1.
[PAST BOARD] A cross section through the center of a football is a circle
x inches in circumference. The football is x-8 inches long from tip to tip
and each seam is an arc of a circle 3/4 of x inches in diameter. Find x.
ANS. 20.69
2.
A coin is so unbalanced that you are likely to get two heads in two
successive throws as you are to get tails in one. What is probability of
getting heads in a single throw?
ANS. 0.618
3.
4.
A parking lot charges X for the first hour or fraction of an hour and 2/3 X
for each hour or fraction thereafter. Smith parks 7 times as long as Jones,
but pays only 3 times as much. How long did each park? (The clock
registers only in 5-minute intervals.)
ANS. 3-1/2
Every year an engineer consultant pays a bonus of $300 to his most
industrious assistant and $75 each to the rest of his staff. After how many
years would his outlay be exactly $6000 if all but two of his staff had
merited the $300 bonus, but none of them more than twice.
ANS. 7
5.
Find the sum of the infinite series 1+1/2+1/3+1/4+1/6+1/8+1/9+1/12+....,
whose terms are the reciprocals of the positive integers which are divisible
by no prime > 3.
ANS. 3
6.
Martian coins are 3-sided (heads, tails, and torsos), each side coming up
with equal probability. Three Martians decided to go odd-man-out to
determine who pays in dinner check. (If two coins come up the same and
one different, the owner of the latter coin foots the bill). What is the
expected number of throws needed in order to determine, the owner of
the latter coins come up the same and one different, the owner of the
latter coin foot the bill). What is the expected number of throws needed in
order to determine a loser?
ANS. 1-1/2
7.
8.
9.
Mr. Field, a speeder, travels on a busy highway having the same rate of
traffic flow in each direction. Except for Mr. Field, the traffic is moving at a
legal speed limit. Mr. Field passes one car for every nine which he meets
from the opposite direction. By what percentage is he exceeding the
speed limit?
ANS. 25%
In California, automobile license plates have three letters followed by
three of the digits from 0 to 9, not necessarily distinct. Is a randomly
chosen car more likely to have all 6 symbols different or at least one
repetition? (The zero and 0 are identical)
ANS. 0.617
There are three families, each with two sons and daughters. In how many
ways can all these young people be married?
ANS. 80
10. How many three digit telephone area codes are possible given that: (a)
the first digit must not be zero or 1; (b) the second digit must be zero or
one; (c) the third digit must not be zero; (d) the third digit may be one only
id the second digit is zero.
ANS. 136
11. Two octopi indulged in a friendly tentacle to tentacle wrestling match.
Each managed to pin 4 of his opponent’s tentacle with 4 of his own. In
how many ways was this possible?
ANS. 2822400
12. Find integer A, B, and C, positive or negative but non-zero, such that the
equation Ax2+Bx+C has roots A and B.
ANS. A=-2, B=4, C=16
13. At a cocktail party a man starts out with a glass of half whiskey and half
soda. After each sip he adds enough soda to fill the glass again.
Assuming that he does this continuously (that is, in infinitesimal sips), how
much whiskey has he consumed by the time he has drunk half a glass?
ANS. 1/5
14. In European countries the decimal point is often written a little above the
line. An American, seeing a number written this way, with one digit on
each side of the decimal point, assumed the numbers were to be
multiplied. He obtained a two-digit number as a result, but was 14.6 off.
What was the original number?
ANS. 5.4
15. A neat computer programmer wears a clean shirt every day. If he drops
off his laundry and picks up the previous week’s load every Monday night,
how many shirts must he own to keep him going?
ANS. 15
16. The teacher marked the quiz on the following basis: one point for each
correct answer, one point off for each question left blank and two points
off each question answered incorrectly. Pat made four times as many
errors as Mike, but Mike left nine more questions blank. If they both got
the same score, how many errors did each make?
ANS. 8 and 2
17. A wall is made of bricks which are twice as long as they are high. The wall
is 13 courses high, with 100 bricks on the odd courses and 99 bricks plus
two half bricks on the even courses. An ant starts at the lower left corner
and walks in a straight line to the upper right corner. Over how many
bricks does he walk?
ANS. 112
18. [MAY 2018] An icicle forming from a dripping gutter is in the shape of a
cone five times as long as it is wide (at the top). A few hours later it has
doubled in length and the generating angle has also doubled. How does
its present weight compare with previous weight?
ANS. 33 times as much
19. There are two barrels, one containing 40 gallons of wine and 60 gallons of
water, the other containing 70 gallons of wine and 30 gallons of water. A
pailful is taken from the first barrel and poured into the second. After
mixing, a pailful is poured back into the first barrel. The proportions of
wine to water in the first barrel are now 19:26. What is the capacity of the
pail?
ANS. 8 gal
20. Very few people are aware of the growth pattern of Jack’s beanstalk. On
the first day it increased its height by 1/2, on the second 1/3, on the third
1/4, and so on. How long did it take to achieve its maximum height (100
times its original height)?
ANS. 198
21. Billy weighs 5 pounds more than Bobby and when they see-saw, Billy has
to sit 1 foot closer to the center in order to balance. When the twins,
Tammy and Tommy, who weigh 35 pounds each get on with them, Billy
and Tammy sit only 6 inches closer to the center in order to balance
Bobby and Tommy. How long is the see-saw?
