Littons part 2-0314

60問 • 2年前

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Prove that each median of a triangle is shorter than the average of the 2 adjacent sides.

Reflect the triangle through the opposite side. The problem now reduces to proving a diagonal of the resulting parallelogram is shorter than the sum of adjacent sides. This follows from the triangle inequality.

Define every point of a the plane with 2 integer coordinates (e.g. [3,0] or [-5, 2]) as a “lattice point.” Let every pair of lattice points I the plane be connected with a “lattice line.” Prove or disprove: “The lattice lines cover the plane.”

Each lattice line is either vertical with equation x = integer, or else it passes through (A, B) and (C, D) and has equation y =................Since the coefficients are rational, y will be rational whenever x is. It follows, for example, that the point (1/2, √2) does not lie on any lattice line. Hence the lattice lines do not cover the plane.

A student beginning the study of Trigonometry came across an expression of the form sin (X + Y). He evaluated this as sin X + sin Y. Surprisingly he was correct. The values of X and Y differed by 10˚; what were these values, assuming that 0˚ < X < Y < 360˚?

x = 175˚, y = 185˚

Above is a map of Lake Puevigi. The cross represents a buried treasure cache. Cover the right hand half of the diagram. Now answer: “Is the treasure in the lake or an land?”

In the lake. It is simple to verify that a point is inside a closed curve if and only if it requires an odd number of “crossings” to be outside. In this case the number is 3.

If the equal sides of an isosceles triangle are given, what length of the third side will provide maximum area? (No calculus, please.)

√2 times the length of the equal sides

One side of the triangle is 10 feet longer than another and the angle between them is 60˚. Two circles are drawn with these sides as diameters. One of the points of intersection of the two circles is the common vertex. How far from the third side is the other point of intersection?

Here’s a rather unusual optical illusion. How many different configurations can you “see”?

1. A little cube nestled in the corner of a big one. 2. A big cube with a cubical chunk removed from one corner. 3. Two cubes meeting externally at a corner. If you perceived all 3, congratulations! If you saw any other configurations, what were they??

The isosceles right triangle shown above has a vertex at the center of the square. What is the area of the common quadrilateral?

12.25

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 he is on the track, one flag is 3 times as far away as the other. How long is the track?

1980π

Through binoculars a bird watcher observed a hummingbird feeder between one and two o’clock of an afternoon. He timed the visits and saw a ruby-throat take a drink at 1, 5, 6, 8, 15, 16, 19, 22, 27, 29, 32, 36, 38, 43, 45, 49, 50, 57, and 58 minutes after the hour of one. The last visit he saw took place at two, at which time he left in perplexity. He knew from experience that a hummer’s “feeding cycle” is remarkably stable and is generally between 5 and 15 minutes long. This one seemed rather erratic, to say the least. Can you advise him on what was going on?

Three hummingbirds were sharing the feeding station with cycles of 7, 11, and 13 minutes, respectively, in the order in which he first observed them.

Find the smallest number (x) of persons a boat may carry so that (n) married couples may cross a river in such a way that no woman ever remains in the company of any man unless her husband is present. Also find the least number of passages (y) needed from one bank to the other. Assume that the boat can be rowed by one person only.

no. of persons x=2 ; no. of passengers y=5

A, B and C participate in a track meet, consisting of at least three events. A certain number of points are given for first place, a smaller number for second place, and a still smaller number for third place. A won the meet with a total score of 14 points; B and C are tied for second with 7 points apiece. B won first place in the high jump. Who won the pole vault assuming no ties occurred in any event?

"A" won the pole vault

Find the simplest solution in integers for the equation 1/x² + 1/y² = 1/z²

x=15, y=20, z=12

Maynard the Census Taker visited a house and was told, “Three people live there. The product of their ages is 1296, and the sum of their ages is our house number.” After an hour of cogitation Maynard returned for more information. The house owner said, “I forgot to tell you that my son and grandson live here with me.” How old were the occupants and what was their street number?

age = 1 and 18, street number = 72

Prove that the produce of 4 consecutive positive integers cannot be a perfect square.

