問題一覧
1
A pupil wrote on the blackboard a series of fractions having positive integral terms and connected by signs which were either all + or all x, although they were so carelessly written it was impossible to tell which they were. It still wasn't clear even though he announced the result of the operation at every step. The third fraction had denominator 19. What was the numerator?
25
2
below
x = 0 or 2
3
Dr. Reed, arriving late at the lab one morning, pulled out his watch and the hour hand are exactly together every sixty-five minutes." Does Dr. Reed's watch gain or lose, and how much per hour?
60/143 minutes gain
4
Represented above is a "magic square" in which the sum of each row, column, or main diagonal is the same. Using nine different integers, produce a "multiplicative" magic square, i.e., one in which the word "product" is substituted for "sum".
Using the rule that powers multiply by adding their exponents, a multiplicative magic square is easily obtained from the given square by substituting 2^n (or k^n where k > 1) for n in each block
5
Which is greater: 3+4/(3+4/(3+...) or 3+(3+(3+.../4)/4 where the respective denominators and numerators continue indefinitely?
Equal to 4
6
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?
4:26.853
7
For X < 1, evaluate the infinite product:
(1+X+X^2+...+X^9)(1+X^10+X^20+X^30+...X^90)
(1+X^100+X^200+...+X^900)...
T1二0^∞ 【x^n=1/（1-x）】
8
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 for 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?
Pat = 8 errors, Mike = 2 errors
9
If v varies as w2, w^3 as x'4, x^5 as y^6, and y^7 as z^4, show that the product does not vary at all.
since each v/w^2, w^3/x^4, x^5/y^6 and y^7/z^4 is a constant, therefore their product is also a constant
10
There are nine cities which are served by two competing airlines. One or the other airline (but not both) has 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)?
[CE BOARD NOV 2016]
12
11
A necklace consists of pearls which increase uniformly from a weight of 1 carat for the end pearls to a weight of 100 carats for the middle pearl. If the necklace weighs altogether 1650 carats and the clasp and string together weigh as much (in carats) as the total number of pearls, how many pearls does the necklace contain?
33 pearls
12
A cubic box with sides 'a' feet long is placed flag 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 and p=sqrt(15), calculate at what height the ladder touches the wall.
1.38 ft or 3.62 ft
13
A parking lot charges & 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 time clock registers only in 5-minute intervals)
Jones = 0.5 hours, Smith = 3.5 hours
14
What is the millionth term of the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ... in which positive integer n occurs in blocks of n terms?
1414
15
There are n points on a circle. A straight-line segment is drawn between each pair of points. How many intersections are there within the circle if no 3 lines are collinear?
764,488
16
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?
pencil = 26, eraser = 19, notebook = 55
17
Without performing any algebraic manipulation at all, Archimedes O'toole remarked that the sum and product of the two expressions (x+y-Ix-yl)/2 and (x+y+|x-yl)/2 are respectively x+y and xy. Why was this obvious?
The two expressions are identically equal, respectively, to the smaller and the larger of the two numbers x and y.
18
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 the 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?
25%
19
A student just beginning the study of logarithms was required to evaluate an expression of the form log A/log B. He proceeded to cancel common factors in both numerator and denominator, (including the "factor" log), and arrived at the result 2/3. Surprisingly, this was correct. What were the values of A and B?
A = 9/4, B = 27/8
20
Citizens of Franistan pay as much income tax (percentage-wise) as they make rupees per week. What is the optimal salary in Franistan?
50 rupees
21
Dr. Irving Weiman, who is always in a hurry, walks up an upgoing 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?
[CE BOARD NOV 2017]
80 steps
22
Depicted above are two interlocked hyperbolas. Impossible? You're right, but can you prove it?
The number of solutions of a set of polynomial equations is the product of their degrees. Since hyperbolas are of 2nd degree, the solutions is to a pair of hyperbolic equations cannot exceed 4. Thus, two hyperbolas can intersect in no more than 4 points.
23
There are four towns at the corners of a square. Four motorists set out, each driving in 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 204, and the second and fourth 212 (beeline) miles apart. How fast is the first car traveling and how far apart are the towns?