ANS. 14 ft
22. In a little known work, the famous geometer of Skalenos proves the
following theorem: “The square of the side opposite the Fandangle is
equal to the sum of the squares of the other two sided added to the
product of those two sides multiplied by the square root of two.” What is a
Fandangle?
ANS. 135°
23. Two wheels in the same plane are mounted on shafts 13 inches apart. A
belt goes around both wheels to transmit power from one to the other. The
radii of the two wheels and the length of the belt not in contact with the
wheels at any moment are all integers. How many times larger is one
wheel than the other?
ANS. 5 in larger
24. [PAST BOARD] In Bristol 90% of the citizens drink tea; 80% drink coffee;
70% drink whiskey; and 60% drink gin. No one drinks all four beverages.
What percent of Bristol’s citizens drink liquor?
ANS. 100%
25. Evaluate the infinite product
∞
2𝑛 + 1 3 5 9
∏ 𝑛
= ∙ ∙
∙⋯
2 + 2 4 6 10
ANS. 1/2
𝑛=1
26. Lazy Levy wishes to toss a snowball over a building 144 ft x 144 ft and
133 ft high with the least expenditure of energy. How far away from the
building should he stand? Hint: Derive constraints to specify the required
parabola.
ANS. 84 ft
27. There are four towns at the corners of a square. Four motorists set out,
each driving to the next (clockwise) town, and each man but the fourth
going 8 mi/hr faster than the car ahead - thus the first car travels 24 mi/hr
faster than the fourth. At the end of one hour the first and third cars are
now 204, and the second and the fourth 212 (beeline) miles apart. How
fast is the first car traveling and how far apart are the towns?
ANS. 180
28. A record enthusiast decided to calibrate his 33 1/3 rpm player by placing
equally spaced dots around the rim. What is the minimum number of dots
required in order that they appear stationary under 60 cycles light?
ANS. 216
29. A motorist rotated his five tires every 500 miles. At the end of 10,000
miles the original spare got a slashed, and was replaced, he continued
rotating every 5000 miles, but avoided using the new tire as a spare until
all five had worn out equally. When the new tire first became a spare,
what was the reading on the mileage gauge?
ANS. 45 000 mi
30. Only two polygons can have a smallest interior angle of 120 degrees with
each successive angle 5 degrees greater than its predecessor. One is the
nonagon. What is the other?
ANS. Pentadecagon
31. A bridge across a river is in the form of an arc of s circle. A boy walking
across the bridge finds that 27 feet from the shore the bridge is 9 feet
above the water. He continues on to the center of the span and finds that
the bridge is now 10 feet above the water. How deep is the river?
ANS. 80
32. A conical drinking cup has a 12-inch rim and is 4 inches at the center. If
ceased flat, what is the vertex angle of the resulting figure?
ANS. 77°33’
33. There are four volumes of an encyclopedia on a shelf, each volume
containing 300 pages, (that is numbered 1 to 600), but these have been
placed on the shelf in random order. A bookworm starts at the first page of
Vol.1 and eats his way through to the last page of Vol 4. What is the
expected number of pages (excluding covers) he has eaten through?
ANS. 500
34. Four spectators, viewing one of a pair of dice from different angles, see
spots totaling 10, 15, 14, and 9 respectively. How many spots are on the
top face of the die?
ANS. 5
35. A hula hoop of circumference 40inches performs one revolution about a
girl with a 20-inch waist. How far has the original point of contact of the
hoop traveled?
ANS. 80/𝜋
36. How many people would you expect to meet before you met one who was
born on a Wednesday?
ANS. 7
37. Some couples plan to hold séances around the round table. Dropping the
usual requirement that men and women alternate, they find the number of
opposite seating arrangements is increased tenfold. How many couples
are there?
ANS. 3
38. A cowboy tied his horse to a hitching post without square-cross section
using a frictionless rope with a slip knot. The horse promptly pulled as far
as he could in a direction straight out from the center and perpendicular to
the side of the post. At what angle did the rope leave the post?
ANS. 30°
39. Solve the equation √𝑥 + √𝑥 + √𝑥 + ⋯ = √𝑥√𝑥√𝑥 ⋯ where both
members represent infinite expressions.
ANS. 0 or 2
40. Commander Whitebread’s yacht can do 4 knots per hour. If he requires 3
hours to sail the English Channel at its narrowest, what is the distance
involved?
ANS. 18
41. Mr. X veers to the right when he walks. The curvature of his path is
proportional to his latitude. He starts walking North from point A on the
equator, in the area of a large level plain, and finds he is proceeding East
when he is one mile North of the equator. He continues walking and
arrives back at the equator at point B. What is the straight line distance
from A to B?
ANS. a trifle less than 1.2 miles
person. If he has on board now the number of passengers that maximizes
the total collected, what is the boat owner’s profit?
ANS. 0
45. A billiard table is in the form of an ellipse with one axis two feet longer
than the other. A ball is stuck from one focus and after bounding against
two cushions returns to its starting point. At the halfway point in its trip the
ball is eight feet from the source. How big is the table?
ANS. 15 and 17 ft
46. A number of 5x8 cards have been divided into 1-inch squares numbered 1
to 40. It is desired to use these for window cards with exactly six interior
18 squares cut out. How many different cards can be made?
ANS. 18,249
47. In the final seconds of the game, your favorite NBA team is behind 117 to
118. Your center attempts a shot and is fouled for the 2nd time in the last
2 minutes as the buzzer sounds. Three to make two in the penalty
situation. Optimistic? Note: the center is only a 50% free-thrower. What
are your team’s overall chances of winning?
