square yard is equivalent to how many square feet?
(a) 1/27
(b) 1/3
(c) 27
(d) 9
5. When 1.00234152 is converted to a fraction and simplified, let the numerator of the
resulting fraction be called n and the denominator d. Which of the following is true?
(a) n < d
(b) n > d
(c) n = d
(d) n = 100d
6. Which of the answer choices is the largest?
(a) 4.1 × 10−25
(b) 9 × 10−26
(c) 4.1 × 10−25 − 10−50
(d) 40 × 10−24
7. Find the probability of rolling a prime number with the roll of a die (singular of dice).
(a) 1/6
(b) 1/3
(c) 1/2
(d) 2/3
8. Which of the following numbers is a solution to the equation
x6 + x − 2
= 0?
24
(a) 1
(b) 0
(c) -1
(d) -2
9. The area of a 45◦ sector of a circle of radius 2 is nearest to which of the following?
(a) 0
(b) 1
(c) 2
(d) 3
10. In right triangle 4ABC with right angle at vertex C and m∠B = 30◦ find the greatest
integer that is not greater than 100 sin A cos A tan A, csc A sec A cot A, and call it r.
Which of the following is true?
(a) r = 0
(b) 0 < r ≤ 100
(c) 100 < r ≤ 200
(d) 200 < r
11. The x-axis and the y-axis are both lines of symmetry for a particular square in the
coordinate plane. If the square has area 1, what area of the square is in the third
quadrant?
(a) 0
(b) 1/4
(c) 1/8,
(d) 1
12. If a bowler has a 9/10 probability of bowling a strike, then what is the probability of
the bowler bowling at least one strike on three particular attempts?
(a) 0.001
(b) 0.99
(c) 0.729
(d) 0.999
13. What is the tens digit in the product 7,234,015,542×345,106?
(a) 4
(b) 5
(c) 6
(d) 7
14. What is the sum of the values of x for which x2 = 1967099904?
(a) 44,352
(b) 88,704
(c) 68,682
(d) 0
kg·m2
15. The speed of light c is 3 × 108 m/s Given E = mc2 and E = 9 × 1015 s2 , find m.
(a) 0.33 kg
(b) 1 kg
(c) 1/10 kg
(d) 0.005 kg
16. The set {0, 0, 1, a, b, c} has a distinct mode. If the difference between the median and
the mode is 2, then what is a + b + c?
(a) 1
(b) 6
(c) 10
(d) 11
17. The following code is applied to a computer monitor with resolution 1024 by 768
(horizontal dimension by vertical dimension).
for y=1 to 768
plot pixel(y,y)
Describe the output.
(a) A horizontal line
(b) A vertical line
(c) A diagonal line that begins in one corner of the screen but does not end in the
other.
(d) A diagonal line that begins in one corner of the screen and ends in another.
18. If x2 − y 2 = 1, which of the following is equal to x + y?
(a) x2 + y 2
(b) x2 + xy + y 2
(c) x − y
(d) (x − y)−1
19. A circle is inscribed in a square whose area is 4. What is the area of the region that is
inside the square and outside the circle?
(a) π
(b) 4 − π
(c) 4 − 4π
(d) π/4
20. How many ordered pairs (a, b) of positive integers satisfy the following: a is divisible
by 7, b is divisible by 7, and a + b = 100?
(a) 0
(b) 1
(c) 5
(d) 14
21. Given that 1 + 2 + 3 + · · · + 100 = 5050, what is 2 + 4 + 6 + · · · + 202?
(a) 202
(b) 1,010
(c) 10,010
(d) 10,212
22. How many digits does
(a) 4
(b) 5
(c) 100
(d) 1,000
100100
10098
have?
√
√
80
−
5?
23. Which
of
the
following
is
the
same
as
√
(a) √45
(b) √ 65
(c) √75
(d) 35
qp
√
24. The number
x is an integer greater than 1. What is the smallest number of
digits x may have?
(a) 3
(b) 4
(c) 5
(d) 6
25. Consider the sequence defined by a1 = 3 and an = an−1 ! for n > 3. What is the
smallest number n such that an is divisible by the prime number 929? (Note: n! is
defined by n! = n · (n − 1) · (n − 2) · · · · · 2 · 1).
(a) 3
(b) 4
(c) 5
(d) 6
√ √
2
2
26. Let the foci of the ellipse x16 + y8 = 1 be A and B, and consider C(2 3, 2). What is
AC + BC?
(a) 8√
(b) 8
(c) 4
(d) 16
27. A circle in the plane has one chord whose endpoints are (0,10) and (8,8) and another
chord whose endpoints are (−4, −2) and (4, −1). What is the x-coordinate of its
center?
(a) 0
(b) 7/20
(c) 9/17
(d) 11/24
28. How many different integer values may
ative real numbers?
(a) 0
(b) 1
(c) 2
(d) infinitely many
mn
take on, where m and n are nonnegm2 + n2
29. Denote the side lengths of triangle 4ABC by a, b, and c where the side of length a is
across from vertex A, and that of length b is across from B. Given that sin A sin B =
sin(A + B), which of the following gives a formula for the area of 4ABC?
(a) ab sin C
(b) c2 /2
(c) abc/4(a + b + c)
(d) ab cos C/2
p
√
30. p
What is the smallest value that the function f (a, b) = 22 + a2 + 22 + (a − b)2 +
12 + (12 − b)2 takes on over real values for a and b?
(a) 13
(b) √
14
(c) √200
(d) 216