Download Lehigh 2006 (no calculators)

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?
9
x2 + 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?
4
Set 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?
71
x + 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
Question
Amy Bob Chris
Correct
1
F
T
T
T or F
2
F
F
T
F
3
T
F
F
T or F
4
T
T
F
T
5
T
T
T
T
6
T
T
T
T
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 5 . Let x be the
smallest angle of the triangle. What is cos x ?
5 / 3 . 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 1011 will 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.
3 2 2  3 2 2 .
Let x =
3  2 2  3  2 2 and square both sides.
Thus x2 = 4. The answer must be positive because 3  2 2 is larger than
3  2 2 , 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?
4
2 3 2 or 2 3 .
17. Which integer is closest to ½ ( 3 829  log10 829 )?
6.
Since 93 = 729 and 103 = 1000, 3 829 is slightly greater than 9. Since
log10 100  2 and log10 1000  3 , log10 829 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 k is a b , and the correct simplification of 3 k is
b3 a ?
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.