Ohio Council of Teachers of Mathematics
Solutions to Thirty-First Annual Contest
February 28, 2004
1) 16
2) If not Reds, then Indians. Answer to the question “Which major league baseball team is least likely to win the World Series within your lifetime?”
3) Whole numbers are the positive integers and 0. and
4) 64 The simplest solution is to do this by brute force. Alternatively, if is n factored into unique primes, then the number of distinct positive factors of is . For this to equal 7, must be the only nonzero power of a prime. Thus, to make n (and hence ) as small as possible, choose .
5) The midpoint is .
6) N, A, S By definition, skew lines don’t intersect. The diagonals of a rhombus are ALWAYS perpendicular. Consecutive angles of a parallelogram are supplementary, and thus only congruent if the parallelogram is a rectangle.
7) 4 .
8) All real numbers, or .
9) 496
10) C For a line to contain another side of the square, it must be parallel or perpendicular to the given side , and thus must have a slope of 2 or .
11) Dan; 296 Note Dan received 40% of the votes, so George must have received 25% of the votes. If V is the total number of votes, then Dan had the highest percentage of votes and received votes.
12) 7/5 The probability that Mr. Knapke wins is . Thus, the odds he wins is .
13) 51 For an equation of the form to have two distinct real number solutions, the discriminant must be >0. For this problem, we need . Since k is positive, k is the smallest integer >50.2.
14) 5 Using the Principle of Inclusion-Exclusion, the number who like math or science is , leaving 5 students who like neither.
15) The term of has coefficient . The constant term of this expression is .
16) 90 games The number of games between distinct teams is the number of ways to select two teams to play, which is . Since each of these will occur twice, the total number of games is 90.
17) 8/9 If , cross-multiply given equation to get .
18) 2004 cm2 The surface area of the box with the hole is cm2.
19)
20)
Case I: – Then the inequality becomes or or (noting x is positive) , so the inequality holds for.
Case II: – Then the inequality becomes or which holds for or . Only the latter part of this solution set satisfies , so the inequality holds for .
21) 29.1% The original values of investments A, B, and C were and , respectively for a total original value of $313562.381. The current value is $222,444, so the total percent decrease is , or to the nearest tenth, 29.1%.
22) 49, 81, 100 The set of perfect squares less than 112 is {0,1,4,9,16,25,36,49,64,81,100} and the set of perfect cubes less than 112 is {0,1,8,27,64}. There are only 3 ways to pick three numbers from these two sets that sum to 112, namely 100+8+4, 81+27+4, and 64+49+9.
23) 1336 cm3 The volume of a cylinder with height h and radius r is . In this case, h=2r , so . The inscribe sphere will have the same radius and thus have a volume of cm3.
24) 9 marbles If only 8 marbles have been drawn, there could be two marbles of each color. However, a 9th marble selected guarantees 3 marbles of some color. This is an application of the Generalized Pigeonhole Principle.
25) . If the three roots are r, 2r, and s, then . In particular, and . Note, the sum of the roots gives and the product of the roots gives . Thus, , or . Graph to find it has one real zero at . Thus, and .
26) 207 km/hr Rotate the axes so that the air speed of the plane is 200km/hr, and its heading is along the positive x-axis. Then the wind of 30km/hr is blowing along at 80°. Then, the plane’s velocity vector would be , which has magnitude km/hr.
27) , so
28) 7 .
29) 2 Note , so is isosceles with one base angle at A. is also isosceles with the same base angle, and thus these two triangles are similar. Thus, .
30) Since , it is clear that the sum won’t have many terms and thus the easiest solution is to just notice , and hence .
31) (2004, ##) Finding the determinants, the equation becomes. Thus, . Since the determinant equation does not depend on b, b can be any number. An easy way to find one solution is to remove the determinants from the determinant equation to form an equation relating matrices. This equation yields and , so . Note this uses the property that the determinant of the product of two matrices is the product of the determinants.
32) The area of an ellipse is where a is half the length of its major axis and b is half the length of its minor axis. Thus, the given ellipse has area . The area above the x-axis is thus . The area of the big triangle above the x-axis is . Thus, the shaded area is .
33) Note for any m, , so in particular, . Thus, we need to solve . Thus, and n can be any odd number, so the smallest possible value for n is 3.
34) Note . We need to find b such that . This is most easily done be inspection since we know .
35) 300 monograms The last initial will be Z. The number of ways to have the monogram in alphabetical order is just the number of ways to select two of the remaining 25 letters and place them in order. This can be done in ways.
36) (with domain restricted to ) .
37) Draw in the radius OD. Then is a right angle and hence is a right triangle. If r is the length of the radius, then . Hence, . Thus, the diameter AB has length
38) Leonhard Euler His last name if pronounced “Oiler.” He lived from 1707 to 1783, earned a Masters degree in Philosophy at the age of 16, published his first paper at the age of 19, published over 380 articles during a span of 25 years while he was in Berlin, and had works published listing him as an author for 50 years after his death.
39) 358 feet Note . Thus, the possible dimensions of a rectangle with integral sides enclosing an area of 2004 ft2 are 1×2004, 2×1002, 3×668, 4×501, 6×334, 12×167. Of these, the one with smallest perimeter has dimensions 12×167 and thus the perimeter is feet. Note that the one of smallest perimeter is the one that has the two dimensions as close to each other as possible, thus, in some sense, making the rectangle as close to a square as possible.
40) 2:45 P.M. Since Mr. Abineri is canoeing upstream, his speed relative to the ground will be 6–3=3mph. Since it took him two hours to reach point B, point A and point B are 6 miles apart. Since Ms. Depoe is canoeing downstream, her speed relative to the ground will be 5+3=8mph. Thus, she can canoe the 6 miles to point A in hours, or 45 minutes. Hence, she arrives at point A at 2:45 P.M.
Solutions provided by: Dr. Gordon Swain, Ashland University
Dr. Christopher Swanson, Ashland University