Let N be the smallest integer. The product is then N(N + 1) (N + 2) (N + 3) = (N2 + 3N) (N2 + 3N + 2) = (N2 + 3N + 1)2 – 1. This is not a perfect square since 2 positive squares cannot differ 1.

In Byzantine basketball there are 35 scores which are impossible for a team to total, one of them being 58. Naturally a free throw is worth fewer points than a field goal. What is the point value of each?

free throw = 8, field goal = 11

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?

1, 2, 3

Three farmers, Adams, Brown and Clark all have farms containing the same number of acres. Adams’ farm is most nearly 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. Brown’s farm is intermediate between these two, the length being 28 miles longer than the width. If all the dimensions are in exact miles, what is the size of each farm?

Adam = (40 x 48), Brown = (32 x 60), Clark = (30 x 64)

1960 and 1961 were bad years for ice cream sales but 1962 was very good. An accountant was looking at the tonnage sold in each year and noticed that the digital sum of the tonnage sold in 1962 was three times as much as the digital sum of the tonnage sold in 1961. Moreover, if the amount sold in 1960 (346 tons), was added to the 1961 tonnage, this total was less than the total tonnage sold in 1962 by the digital sum of the tonnage sold in that same year. Just how many more tons of ice cream were sold in 1962 than in the previous year?

361 tons

Three rectangles of integer sides have identical areas. The first rectangle is 278 feet longer than wide. The second rectangle is 96 feet longer than wide. The third rectangle is 542 feet longer than wide. Find the area and dimensions of the rectangles.

Area is 1,466,690 square feet Rectangular dimensions are: (1080 x 1358); (1164 x 1260); (970 x 1512 )feet

When little Willie had sold all his lemonade he found he had $7.95 in nickels, dimes and quarters. There were 47 coins altogether and, having just started to study geometry, he noticed that the numbers of coins satisfied a triangle inequality, i.e., the sum of any two denominations was greater than the third. How many of each were there?

D = 20 dimes, Q = 23 quarters, N = 4 nickels

There are 100 coins in a piggy bank totaling $5.00 in value, the coins consisting of pennies, dimes and half dollars. How many of each are there?

Half dollars = 1, Dimes = 39, Pennies = 60

Every year an engineering 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 $6,000 if all but two of his staff had merited the $300 bonus, but none of them more than twice?

8 years

n 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?

5.4 = 20

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.

43814

Two wheels in the same plane are mounted on shafts 13 in. 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 much larger is one wheel than the other?

5 inches larger

Five points are located in or on the perimeter of an equilateral triangle with 9-inch sides. If d is the distance between the closest pair of points, what is the maximum possible value of d?

4.5 in.

If THAT = (AH)(HA), what is THAT?

6786

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 “arrangements” is the same. The girls outnumber the boys. How many are there?

4 girls, 2 boys

Dad and his son have the same birthday. One the last one, Dad was twice as old as Junior. Uncle observed that this was the ninth occasion on which Dad’s birthday age has been an integer multiple of Junior’s. How old is Junior?

Junior = 36 years old, Dad = 72 years old

The undergraduate of a School of Engineering wished to form ranks for a parade. In ranks of 3 abreasts, 2 m2n 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?

1154

What is the remainder upon dividing 5⁹⁹⁹,⁹⁹⁹ by 7?

A pet store offered a baby monkey for sale at $1.25. The monkey grew. Next week it was offered at $1.89, then $5.13, then 5.94, then $9.18 and on the sixth week a Ph.D in Aeronautics bought it for $12.42. How were the new prices figured?

The price was figured by adding the square of the sum of the digits of the previous price TO the previous price.

The odd digits 1, 3, 5, 7,and 9 add up to 25, while the even figures, 2, 4, 6, and 8, only add up 20. Arrange these figures so that the odd ones and the even ones add up alike. Complex and improper fractions and recurring decimals are not allowed.

84 ⅓

Assume the universe is a billion billion light years in diameter and is packed solidly with matter weighing a billion billion tons per cubic inch and each gram of this matter contains a billion billion atoms. Also, every second during the past billion billion years, a billion billion similar universes were created. Without using any symbols and restricting yourself to a total of three digits, write a number that far exceeds the total atoms of all these universes.