[CE BOARD MAY 2017 / NOV 2022]
50 miles/hour, 180 miles
24
A mathematician whose clock had stopped wound it but did not bother to set It correctly. Then he walked from his home to the home of a friend for an evening of hi-fi music. Afterwards, he walked back to his own home and set his clock exactly. How could he do this without knowing the time his trip took?
t2+1/2[(T2-T1)-(t2-t1)]
25
An expert on transformer design relaxed one Saturday by going to the races. At the end of the first race, he had doubled his money. He bet $30 on the second race and tripled his money. He bet $54 on the third race and quadrupled his money. He bet $72 on the fourth race and lost it, but still had $48 left. With how much money did he start?
29
26
To stimulate his son is the pursuit of partial differential equations, a math professor offered to pay him $8 for every equation correctly solved and to fine him S5 for every incorrect solution. At the end of 26 problems, neither owed any money to the other. How many did the boy solve correctly?
[CE BOARD MAY 2019]
10
27
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?
50 flags
28
Jai Alai balls come in boxes of 8 and 15; so that 38 balls (one small box and two large) can be bought without having to break open a box, but not 39. What is the maximum number of balls which cannot be bought without breaking boxes?
97
29
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 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.)
5562
30
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 direction the other took, the first snail would have reached his food 35 minutes after the second How do their speeds compare?
V1 = ¾ V2
31
A yang, ying, and yung is constructed by dividing a diameter of a circle, AB, into three parts by points C and D, then describing on one side of AB semicircles having AC and AD as diameters and on the other side of AB semicircles having BD and BC as diameters. Which is larger, the central portion or one of the outside pieces?
Same
32
A forgetful physicist forgot his watch one day and asked an E.E. on the staff what time it was. The E.E. looked at his watch and said: "The hour, minute, and sweep second hands are as close to trisecting the face as they ever come. This happens only twice in every 12 hours, but since you probably haven't forgotten whether you ate lunch, you should be able to calculate the time." What time was it to the nearest second?
2:54:35 and 9:05:25
33
How can seven points be placed, no three on the same line, so that every selection of three points constitutes the vertices of an isosceles triangle?
Place five at the vertices of a regular pentagon, the sixth at the center of the pentagon, and the seventh above the center at a distance equal to the radius of the pentagon.
34
A cowboy is five miles south of a stream which flows due east. He is also 8 miles west and 6 miles north of his cabin. He wishes to water his horse at the stream and return home. What is the shortest distance he can travel and accomplish this?
17.9 miles
35
In Puevigi numbers such as 2, 5, 8, 10, etc., that are the sum of two squares, are considered sacred. Prove that the product of any number of sacred numbers is sacred.
It suffices to prove it for two sacred numbers, since the theorem then follows by induction. Let M = A^2 + B^2 and N = C^2 + D^2. Then MN = (A^2 + B^2)(C^2 + D^2) = (AC + BD)^2 + (AD - BC)^2. Amen.
36
Given a point P on one side of a general triangle ABC, construct a line through P which will divide the area of the triangle into two equal halves.
From P draw a line, L, to the opposite vertex, say A. Now construct a line parallel to L from the midpoint of BC, intersecting the side of the triangle at Q. The line PQ divides the triangle into two equal areas.
37
Let c be the hypotenuse of a right triangle with legs a and b. Prove that if x > 2, then a'x + b^x < c^x
a^x+b^x=a^2 (a^(x-2))+b^2 (b^(x-2)) < a^2 c^(x- 2) +b^2 c^(x-2)=(a^2+b^2) c^(x-2) =c^2 c^(x-2)=c^x
38
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?
24 inches
39
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 corner on his left, thus following a curved course which terminates in the middle of the field, all three hares arriving there together. The hares obviously ran at the same speed, but just how far did they run?
100 yards
40
A wizard in Numerical Analysis has a gold chain with 7 links. A Lady Programmer challenges him to use the chain to buy 7 kisses, each kiss to be paid for, separately, with one chain link. What is the smallest number of cuts he will have to make in the chain? What is his sequence of payments?
One cut in the third link will allow two links to be swapped for a kiss and a link on the second transaction, and 3 links for a kiss and 2 links on the third and so on.
41
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 two tangents. Each tangent and the arc of the circle is 1 mile long. What is the radius of the circle?
[CE BOARD NOV 2017]
1437.45 ft
42
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 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 sidewalls. Having no wings, the bug must take this trip by sticking to the surfaces of the room. What is the shortest route that the bug can take?