ANS. 11/16
48. A novice in calculus was required to differentiate an expression of the
form Ax and evaluate at x=3. Naively using A(x-1) as the derivative, he
nevertheless obtained the correct value. What was A?
ANS. 2.8564
49. By the time the radius of a certain pearl has increased 1 mm, the area will
have increased as much (in square mm) as the volume (in cubic mm). If
the pearl is an exact sphere, what is the radius now?
ANS. 1.457
50. If all 720 permutations of the digit 1 through 6 are arranged in numerical
order, what is the 417th term?
ANS. 432516
51. A pencil, eraser and notebook together cost $1.00. A notebook costs
more than two pencils, and three pencils cost more than four erasers. If
three erasers cost more than a notebook, how much does each cost?
ANS. 19, 55
52. A car accelerates uniformly from rest to 8k mi/hr in k/5 minutes. It
continues at that speed from k minutes, then decelerates uniformly and
takes k/5 minutes to come to rest, having traveled k-1 miles altogether.
The trip took an exact number of minutes. How many?
ANS. 7
53. A kidney-shaped swimming pool is laid out by describing two tangent
circles, drawing a circular arc 40 feet long tangent to both of these circles
on one side and a parallel circular arc 20 feet long tangent to both of them
on the other side. What is the longest (straight line) distance one can
swim in this pool?
ANS. 58 ¾
54. The local weather forecaster says “no rain” and his record is 2/3 accuracy
of prediction. But the Federal Meteorological Service predicts rain and
their record is 3/4. With no data available, what is the chance of rain?
ANS. 3/5
55. A group of hippies are pondering whether to move to Patria, where
polygamy is practiced but polyandry and spinsterhood are prohibited, or
Matria, where polyandry is permitted, and polygamy and bachelorhood
are proscribed. In either event the possible number of “arrangement” is
the same. The girls outnumber the boys. How many are there?
ANS. 4G 2B
56. A sharp operator makes the following deal. A player is to toss a coin and
receive 1, 4, 9, ... n2 dollars if the first hand comes up on the first, second,
third, ... nth toss. The sucker pays ten dollars for this. How much can the
operator expect to make if this is repeated a great many times?
ANS. $4/game
42. If eggs where x ₵ a dozen less, one would pay 1₵ less for x+1 eggs than
if there were x₵ dozen more. Find x.
ANS. 2
57. What is the smallest positive integer which, when divided by any N in the
range 2, 3, ..., 10, leaves a remainder of N-1?
ANS. 2519
43. A student beginning the study of trigonometry came across an expression
of the form sin (X+Y). He evaluated this as sinX + sinY? Surprisingly he
was correct. The values of X and Y differed by 10°; what were these
values assuming that 0° < X < Y < 360°?
ANS. 175° and 185°
58. The price per cubic inch for platinum trays is the same as that per square
inch for platinum sheets. A metal supply house has a square of platinum
which will yield the same amount whether sold as a sheet, or fashioned
into a tray of maximum volume with the four cut-out corners sold as
sheets. How big is the square?
ANS. 1 ft
44. A boat owner agrees to take a group on an outing at $4.50 apiece if the
number of passengers is equal to or less that his breakeven point. For
each person above this he reduces the fare for all passengers 3 cents per
59. Solve for x: log x2 = 1 + (logx)2
ANS. 10
60. For what positive integer k is k+5 a factor of k 2+11 while k+11 is a factor if
k2+5?
ANS. 7
61. There is one flag at the entrance to a racetrack and another inside the
track, half a mile from the first. A jockey notes that no matter where is on
the track, one flag is 3 times as far away as the other. How long is the
track?
ANS. 1980pi
62. For x, y and z real numbers, solve the equation 3x2 + y2 + z2 = 2x (y+z)
ANS. x=y=z=0
63. A housewife wishes to make 2 circular doilies of equal radius from a
rectangular tablecloth of dimensions 78” by 108”. What are the largest
doilies she can obtain by cutting and stitching not more than 4 segments.
ANS. 34.04”
64. In a carnival game 5 balls are tossed into a square box divided into 4
square cells, with baffles to ensure that every ball has an equal chance of
going in any cell. The player pays $1 and receives $1 for every cell which
is empty after the 5 balls are thrown. How much does the operator expect
to make per game?
ANS. 95 cents
65. Solve for the real values of x: (7 + 4√3)𝑋 − 4(2 + √3)𝑋 = −1
ANS. 1
66. A lighthouse shows successive one-second flashes of red, white, green,
green, white, red. A second lighthouse does the same only with twosecond flashes. The six-second sequence of the first lighthouse is
repeated steadily, as is the twelve-second sequence of the other
lighthouse. What fraction of the time do the two lights show the same
color if the given sequences start at the same time?
ANS. 1/6
67. The sum of the reciprocals of a, b, and c is 5/8. So is the sum of the
reciprocals of d, e, and f. The sum of the reciprocals of a, d and e is four
times the reciprocals of the other quantities. What combination of
reciprocals will sum to 1?
ANS. a, d, e
68. An electric iron is in the form of two arcs of circles 20 inches in diameter,
each are being 9inches long. The back of the iron is on the line of centers
of the two circles. What is the angle at the point?
ANS. 52°34’
69. A certain magic square contains nine consecutive 2-digit numbers. The
sum of the numbers in any line is equal to one of the numbers in the
square with digits reversed. This is still the case if 7 is added to each
entry. What is the number in the center square?