9⁹⁹

The sum of the digits on the odometer in my car (which reads up to 99999.9 miles) has never been higher than it is now, but it was the same 900 miles ago. How many miles must I drive before it is higher than it is now?

100 miles

How many primes are in the following infinite series where the digits are arranged in declining order? 9; 98; 987; 9876; …………….; 987654321; 9876543219; 98765432198; ... etc.

What is the largest number which can be obtained as the product of positive integers which add up to 100?

3³²• 2²

The first expedition to Mars found only the ruins of a civilization. The explorers were able to translate a Martian equation as follows: ................ . This was strange mathematics. The value x = 5 seemed 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 Martians had?

A rectangular picture, each 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.

3 x 10 or 4 x 6

Find unequal rational numbers, a, b, (other than 2 and 4) such that a^b= b^a.

a = 9/4, b = 27/8

Find a five-digit number whose first two digits, central digit, and last two digits are perfect squares and whose square root is a prime palindrome.

191² or 36481

My house is on a road where the numbers run 1, 2, 3, 4… consecutively. My number is a three digit one and, by a 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 my road?

house number = 204, no. of houses = 208

The sum and difference of two squares may be primes: 4 – 1 = 3 and 4 + 1 = 5; 9 – 4 = 5 and 9 + 4 = 13, etc. Can the sum and difference of two primes be squares? If so, for how many different primes is this possible?

p = 2, q = 2

On what days of the week can the first day of a century fall? (The first day of the twentieth century was Jan. 1, 1901)

Monday

Solve for A and B, both triangular numbers: 799³ = A² – B².

318,801

A certain 6-digit number is a square in both the scale of 5 and the scale of 10. What is it?

232324

Starting with one, place each succeeding integer in one of two groups such that neither group contains three integers in arithmetic progression. How far can you get?

First 8 integers

In a lottery the total prize money available was a million dollars, paid out in prizes which were powers of $11 viz., $1, $11, $121, etc. Noe more than 6 people received the same prize. How many prize winners were there, and how was the money distributed?

20 winners

In the arithmetic of Puevigi, 14 is a factor of 41. What is the base of the number system?

For what n is åⁿ (k=1)k! square?

1 or 3

Find the only number consisting of five different digits which is a factor of its reversal.

87912

No factorial can end in five zeros. What is the next smallest number of zeros in which a factorial can not end?

One is the smallest integer which is simultaneously a perfect square, cube and fifth power. What is the next smallest integer with this property?

2³⁰

Barnie Bookworm bought a thriller – found to his dismay, Just before the denouement a fascicle astray. Instead of counting one through ten, a standard cure for rages. He totalled up the number of the missing sheaf of pages. The total was eight thousand and six hundred fifty-six. What were the missing pages? Try to find them just for kicks,

255 – 286 are missing pages

The reciprocals of the divisors of six sum to two, i.e., 1/1+½+⅓+⅙= 2 . Find another number with this property.

the next two perfect numbers after 6 being 28 and 496.

The Sultan arranged his wives in order of increasing seniority and presented each with a golden ring. Next, every 3rd wife, starting with the 2nd, was given a 2nd ring; of these every 3rd one starting with the 2nd received a 3rd ring, etc. His first and most cherished wife was the only one to receive 10 rings. How many wives had the Sultan?

9842 wives

If you solve the alphametic WATER – HEAT = ICE, you will have the solution to this double riddle: “This bird’s assured of his breakfast/and these before steeds cause a wreck fast.” Curiously, 70243 is the answer to both riddles!

Decoding 70243 with respect to each solution the two riddles : EARLY and CARTS

(L I X + L V I) devided by C X V ; x²=C The above alphametic involving Roman numerals is correct. It will still be correct if the proper Arabic numerals are substituted. Each letter denotes the same digit throughout and no 2 letters stand for the same digit. Find the unique solution.

the unique solution: 453 + 485 = 938 is obtained.

Find a permuation of the numbers 1 through 7 with the property that when placed in both the first and third rows, the seven row totals will alll be perfect squares.

4736251

PSAD TERMS 2011-2018

Ian Calasang · 3回閲覧 · 60問 · 2年前

PSAD TERMS 2011-2018