[CE BOARD NOV 2017]
58
43
A pirate buried his treasure on an island, a conspicuous landmark of which were three palm trees, each one 100 feet from the other two. Two of these trees were in a N-S line. The directions for finding the treasure read: "Proceed from southernmost tree 15 feet due north, then 26 feet due west." Is the treasure buried within the triangle formed by the tree?
no, outside the triangle
44
A coffee pot with a circular bottom tapers uniformly to a circular top with radius half that of the base. A mark halfway up the side says "2 cups." Where should the "3 cups" mark go?
[CE BOARD NOV 2016]
2% down from the top of cup
45
A spider and a fly are located at opposite vertices of a room of dimensions 1, 2 and 3 units. Assuming that the fly is too terrified to move, find the minimum distance the spider must crawl to reach the fly.
sqrt(18)
46
While still at a sizable distance from the Pentagon building, a man first catches sight of it. Is he more likely to be able to see two sides or three?
Equal
47
The Ben Azouli are camped at an oasis 45 miles west of Tagaba. They decided 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?
2 days
48
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 ¾ x inches in diameter. Find x.
[CE BOARD NOV 2017]
20.69 inches
49
Solve for real values of x: (7+4sqrt(3))^×-4(2+sqrt(3))^×=-1
1
50
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=25,b=16,c=39
51
A diaper is in the shape of a triangle with sides 24, 20 and 20 inches. The long side is wrapped around the baby's waist and overlapped two inches. The third point is brought up to the center of the overlap and pinned in place. The pin is to go through three thicknesses of material. What is the area in which the pin may be placed?
2.5 in^2
52
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?
5pi
53
Near the town of Lunch, Nebraska there is a large triangular plot of lands bounded by 3 straight roads which are 855, 870, and 975 yards long respectively. The owner of the land, a friend of mine, told me that he had decided to sell half the plot to a neighbor, but that the buyer had stipulated that the seller of the land should erect the fence which was to be a straight one. The cost of the fences being high, my friend naturally wanted the fence to be short as possible. What is the minimum length the fence can be?
600 yards
54
A man leaves from the point where the prime meridian crosses the equator and moves forty-five degrees northeast by geographic compass which always points toward the north geographic pole. He constantly corrects his route. Assuming that he walks with equal facility on land and sea, where does he end up and how far will he have travelled when he gets there?
North pole, (sqrt(2))×10^7 meters
55
The faces of a solid figure are all triangles. The figure has nine vertices. At each of six of these vertices, four faces meet, and at each of the other three vertices, six faces meet. How many faces does the figure have?
[CE BOARD NOV 2016]
14
56
Show that tan n/10 is a root of the equation 5x^4 - 10x^2 + 1 = 0
The angle a = pi/10 satisfies the relation tan 3a = cot 2a
57
A farmer owned a 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 filed 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?
[CE BOARD NOV 2018]
Area = 939120, Length = 2018
58
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?
one acre
59
Given five points in or on a unit square, prove that at least two points are no farther than sqrt(2/2) units apart.
Draw two lines through the center of the square, perpendicular to each other, such that each is parallel to a side of the unit square. These two lines partition the unit square into four ½ unit squares. At least 2 of the 5 points must be in (or on the perimeter of) one of these smaller squares. This pair of points cannot be farther apart than the length of the small square's diagonal sqrt(2/2) units
60
A divided highway goes under a number of bridges, the arch over each lane being in 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?
13 ft. and 5 inches
61
A certain 3-digit number in base 10 with no repeated digits can be expressed in base R by reversing the digits. Find the smallest value of R.
