Lehigh 2006 (no calculators)
1. 1/(1/3-1/4)=
12.
2. Dick is 6 years older than Jane. Six years ago he was twice as old as she was. How old is Jane now?
12 years. D = J + 6. D - 6 = 2(J-6). Solve to get J = 12.
3. A bicyclist riding against the wind averages 10 mph from A to B, but with the wind averages 15 mph returning from B to A . What is his average speed for the trip?
12 mph. Let d = distance between A and B.
Average speed = total distance/total time = 2d/total time
Total time = d/10 + d/15 = d /6
Avg. speed = 2d / d/6 = 12.
4. What is the largest possible value for the sum of two fractions such that each of the four 1-digit prime numbers occurs as one of the numerators or denominators?
31/6.The primes are 2,3,5,7. Put the largest on top.
5/2 + 7/3 = 29/6 5/3 + 7/2 = 31/6. which is larger.
5. How many integers x in {1,2,3,...,99,100} satisfy that x2+ x3 is the square of an integer?
9x2 + x3 is square of an integer, so x2 (x+1) is a square, so (x+1) is a square. One less than all perfect squares up to 100: 3,8,15,24,35,48,63,80,99
6. What is the number of real numbers x such that 25|x| = x2 + 144?
4Set to zero and factor, writing pos and neg case.
x2 -25x + 144 = 0 or x2 +25x + 144 = 0
Either factor and solve or just check that the discriminant (b2-4ac) is greater than zero. The 4 solutions are -9,-16,9,16.
7. How many pairs (x,y) of positive integers satisfy 2x + 7 y = 1000?
71x + 7/2 y = 500. So y must be even to be an integer.
Let y = 2z, then x + 7z = 500.
z can be 1 through 71, then subtract to find x.
8. A ladder is leaning against a house with its bottom 15 feet from the house. When its bottom is pulled 9 feet farther away from the house, the upper end slides 13 feet down. How m any feet long is the ladder?
25 feet.Let x be the length of the ladder and h be the height up the side of the house. Initially the triangle gives x2=h2+152, then after moving, the triangle gives x2=(h-13)2+242. Set the 2 equations equal to each other and solve for h. Hence h = 20 and x = 25.
9. What is the sum of the three smallest prime numbers each of which is two more than a positive perfect cube?
159.When you start cubing the smallest primes and adding 2, you realize you need odd perfect cubes because an even cube plus 2 equals an even number. 1+2 = 3. 27+2= 29, 125 + 2 = 127. Adding these primes, 3 + 29 + 127 = 159.
10. Amy, Bob, and Chris each took a 6-question true-false exam. Their answers to the six questions in order were Amy: FFTTTT, Bob: TFFTTT, and Chris: TTFFTT. Amy and Bob each got 5 right. What is the most that Chris could have gotten right?
3.Make a table of answers
QuestionAmyBobChrisCorrect
1FTTT or F
2FFTF
3TFFT or F
4TTFT
5TTTT
6TTTT
Amy and Bob each only got one wrong. They agreed on 2, 4-6, so for questions 1 and 3 they each got one right and one wrong. Chris got 2 wrong out of questions 2 and 4-6, and he agreed with Bob for 1 and 3, so he got one additional question wrong.
11. The two shortest sides of a right triangle have lengths 2 and . Let x be the smallest angle of the triangle. What is cos x ?
. Us the Pythagorean Theorem to get the hypotenuse is 3. 2 is the smaller side, so it would be opposite the smaller angle. Then adj/hyp gives the answer.
12. From a point P on the circumference of a circle, perpendiculars PA and PB are dropped to points A and B on two mutually perpendicular diameters. If AB = 8, than what is the diameter of the circle?
16. Let the circle have a center at the origin, with x2+ y2= r2. Then P, A, and B form a rectangle with a corner at the origin (O). Thus the diagonal AB = PO, the radius of the circle. Since the radius is 8, the diameter is 16.
13. How many 9’s are in the decimal expansion of 999999899992 ? (This is the square of an 11-digit number.)
9. Hint: Rewrite as (1011-10001) and square it.
(1011-10001)2= 1022-(20002)1011 + (10001)2
= 1011 (1011 - 20002) + 100020001
Multiplying by 1011will give 11 zero digits, to which 100020001 will be added.
So all the “9” digits will come from 1011 - 20002 = 99999979998.
14. Let A be the point (7,4) and D be (-5,-3). What is the length of the shortest path ABCD, where B is a point (x,2) and C is a point (x,0)? This path consists of three connected segments, with the middle one vertical.
15.
15. Simplify .
2.Let x = and square both sides.
Thus x2 = 4. The answer must be positive because is larger than , so x = 2.
16. A square has its base on the x-axis, and one vertex on each branch of the curve y=1/x2. What is the area?
.
17. Which integer is closest to ½ ( )?
6.Since 93 = 729 and 103 = 1000, is slightly greater than 9. Since , is slightly less than 3. Thus, ½ (9. xxx + 2. xxx) is either ½(11.xxx) or ½(12.xxx). Both are close to 6.
18. In a 9-12-15 right triangle, a segment is drawn parallel to the hypotenuse one third of the way from the hypotenuse to the opposite vertex. Another segment is drawn parallel to the first segment one third of the way from it to the opposite vertex. Each segment is bounded by sides of the triangle on both sides. What is the area of the trapezoid inside the triangle between these two segments?
40/3.
19. A rhombus has sides of length 10, and its diagonals differ by 4. What is its area?
96.
20. What is the smallest positive integer k for which there exist integers a>1 and b>1 for which the correct simplification of is a, and the correct simplification of is ?
32. Solving for k, we have k = a2b and k = ab3. Setting them equal to each other, a = b2. The smallest values that work are b = 2 and a = 4. Thus k = 16x2=32.