ANS. 17
70. What is the longest 6’-wide shuffle board court which will fit in a 20’ x 30’
rectangular room?
ANS. 30-7/8
71. A ribbon clerk in a department store has on her counter a cylinder 31
inches in circumference with six marks around the rim. These are so
spaced that she can measure any integral length of ribbon from 1 to 31
inches by starting at one mark, wrapping the ribbon around the cylinder
and cutting it off at another mark. How are the marks spaced on the
cylinder?
ANS. The marks are spaced at intervals of 1, 3, 2, 7, 8, and 10 inches
respectively
72. The planet Arida in the Magellanic Clouds is water-starved. It is one vast
desert containing a single fresh-water oasis about the size of a backyard
swimming pool. What is the a priori probability that there is an island in
that lonely pond?
ANS. Unity
73. A certain hexahedron has 4 triangular and a quadrilateral face. How many
edges does the 6th face have?
ANS. 3, 4 or 5
74. Following the big game, a college freshman tore down one of the goal
posts and spirited away an upright. In the dormitory corridor he was just
able to get the post around the corner without bending it. Assuming the
corridor is 17 feet high and 10 feet wide, and the post was extremely
slender, how long was it?
ANS. 33 ft
75. The sum of two numbers is 48, and the sum of their reciprocals is 16. Find
their product.
ANS. 3
76. A carpenter made a box of maximum volume out of a 4’x8’ plywood sheet.
Each of the six rectangular faces was a single piece of plywood. What
were the dimensions? (Ignore the thickness of the material, the loss due
to cutting, and the precise techniques of cornering.)
ANS. 2x2x3’
77. Two players matching coins find that the coins have matched 13 times in
a sequence of 25. The first player has had 11 head, the second player 15.
How many of these matches were tails-tails?
ANS. 6
78. In the battle of Small Tuba, the troopers always outnumbered 3 to 1,
made every shot count. Ultimately both sides were wiped out. What
fraction of braves’ shot hit their mark?
ANS. 1/8
79. The numbers 1, 3, 5, 8, and 9 are written down in random order. What is
the probability that the resulting five-digit number is a square.
ANS. 1/24
80. As streetlamp bulbs burnt out, the City Council replaces them. Through
the reliability of Brand A bulbs is only 0.01 more than that of Brand B, the
council is willing wisely to pay twice as much for Brand A. How reliable is
Brand B?
ANS: Brands A and B have respective reliabilities of 0.99 and 0.98
81. In “Cut-Throat Hearts”, each player pays to each opponent with a smaller
deficit the difference in their totals. In the high-stake, 5-man game, one
opponent threatens to “shoot the moon”. Only you can stop him. How
many points are you willing to take to do so?
ANS. Less than 10
82. If the probability of a spacecraft being struck by exactly one cosmic
particle during the Earth-Neptune roundtrip identical to its probability of
not being struck at all, what is the probability?
ANS. 0.368
83. After learning that the equation (x+a)(x+b) = 0 could be solved by writing
x+a = 0 and x+b = 0 and solving this separately, a student tackled an
equation of the form (x+7)(4-x) = c with c not equal to zero by writing x+7
= c and 4-x = c and solving each of these. Surprisingly, he got the correct
roots. What is the value of c?
ANS. 10
84. To stimulate his son in the pursuit of partial differential equations, a math
professor offered to pay him $8 for every equation correctly solved and to
fine him $5 for every incorrect solution. At the end of 26 problems, neither
owed any money to the other. How many did the boy solve correctly?
A. 10
B. 12
C. 14
D. 16
85. A man enters a bank and has check cashed. The teller mistakes the
figures and pays cents for dollars and dollars for cents. The man then
pays a bill for $24.11 after which he finds he has twice as much money as
the face value of the original check. What was the face value?
A. 12.25
B. 14.58
C. 13.51
D. 15.82
86. An expert on transformer design relaxed one Saturday by going to the
races. At the end of the first race he doubled his money. He bet $30 on
the second race and tripled his money. He bet $54 on the fourth race and
quadrupled his money. He bet $72 on fourth race and lost it, but still had
$48 left. With how much money did he start?
A. 25
B. 29
C. 32
D. 35
87. John, a Computer Engineer, is twice as old as his wife was when he was
as old as his wife is now. He is 24 years old. How old is his wife?
A. 16
B. 17
C. 18
D. 20
88. Suppose a passenger rocket leaves Earth for Planet X every day at noon.
At precisely the same time a rocket leaves Planet X for Earth. Each trip
lasts exactly 192 hours (8 days). How many rockets from Planet X will the
nth rocket from Earth meet?
A. 18
B. 15
C. 21
D. 24
89. Three men play a game with the understanding that the loser is to double
the money of the other two. After three games, each has lost just once;
and each has $24. How much did each have to start?
A. 12, 21, 39
B. 13, 24, 31
C. 12, 11, 10
D. 10, 23, 20
90. The undergraduates of a School of Engineering wished to form ranks for a
parade. In ranks of 3 abreast, 2 men were left over; in ranks of 5, 4 over;
in 7’s, 6 over; and 11’s, 10 over. What is the least number of marchers
there must have been?
A. 1145
B. 1154
C. 1415
D. 1514
91. An aerodynamist out for a stroll walks eastwards at a rate of 3 mph. He
notices that the wind appears to blow directly from the north. He doubles
his speed and the wind appears to blow from the northeast. What was the
wind velocity?