R= 438.14
MSTE
MSTE
Justin · 100問 · 2年前MSTE
MSTE
100問 • 2年前Justin
MSTE 2
MSTE 2
Justin · 23問 · 2年前MSTE 2
MSTE 2
23問 • 2年前Justin
PSAD
PSAD
Justin · 50問 · 2年前PSAD
PSAD
50問 • 2年前Justin
HGE
HGE
Justin · 47問 · 2年前HGE
HGE
47問 • 2年前Justin
MPLE
MPLE
Justin · 41問 · 1年前MPLE
MPLE
41問 • 1年前Justin
Weekly Test 1
Weekly Test 1
Justin · 50問 · 1年前Weekly Test 1
Weekly Test 1
50問 • 1年前Justin
Weekly Test 2
Weekly Test 2
Justin · 50問 · 1年前Weekly Test 2
Weekly Test 2
50問 • 1年前Justin
Weekly Test 3
Weekly Test 3
Justin · 50問 · 1年前Weekly Test 3
Weekly Test 3
50問 • 1年前Justin
Refresher SPDI 1
Refresher SPDI 1
Justin · 100問 · 1年前Refresher SPDI 1
Refresher SPDI 1
100問 • 1年前Justin
Refresher SPDI 1
Refresher SPDI 1
Justin · 9問 · 1年前Refresher SPDI 1
Refresher SPDI 1
9問 • 1年前Justin
Refresher Plumbing Code
Refresher Plumbing Code
Justin · 28問 · 1年前Refresher Plumbing Code
Refresher Plumbing Code
28問 • 1年前Justin
Definition of Terms 1
Definition of Terms 1
Justin · 90問 · 1年前Definition of Terms 1
Definition of Terms 1
90問 • 1年前Justin
Definition of Terms 2
Definition of Terms 2
Justin · 90問 · 1年前Definition of Terms 2
Definition of Terms 2
90問 • 1年前Justin
Definition of Terms 3
Definition of Terms 3
Justin · 90問 · 1年前Definition of Terms 3
Definition of Terms 3
90問 • 1年前Justin
問題一覧
1
A pupil wrote on the blackboard a series of fractions having positive integral terms and connected by signs which were either all + or all x, although they were so carelessly written it was impossible to tell which they were. It still wasn't clear even though he announced the result of the operation at every step. The third fraction had denominator 19. What was the numerator?
25
2
below
x = 0 or 2
3
Dr. Reed, arriving late at the lab one morning, pulled out his watch and the hour hand are exactly together every sixty-five minutes." Does Dr. Reed's watch gain or lose, and how much per hour?
60/143 minutes gain
4
Represented above is a "magic square" in which the sum of each row, column, or main diagonal is the same. Using nine different integers, produce a "multiplicative" magic square, i.e., one in which the word "product" is substituted for "sum".
Using the rule that powers multiply by adding their exponents, a multiplicative magic square is easily obtained from the given square by substituting 2^n (or k^n where k > 1) for n in each block
5
Which is greater: 3+4/(3+4/(3+...) or 3+(3+(3+.../4)/4 where the respective denominators and numerators continue indefinitely?
Equal to 4
6
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?
4:26.853
7
For X < 1, evaluate the infinite product:
(1+X+X^2+...+X^9)(1+X^10+X^20+X^30+...X^90)
(1+X^100+X^200+...+X^900)...
T1二0^∞ 【x^n=1/（1-x）】
8
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 for 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?
Pat = 8 errors, Mike = 2 errors
9
If v varies as w2, w^3 as x'4, x^5 as y^6, and y^7 as z^4, show that the product does not vary at all.
since each v/w^2, w^3/x^4, x^5/y^6 and y^7/z^4 is a constant, therefore their product is also a constant
10
There are nine cities which are served by two competing airlines. One or the other airline (but not both) has 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)?
[CE BOARD NOV 2016]
12
11
A necklace consists of pearls which increase uniformly from a weight of 1 carat for the end pearls to a weight of 100 carats for the middle pearl. If the necklace weighs altogether 1650 carats and the clasp and string together weigh as much (in carats) as the total number of pearls, how many pearls does the necklace contain?
33 pearls
12
A cubic box with sides 'a' feet long is placed flag 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 and p=sqrt(15), calculate at what height the ladder touches the wall.
1.38 ft or 3.62 ft
13
A parking lot charges & 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 time clock registers only in 5-minute intervals)
Jones = 0.5 hours, Smith = 3.5 hours
14
What is the millionth term of the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ... in which positive integer n occurs in blocks of n terms?
1414
15
There are n points on a circle. A straight-line segment is drawn between each pair of points. How many intersections are there within the circle if no 3 lines are collinear?
764,488
16
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?
pencil = 26, eraser = 19, notebook = 55
17
Without performing any algebraic manipulation at all, Archimedes O'toole remarked that the sum and product of the two expressions (x+y-Ix-yl)/2 and (x+y+|x-yl)/2 are respectively x+y and xy. Why was this obvious?
The two expressions are identically equal, respectively, to the smaller and the larger of the two numbers x and y.
18
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 the 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?
25%
19
A student just beginning the study of logarithms was required to evaluate an expression of the form log A/log B. He proceeded to cancel common factors in both numerator and denominator, (including the "factor" log), and arrived at the result 2/3. Surprisingly, this was correct. What were the values of A and B?
A = 9/4, B = 27/8
20
Citizens of Franistan pay as much income tax (percentage-wise) as they make rupees per week. What is the optimal salary in Franistan?
