Littons

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Math professor offered to pay him $8 for every equation correctly solved. and fine him $5 for every incorrect. At end of 26 problems, How many did the boy solve correctly?

An expert on transformer design relaxed one Saturday by going to races. At end of first race, He doubled his money. He bet $30 on second race, $54 on third race, $72 on fourth race and lost but still had $48 left. With how much money did He start?

A mathematician whose clock had stopped 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. He walked back 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)]

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

For X < 1 evaluate the infinite product:

aXn=1/(1-x)

If v varies as w2 , w3 as x4 , x5 as y6 , and y7 as z4 , show that the product (v/z)*(w/z)*(x/z)*(y/z) does not vary at all.

constant

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?

gains 60/143

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

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 sec or 1:32.7 hours

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

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 √15 calculate at what height the ladder touches the wall, using quadratics only.

1.38 ft or 3.62 ft

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?

80 steps

Citizens of Franistan pay as much income tax (percentagewise) as they make rupees per week. What is the optimal salary in Franistan?

50 rupees

There are nine cities which are served by two competing airlines. One or the other airline (but not both) has a flight between avery pair of cities. What is the minimum number of traingular flights (i.e., trips from A to B to C and back to A on the same airline)?

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 =(3/4) *v2

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

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?

Numerator = 25

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 balls

Without performing any algebraic manipulation at all, Archemedes O’toole remarked that the sum and product of the two expressions .........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.

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 time clock registers only in 5-minute intervals).

Jones= 0.5 hrs , Smith 3.5 hrs

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%

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

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

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

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?

50 mph, 180 mph

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 2n (or kn where k > 1) for n in each block

Solve the equation ............where both members represent infinite expressions.

x=0 or 2

Which is greater:............. where the respective denominators and numerators continue indefinitely?

equal to 4

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

Depicted above are two interlocked hyperbolas. Impossible? You’re right, but can you prove it?

two hyperbolas can intersect in no more than 4 points.

Solve for real values of x (7+4√3)^x -4(2+√3)^x=-1

x=1

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.

t suffices to prove it for two sacred numbers, since the theorem then follows by induction. Let M = A2 + B2 and N = C2 + D2 . Then MN = (A² + B²)(C² + D²) = (AC + BD)² + (AD – BC)²

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 hve 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.

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

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?

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?

1437.45 ft.

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.

√18

Show that tan(π/10) is a root of the equation 5x⁴ - 10x²+1=0

x = tan a

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 walll 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?

58 ft.

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?

5π

A farmer owned a square field measuring exactly 2261 yards on each side. 1898 yards from one corner and 1009 yards from an andjacent corner stood a beech tree. A neightbor offered to purchase a triangular portion of the filed stipulating that a fence should be erected in a striaght 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 offere but msde 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?

Area = 939,120, Length = 2018

Given five points in or on a unit square, prove that at least two points are no farther than (√2/2) units apart.

This pair of points cannot be farther apart than the length of the small square’s diagonal.

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

A man leaves from the point where the prime meridian crosses the equator and moves 45 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 traveled when he gets there?

North Pole, √2 *10⁷ meters

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?

100 yards

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

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

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

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

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

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?

outside the triangle

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

The Ben Azouli are camped at an oasis 45 miles west of Taqaba. 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 distance 36 miles

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.

20.69 inches

Let c be the hypotenuse of a right triangle with legs a and b. Prove that if x > 2, then ax + bx < cx

ax + bx = a2 ax–2 +b2 bx–2 < a2 cx–2 + b2 cx–2 = (a2 + b2 ) cx–2 = c2 cx–2 = cx.

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

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 squared inch

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?

2% down from the top of cup

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.

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 its previous weight?

Weight of new = 33 times weight of old

A hostess plans to serve a square cake with icing on top and sides. Upon determining how many guests want cake, what method should she use to insure that each guest will receive the same amount of cake and icing?

Top areas are equal to receive the same amount of cake. Side areas are equal to receive equal icing areas.

Littons part 2-0314

Ian Calasang · 60問 · 2年前

Littons part 2-0314