A. 3 mph
B. 3/2
C. 3√2
D. 3/√2
92. What is the remainder upon dividing 5999,999 by 7?
A. 5
B. 4
C. 6
D. 2
93. Two candles have equal lengths. One is consumed uniformly in four
hours, the other in five hours. If they are lighted at the same time, when
will one be three times as long as the other?
A. 3-6/11 hrs
B. 3-7/11
C. 5-5/7
D. 5-6/7
94. Each face of a regular dodecahedron is painted with a different color.
Using the same 12 colors, how many dodecahedrons with different
arrangements are possible?
A. 479001600
B. 7983360
C. 792
D. 3991680
95. A ball was dropped from the height of 10 feet. It rebounds one-half the
distance on each bounce. What is the total distance it travels?
A. 30 feet
B. 20
C. 25
D. 10
96. A and B live at two opposite corners of a square lot. C and D live at the
other two corners. They all carry water from a spring located within a lot,
which is 5 rods from A; 4 rods from B; 3 rods from C. How far must D
carry water?
A. 4.243 rods
B. 5.657
C. 4.432
D. 5.576
97. Seven singular scientists shared a grindstone the diameter of which was
sixty inches. What part of the diameter was the rightful share of each
singular scientist? Assume that each grinds his share sequentially.
A. 1/√7
B. 1/7
C. 2/√7
D. 2/7
98. A candle 15 inches long will burn in 9 hours, 1 inch at the lesser end will
be consumed in 20 minutes less than the same length at the larger end.
How long will it take for an inch at the lesser end to be consumed?
A. 23.64 min
B. 24.63
C. 26.43
D. 32.46
99. From the equation (x+1)(x2+1)(x3+1)=30x3, find the real values of x by
means of a quadratic only.
A. 3 ± √5⁄2
B. 2 ± √5⁄2
C. 2 ± √3⁄2
D. 5 ± √3⁄2
100. Stations A and B are 120 miles apart on a single-track railroad. At the
same time that a train leaves A for B at 25 mph, a train leaves A for A at
15 mph. Just as the first train leaves a, a South American botfly flies from
the front of the engine straight toward the other train at 100 mph. On
meeting the second train it immediately turns back and flies straight for
the first train. It continues to fly back and forth with undiminished speed
until it is crushed in the eventual collision. How far had the fly flown?
A. 300 mi
B. 320
C. 350
D. 380
101. A farmer has 600 plants arranged in rows, but they require an irrigation
ditch. He finds that he must take out 5 plants, from each row. He can then
make 6 more rows. Find the original number of plants in each row.
A. 20
B. 25
C. 30
D. 35
102. Three regimens move north as follows: B is 20 miles east of A; C is 20
miles south of B, and each marches 20 miles between the hours of 5 AM
and 3 PM. a horseman with a message from C starts at 5 AM and rides
north until it overtakes B, then sets a straight course for the point at which
he will again overtake B, then rides south to the point from which B
Started, reaching that point at the same time as C, namely 3 PM. What
uniform rate of travel enabled the messenger to do this?
A. 6.38 mph
B. 6.83
C. 8.36
D. 8.63
106. When I am as old as my father is now, I shall be five times as old as my
son is now. By then my son will be eight years older than I am now. The
combined ages of my father and me are 100 years. How old is my so?
A. 10
B. 12
C. 14
D. 13
107. Assuming that each packet of cigarettes from a certain manufacturer
contains, as a premium, one set of 52 playing cards, and that these cars
are distributed among packets at random (the number of packets
available being infinite), what is the average minimum number of packets
that must be purchased in order to obtain a complete set of cards?
A. 236
B. 326
C. 263
D. 362
108. Smith and Jones, both 50% marksmen, decide to fight a duel in which
they exchange alternate shots until one is hit. What are the odds in favor
of the man who shots first?
A. 1/3
B. 2/3
C. 1/4
D. 3/4
109. What is the lowest number that the sum of two cubes in two different
ways?
A. 1297
B. 1972
C. 2179
D. 1729
110. A ladder is leaning against a wall at an angle steeper than 45 degrees.
Under there is a barrel which touches both the ladder and the wall. If the
vertical distance, in feet, between the top of the ladder and the ground I 4
times the diameter of the barrel, what is the shortest integral number of
feet the ladder can be?
A. 15
B. 20
C. 25
D. 30
111. While visiting Cape Canaveral, we came upon an engineer digging a hole.
“How deep is that hole?” We asked. “Guess,” said the engineer, being
evasive. “My height is exactly 5’10”.” “How much deeper are you going?”
we inquired. “I am one-third done,” was the answer, “and then my head
will be twice as far as below ground as it is now above the ground.” How
deep will that hole be when finished?
A. 10’6”
B. 11’
C. 12’2”
D. 12’10”
112. At this moment, the hands of a clock in the course of normal operation
describe a time somewhere between 4:00 and 5:00 on a standard clock
face. Within one hour or less, the hands will have exactly exchanged
positions. What time is it now?
A. 4:26.85
B. 4:23.45
C. 4:29.82
D. 4:20.24
113. In a room, 40 feet long, 20 feet wide, and 20 feet high, a bug sits on an
end wall at a point one foot from the other floor, midway between the
sidewalls. He decides to go on a journey to a point on the other end wall
which is one foot from the ceiling midway between the sidewalls. Having
no wings, the bug must make his trip by sticking to the surfaces of the
room. What is the shortest route that the bug can take?