50 rupees
21
Dr. Irving Weiman, who is always in a hurry, walks up an upgoing 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?
[CE BOARD NOV 2017]
80 steps
22
Depicted above are two interlocked hyperbolas. Impossible? You're right, but can you prove it?
The number of solutions of a set of polynomial equations is the product of their degrees. Since hyperbolas are of 2nd degree, the solutions is to a pair of hyperbolic equations cannot exceed 4. Thus, two hyperbolas can intersect in no more than 4 points.
23
There are four towns at the corners of a square. Four motorists set out, each driving in 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 204, and the second and fourth 212 (beeline) miles apart. How fast is the first car traveling and how far apart are the towns?
[CE BOARD MAY 2017 / NOV 2022]
50 miles/hour, 180 miles
24
A mathematician whose clock had stopped wound it but did not bother to set It correctly. Then he walked from his home to the home of a friend for an evening of hi-fi music. Afterwards, he walked back to his own home and set his clock exactly. How could he do this without knowing the time his trip took?
t2+1/2[(T2-T1)-(t2-t1)]
25
An expert on transformer design relaxed one Saturday by going to the races. At the end of the first race, he had doubled his money. He bet $30 on the second race and tripled his money. He bet $54 on the third race and quadrupled his money. He bet $72 on the fourth race and lost it, but still had $48 left. With how much money did he start?
29
26
To stimulate his son is the pursuit of partial differential equations, a math professor offered to pay him $8 for every equation correctly solved and to fine him S5 for every incorrect solution. At the end of 26 problems, neither owed any money to the other. How many did the boy solve correctly?
[CE BOARD MAY 2019]
10
27
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?
50 flags
28
Jai Alai balls come in boxes of 8 and 15; so that 38 balls (one small box and two large) can be bought without having to break open a box, but not 39. What is the maximum number of balls which cannot be bought without breaking boxes?
97
29
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 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.)
5562
30
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 direction the other took, the first snail would have reached his food 35 minutes after the second How do their speeds compare?
V1 = ¾ V2
31
A yang, ying, and yung is constructed by dividing a diameter of a circle, AB, into three parts by points C and D, then describing on one side of AB semicircles having AC and AD as diameters and on the other side of AB semicircles having BD and BC as diameters. Which is larger, the central portion or one of the outside pieces?
Same
32
A forgetful physicist forgot his watch one day and asked an E.E. on the staff what time it was. The E.E. looked at his watch and said: "The hour, minute, and sweep second hands are as close to trisecting the face as they ever come. This happens only twice in every 12 hours, but since you probably haven't forgotten whether you ate lunch, you should be able to calculate the time." What time was it to the nearest second?
2:54:35 and 9:05:25
33
How can seven points be placed, no three on the same line, so that every selection of three points constitutes the vertices of an isosceles triangle?
Place five at the vertices of a regular pentagon, the sixth at the center of the pentagon, and the seventh above the center at a distance equal to the radius of the pentagon.
34
A cowboy is five miles south of a stream which flows due east. He is also 8 miles west and 6 miles north of his cabin. He wishes to water his horse at the stream and return home. What is the shortest distance he can travel and accomplish this?
17.9 miles
35
In Puevigi numbers such as 2, 5, 8, 10, etc., that are the sum of two squares, are considered sacred. Prove that the product of any number of sacred numbers is sacred.
It suffices to prove it for two sacred numbers, since the theorem then follows by induction. Let M = A^2 + B^2 and N = C^2 + D^2. Then MN = (A^2 + B^2)(C^2 + D^2) = (AC + BD)^2 + (AD - BC)^2. Amen.
36
Given a point P on one side of a general triangle ABC, construct a line through P which will divide the area of the triangle into two equal halves.
From P draw a line, L, to the opposite vertex, say A. Now construct a line parallel to L from the midpoint of BC, intersecting the side of the triangle at Q. The line PQ divides the triangle into two equal areas.
37
Let c be the hypotenuse of a right triangle with legs a and b. Prove that if x > 2, then a'x + b^x < c^x
a^x+b^x=a^2 (a^(x-2))+b^2 (b^(x-2)) < a^2 c^(x- 2) +b^2 c^(x-2)=(a^2+b^2) c^(x-2) =c^2 c^(x-2)=c^x
38
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?