A. 40
B. 46
C. 58
D. 62
114. The first expeditions to Mars found only the ruins of civilization. The
explorers were able to translate a Martian equation as follows: 5𝑥 2 −
5
50𝑥 + 125 = 0, x= { . This was strange mathematics. The value of
8
x=5 seem legitimate enough but x=8 required some explanation. If the
Martian number system developed in a manner similar to ours, how many
fingers would you say the Martian had?
A. 3
B. 5
C. 8
D. 13
115. An engineer must take three space suits in two test chambers. Each suit
must be tested for 1 hour at each f two low pressures. He takes ten
minutes to load a suit in a chamber, set the pressure, and start the test; 4
min. to change the pressure, and ten min. to unload the suit from a
chamber. What is the minimum time to complete the tests?
A. 3hr45m
B. 3hr54m
C.3hr30m
D.3hr24m
103. The faces of a solid figure are al triangle. The figure has nine vertices. At
each of six these six vertices, four faces meet. How many faces does the
figure have?
A. 10
B. 12
C. 14
D. 16
116. A circle of radius 1 inch is inscribed in an equilateral triangle. a smaller
circle is inscribed at each vertex, tangent to the circle and two sides of the
triangle. The process is continued with progressively smaller circles. What
is the sum of the circumference of all circles?
A. 5pi
B. 6pi
C. 7pi
D. 8pi
104. A new kind of atom smasher is to be composed of two tangents and a
circular arc which is concave towards the point of intersection of the two
tangents. Each tangent and the arc of the circle is 1 mile long. What is the
radius of the circle?
A. 1374.5
B. 1437.5
C. 1547.34
D. 1745.3
117. A man passed one sixth of his life in childhood, on twelfth in youth and
one seventh more as a bachelor. Five years after his marriage, a son was
born who dies for years before his father at half his father’s final age.
What is the man’s final age?
A. 76
B. 80
C. 84
D. 88
105. A’s age equals B’s age plus the cube root of C’s age. B’s age equals C’s
age plus the cube root of A’s age, plus 14 years. C’s age equals the cube
root of A’s age plus the square root of B’s age. What is the age of each?
A. A is 28 years old, B is 26 and C is 9
B. A is 25 years old, B is 25 and C is 7
C. A is 27 years old, B is 24 and C is 6
D. A is 27 years old, B is 25 and C is 8
118. A rectangular picture, etch of whose dimensions is an integral number of
inches, has an ordinary rectangular frame 1 inch wide. Find the
dimensions of the picture if the area of the picture and the area of the
frame are equal.
A. 3x8, 5X6
C. 3x8, 4X7
B. 3x12, 4X6
D. 3x10, 4X6
7
119. What is the rightmost digit of 77 ?
A. 7
B. 9
C. 3
D. 1
120. A castle and a bishop are placed at random on different squares if the
chessboard. What is the probability that one piece threatens the other?
A. 1/6
C. 7/36
B. 13/36
D.17/36
121. A farmer owned s square field measuring exactly 2261 yards on each
side. 1898 yards from one corner and 1009 yards from an adjacent corner
stood a beech tree. A neighbor offered to purchase a triangular portion of
the field stipulating that a fence should be erected in a straight line from
one side of the field to an adjacent side so that the beech tree was part of
the fence. The farmer accepted the offer but made sure that the triangular
portion was of minimum area. What was the area of the field the neighbor
received, and how long was the fence?
A. 2015 yards
C. 2016
B. 2017
D. 2018
122. Two motorists set out at the same time to go from A to B, a distance 100
miles. They both followed the same route and traveled at different, though
uniform, speeds of an integral number of mile per hour. The difference in
their speeds was a prime number of miles per hour, and after they had
been driving for two hours the slower car was five times further from A
than the faster car was from B. How fast did the two motorists drive?
A. 41 and 40
B. 42 and 40
C. 42 and 36
D. 45 and 40
123. Two men are walking towards each other at the side of a railway. A freight
train overtakes one of them in 20 seconds and exactly ten minutes later
meets the later meets the other man coming in the opposite direction. The
train passes this man in 18 seconds. How long after the train has passed
the second man will the two men meet? (Constant speeds are to be
assumed throughout)
A. 5562 sec
B. 6255
C. 6552
D. 5625
124. A town owns three snowplows which plow at a rate inversely proportional
to the depth of the snow. One evening snow starts falling and continues at
a constant rate throughout the night. At midnight one plow starts down the
highway. At 1:00 AM a second plow starts and plows the same track as
the first. At 2:00 AM the third plow follows the other two. The third plow
catches up to the second at the same time that the second catches to the
first. What time did it start snowing?
A. 11:15pm
B. 11:30pm
C. 10:30pm
D. 10:45pm
125. Four cities, A, B, C, and D, lie at the vertices of a rectangle. Inside this
imaginary rectangle there is a fifth city, E, which is exactly 33 miles from A
and 56 miles from CE. Also happens to be an integral number of miles
from the other two cities, being further from B than from D. If the distance
between B and C is 3 times that of E from D, how far to the nearest half
mile is A from B?
A. 58.5
B. 60
C. 63.5
D. 65
126. A drawer contains an odd number of plain brown socks and an even
number of plain black socks. What is the least number of brown and black
socks such that the probability of obtaining two brown socks is 1/2 when
two socks are chosen at random from the complete collection?
A. 13 brown 6 black
C. 15 brown 6 black
B. 21 brown 8 black
D. 15 brown 8 black
127. Using the French Tricolor as a model, how many flags are possible with
five available colors if two adjacent rows must not be colored the same?
A. 20 flags
B. 30
C. 40
D. 50
128. Lottie and Lucy Hill are both 90 years old. Mary Jones, on the other hand,
is half again as old as she was when she was half again as old as she
was when she lacked 5 years of being half as old as she is now. How old
is Mary?
A. 60
B. 70
C. 80
D. 90
132. Johann Jungfrau, the famous mountain climber, was traveling through the
Trondheim timber country one day. Quite by accident he dropped his
trusty Alpenstock, an unusually straight stick, near the buzz saws where,
in two shakes of a yak’s tail, it was nearly cut into three pieces. What is
the probability that these three pieces can be placed together to form a
triangle?
A. 1/2
B. 1/3
C. 1/4
D. 1/6
133. My house is on a road where the numbers run 1, 2, 3, 4... consecutively.
My number is a three digit one and, by curious coincidence, the sum of all
house numbers less than mine is the same as the sum of all house
numbers greater than mine. What is my number and how many houses
are there on the road?
A. 204288
B. 205298
C. 215300
D. 225500
134. Three hares are standing in a triangular field which is exactly 100 yards
on each side. One hare stands at each corner; and simultaneously all
three set off running. Each hare runs after the hare in the adjacent corners
on his left, thus following a curved course which terminates in the middle
of the field, all three hares arriving there are together. The hares obviously
ran at the same speed, but just how far did they run?
A. 100 yards
B. 200
C. 300
D. 400
135. A set of items sells for $1122.00, and another set of like items sells for
$2210.00. What is the cost of each item?
A. 63150
B. 65140
C. 66130
D. 67120
136. A cubic box with sides ‘a’ feet long is placed flat against a wall. A ladder
‘p’ feet long is placed in such a way that it touches the wall as well as the
free horizontal edge of the box. If a=1 p=√15 and, calculate at what
height the ladder touches the wall, using quadratics only.
A. 3.59 or 1.23
B. 3.22 or 1.34
C. 3.62 or 1.38
D. 5.32 or 2.31
137. How many nine-digit numbers are divisible by 11, no digit equal to zero
and no two digits alike?
A. 29650
B. 30150
C. 31680
D. 32790
138. Dr. Irving Weiman, who is always in a hurry, walks up an up going
escalator at the rate of one step per second. Twenty steps bring him to
the top. Next day he goes up at two steps per second, reaching the top in
32 steps. How many steps are there in the escalator?
A. 24
B. 32
C. 56
D. 80
139. Two similar triangles with integral sides have two of their sides the same.
The third sides differ by 387. What are the lengths of the sides?
A. 320, 200, 125
C. 512, 320, 200
B. 315, 150, 180
D. 330, 300, 250
140. A scalene triangle ABC which is not a right triangle has sides which are
integers. If sin A=5/13, find the smallest values for its sides, i.e., those
values which make the perimeter a minimum.
A. 21, 17, 36
B. 24, 20, 33
C. 24, 16, 38
D. 25, 16, 39
141. The planet Octerra is divided into eight countries, each occupying an
octant, (this each country borders three others). In how many ways can a
traveler visit each of the other countries once and only once, returning to
his home country only at the end of his trip?
A. 9
B. 10
C. 11
D. 12
142. Gherkin Gesundheit, a brilliant graduate mathematics student, was
working on an assignment but, being a bit absent-minded, he forgot
whether he was to add or to multiply the three different integers on his
paper. He decided to do it both ways and, much to his surprise, the
answer was the same. What were the three different integers?
A. 1, 2, 3
B. 2, 3, 4
C. 2, 4, 6
D. 4, 5, 6
129. If the hour and minute hand of a watch are interchanged, how many
different possible times could the watch show?
A. 126
B. 134
C. 143
D. 162
143. A farmer used 139 yards of fencing to enclose a rectangular field and to
construct a fence along one of the diagonals of length 41 yards. He then
found that a neighbor had fenced a one-third larger rectangular area in the
same manner with less fencing. If all dimensions are integral yards, what
are the dimensions of the neighbor’s field?
A. 9x40
B. 10x24
C. 15x32
D. 16x30
130. A, B, and C are three towns, each pair being connected by a network of
roads. A motorist notices that there are 82 routes from A to B, including
those via C and 62 routes from B to C, including those via A. He also
notices that there are fewer than 300 routes from A to C, including those
via B. How many are there?
A. 45
B. 46
C. 47
D. 48
144. The consumption of coal by a locomotive varies as the square of the
speed. Other operating expenses (exclusive of coal) are $72 per hour. If
the price of coal is $10 per ton and at a speed of 25 mi/hr the locomotives
uses 5 tons per hour, what is the minimum cost of a trip of 100 miles?
A. $480 at 30 mph C. $500 at 50 mph
B. $400 at 35 mph D. $350 at 60 mph
131. Six grocers in a town each sell a different brand of tea in four-ounce
packets at 25 cents per packet. One of the grocers gives short weight,
each packet of his brand weighing only 3 and 3/4 ounces. If I can use a
balance for only one weighing, what is the minimum amount I must spend
to be sure of finding the grocer who gives short weight?