24 inches
39
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 corner on his left, thus following a curved course which terminates in the middle of the field, all three hares arriving there together. The hares obviously ran at the same speed, but just how far did they run?
100 yards
40
A wizard in Numerical Analysis has a gold chain with 7 links. A Lady Programmer challenges him to use the chain to buy 7 kisses, each kiss to be paid for, separately, with one chain link. What is the smallest number of cuts he will have to make in the chain? What is his sequence of payments?
One cut in the third link will allow two links to be swapped for a kiss and a link on the second transaction, and 3 links for a kiss and 2 links on the third and so on.
41
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 two tangents. Each tangent and the arc of the circle is 1 mile long. What is the radius of the circle?
[CE BOARD NOV 2017]
1437.45 ft
42
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 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 sidewalls. Having no wings, the bug must take this trip by sticking to the surfaces of the room. What is the shortest route that the bug can take?
[CE BOARD NOV 2017]
58
43
A pirate buried his treasure on an island, a conspicuous landmark of which were three palm trees, each one 100 feet from the other two. Two of these trees were in a N-S line. The directions for finding the treasure read: "Proceed from southernmost tree 15 feet due north, then 26 feet due west." Is the treasure buried within the triangle formed by the tree?
no, outside the triangle
44
A coffee pot with a circular bottom tapers uniformly to a circular top with radius half that of the base. A mark halfway up the side says "2 cups." Where should the "3 cups" mark go?
[CE BOARD NOV 2016]
2% down from the top of cup
45
A spider and a fly are located at opposite vertices of a room of dimensions 1, 2 and 3 units. Assuming that the fly is too terrified to move, find the minimum distance the spider must crawl to reach the fly.
sqrt(18)
46
While still at a sizable distance from the Pentagon building, a man first catches sight of it. Is he more likely to be able to see two sides or three?
Equal
47
The Ben Azouli are camped at an oasis 45 miles west of Tagaba. They decided 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?
2 days
48
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 ¾ x inches in diameter. Find x.
[CE BOARD NOV 2017]
20.69 inches
49
Solve for real values of x: (7+4sqrt(3))^×-4(2+sqrt(3))^×=-1
1
50
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=25,b=16,c=39
51
A diaper is in the shape of a triangle with sides 24, 20 and 20 inches. The long side is wrapped around the baby's waist and overlapped two inches. The third point is brought up to the center of the overlap and pinned in place. The pin is to go through three thicknesses of material. What is the area in which the pin may be placed?
2.5 in^2
52
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?
5pi
53
Near the town of Lunch, Nebraska there is a large triangular plot of lands bounded by 3 straight roads which are 855, 870, and 975 yards long respectively. The owner of the land, a friend of mine, told me that he had decided to sell half the plot to a neighbor, but that the buyer had stipulated that the seller of the land should erect the fence which was to be a straight one. The cost of the fences being high, my friend naturally wanted the fence to be short as possible. What is the minimum length the fence can be?
600 yards
54
A man leaves from the point where the prime meridian crosses the equator and moves forty-five degrees northeast by geographic compass which always points toward the north geographic pole. He constantly corrects his route. Assuming that he walks with equal facility on land and sea, where does he end up and how far will he have travelled when he gets there?
North pole, (sqrt(2))×10^7 meters
55
The faces of a solid figure are all triangles. The figure has nine vertices. At each of six of these vertices, four faces meet, and at each of the other three vertices, six faces meet. How many faces does the figure have?
[CE BOARD NOV 2016]
14
56
Show that tan n/10 is a root of the equation 5x^4 - 10x^2 + 1 = 0
The angle a = pi/10 satisfies the relation tan 3a = cot 2a
57
A farmer owned a 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 filed 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?
[CE BOARD NOV 2018]
Area = 939120, Length = 2018
58
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?
one acre
59
Given five points in or on a unit square, prove that at least two points are no farther than sqrt(2/2) units apart.
Draw two lines through the center of the square, perpendicular to each other, such that each is parallel to a side of the unit square. These two lines partition the unit square into four ½ unit squares. At least 2 of the 5 points must be in (or on the perimeter of) one of these smaller squares. This pair of points cannot be farther apart than the length of the small square's diagonal sqrt(2/2) units
60
A divided highway goes under a number of bridges, the arch over each lane being in 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?
13 ft. and 5 inches
61
A certain 3-digit number in base 10 with no repeated digits can be expressed in base R by reversing the digits. Find the smallest value of R.
R= 438.14