A. 3.7 dollars
B. 3.9
C. 4.2
D. 4.5
145. On a certain day, our parking lot contains 999 cars, no two if which have
the same 3-digit license number. After 5:00 pm, what is the probability that
the license numbers of the first 4 cars to leave the parking lot are in
increasing order of magnitude?
A. 1in24
B. 1in25
C. 1in28
D. 1in32
146. In a certain community there are 1000 married couples. Two thirds of the
husbands who are taller that their wives are also heavier and three
quarters of the husbands who are heavier than their wives are also taller.
If there are 120 wives who are taller and heavier than their husbands, how
many husbands are taller and heavier that their wives?
A. 420
B. 450
C. 480
D. 485
147. The area and volume of a certain sphere are both 4-digit numbers times
𝜋. What is the radius of the sphere?
A. 17
B. 18
C. 19
D. 20
148. Furbisher lives in Canoga Acres and works in Beverly Flats. LaRouche
lives in Beverly Flats and works in Canoga Acres. They usually leave their
respective homes at the same time, and pass each other at Sam’s
Hamburger Shack. (Furbisher drives twice as fast as LaRouche). On a
day when Furbisher’s wife Formica spoiled eggs and he left 5 minutes
late, they passed each other at the gas station, two miles from Sam’s.
How fast do they drive?
A. 48 and 96 mph
C. 36 and 72 mph
B. 32 and 64 mph
D. 50 and 100 mph
149. Three farmers, Adams, Brown and Clark all have farms containing the
same number of acres. Adam’s farm is most neatly square, the length
being only 8 miles longer than the width. Clark has the most oblong farm,
the length being 34 miles longer than the width. If all the dimensions are in
exact miles, what is the size of each farm?
A. 40x48, 32x60, 30x64
C. 45x42, 35x70, 25x50
B. 35x45, 32x40, 38x50
D. 50x50, 35x35, 60x60
150. A one-acre field in the shape of a right triangle has a post at the midpoint
of each side. A sheep is tethered to each of the side posts and a goat to
the post on the hypotenuse. The ropes are just long enough to let each
animal reach the two adjacent vertices. What is the total area the two
sheep have to themselves, i.e., the area the goat cannot reach?
A. exactly half acre
C. exactly one acre
B. exactly one and a half acre
D. exactly three fourths acre
151. A divided highway goes under a number of bridges, the arch over each
lane being the form of a semi-ellipse with the height equal to the width. A
truck is 6 ft wide and 12 ft high. What is the lowest bridge under which it
can pass?
A. 12’3”
B. 13’5”
C. 14’5”
D. 15’5”
152. The algebra teacher wrote on a blackboard a quadratic equation of the
form x2-Ax+B=0. In copying this careless student erroneously transposed
the two digits of B as well as the plus and minus signs. However, one of
the roots was the same. What was this root? (Assume both A and B are
integers)
A. 1
B. -2
C. 3
D. -4
153. Citizens of Franistan pay as much income tax (percentage-wise) as they
make rupees per week. What is the optimal salary in Franistan?
A. 42
B. 45
C. 50
D. 65
154. There are nine cities which are served by two competing airlines. One or
the other airline (but not both) have a flight between every pair of cities.
What is the minimum number of triangular flights (i.e., trips from A to B to
C and back to A on the same airline)?
A. 9
B. 11
C. 12
D. 15
155. Obviously the smaller the compounding period, the greater the interest.
How much does one dollar amount to after one year at 100% per annum
interest, compounded continuously, i.e., instantaneously?
A. 2.67
B. 2.71
C. 2.84
D. 2.96
156. A hospital nursery contains only two baby boys; the girls have not yet
been counted. At 2:00 pm s new baby is added to the nursery. A baby is
then selected at a random to be the first to have its footprint taken. It turns
out to be a boy. What is the probability that the last addition to the nursery
was girl?
A. 1/5
B. 2/5
C. 3/5
D. 4/5
157. Assume that a single depth charge has a probability of ½ of sinking a
submarine, ¼ of damage and ¼ missing. Assume also that two damaging
explosions sink the sub. What is the probability that 4 depth charges sink
the sub?
A. 251/256
B. 252/256
C. 253/256
D. 254/256
158. An Origami expert started making a Nani-des-ka by folding the top left
corner of a sheet of paper until it touched the right edge and the crease
passed through the bottom left corner. He then did the same with the
lower right corner, thus making two slanting parallel lines. The paper was
25 inches long and the distance between the parallel lines was exactly
7/40 of the width. How wide was the sheet of paper?
A. 24 in
B. 25 in
C. 32 in
D. 36 in
159. Two snails start from the same point in opposite directions toward two bits
of food. Each reaches his destination in one hour. If each snail had gone
in the direction the other took, the first snail would have reached his food
35 minutes after the second. How do their speeds compare?
A. x=2/5 y
B. x=3/4 y
C. x=1/4 y
D. x=3/5 y
160. If two marbles are removed at a random from a bag containing black and
white marbles, the chance that they are both white is 1/3. If three are
removed at random, the chance that they are all white is 1/6. How many
marbles are there of each color?
A. 6&4
B. 6&5
C. 8&4
D. 8&5
161. The Ben Azouli is camped at an oasis 45 miles west of Taqaba. They
decide to dynamite the Trans-Hadramaut railroad joining Taqaba to
Maqaba, 60 miles north of the oasis. If the Azouli can cover 18 miles a
day, how long will it take them to reach the railroad?
A. 36
B. 42
C. 40
D. 45