Suppose I randomly choose 4 distinct integers between 1 and 60. Is it true, or is it not true, that at least 2 of them must differ by no more than 19.

Answers

Answer 1

Answer:

It’s true

Step-by-step explanation:

Proof by induction

Let the four chosen numbers be x_1, … x_4 and, without loss of generality, let x_1 < x_2, … < x_4.

For x_1 = 1, the smallest possible choice, let x_2 = x_1 + 20 = 21 be the smallest possible choice that has a difference larger than 19. Likewise, x_3 = x_2 + 20 = x_1 + 40 = 41. Then x_4 has to be at least x_3 + 20 = x_1 + 60 = 61 which is out of range.

For the induction, suppose that x_4(n) = x_1(n) + 60 > 60 and therefore x_4(n) can’t be chosen for a given x_1(n). Then incrementing x_1(n+1) = x_1(n) + 1 would require x_4(n+1) to be at least x_1(n) + 1 + 60 = x_4(n) + 1 > 61, so this is out of range as well.

For another induction, suppose that x_4(n) = x_2(n) + 40 > 60 and therefore x_4(n) can’t be chosen for a given x_2(n). Then incrementing x_2(n+1) = x_2(n) + 1 would require x_4(n+1) to be at least x_2(n+1) + 40 = x_4(n) + 1 > 61, so this is out of range as well.

Likewise, for the induction, suppose that x_4(n) = x_3(n) + 20 > 60 and therefore x_4(n) can’t be chosen for a given x_3(n). Then incrementing x_3(n+1) = x_3(n) + 1 would require x_4(n+1) to be at least x_3(n+1) + 20 = x_4(n) + 1 > 61, so this is out of range as well.

Therefore all choices of x_1, x_2, x_3 with differences greater than 19 leaves no choice of x_4 within range with x_4 > x_3 + 19.


Related Questions

Help please please answer

Answers

Answer:

A

Step-by-step explanation:

x. y = 28

check attachment

When Three friends divide four boxes of baseball trading cards ?evenly, they each get 48 cards. How many cards are in each box?

Answers

Answer:

36 cards in each box

Step-by-step explanation:

Let the number of cards in each box be x.

4 * x / 3 = 48 (4 packs of cards, three friends)

x / 3 = 21

x = 36.

Answer:

36 cards in each box

Step-by-step explanation:

Because each friend gets the same amount of cards, there needs to be 48*3 or 144 trading cards in total. There are 4 boxes so in each box there would be 144/4 or 36 cards in total.

Alex and glen share a 30 ounce bucket of clay by the end of the week alex has used 1/10 of the bucket and glen has used 2/5 of the bucket of clay how many ounces are left in the bucket

Answers

Answer:

21 ounces

Step-by-step explanation:

Answer:

15 ounces

Step-by-step explanation:

we know that the whole is 30  we know that Alex 1/10 and glen has 2/5 we dont know wht left over :: 1/10 + 2/5 = 1/10 + : 4/10 = 5/10 so 1 - 5/10 = 5/10

We need to find how many ounces of clay are left. Which equation can we use to find how many ounces are in 5/10

of the whole bucket?    5/10 x 30  and 5x 30 10

15 onces

Anyone know the middle question ?

Answers

Ab is obtuse acute to AD
DC is perpendicular parallel to AB
parrallel :)))))))))))

Trapezoid STUV is similar to trapezoid NOPQ. What is the length of OP?

Answers

Answer:

F. 4 2/3 centimeters

Step-by-step explanation:

since the trapezoids are similar to each other, that means there is a common ratio between them. so taking into consideration the first trapezoid STUV when we compare that to NOPQ, OP comes out to be 4 and 2/3 centimeters. start by comparing the ratio of ST to NO and then move on to TU and then OP.

. Since the pentagon is a regular pentagon, the measure of each interior angle will be the same.

To find the size of each angle, divide the sum, 540º, by the number of angles in the pentagon.

(which is the same as the number of sides).

540° ÷ 5 = 108° There are 108° in each interior angle of a regular pentagon.

2. 540°

Sum of Interior Angles of a Polygon

A pentagon has five sides, thus the interior angles add up to 540°, and so on.

please help!

thank u!

Answers

Answer:

c

Step-by-step explanation:

A line with a slope of 2 passes through the point (5, 8). What is its equation in slope-intercept form? (I will make u brainleist)

Answers

Answer:

y= 2x -2

Step-by-step explanation:

Find the solution
4y-x=5+2y
3x+7y=24

Answers

Answer:

24

Step-by-step explanation:

Please explain how you got the answer. Don’t just tell me the answer

Answers

Answer: (4 3/5) x (3 1/2)

23/5  (3 1/2)

23/5 (7/2)

161/10 =

your answer is 16 1/10

Step By-step explanation:

A baker melts 80 grams of chocolate that contains 40% cocoa and mixes it with 60
grams
of his secret mixture. The result is a chocolate with 55% cocoa. What is the
percent of cocoa in the baker's secret mixture?

Answers

Answer:

15%

Step-by-step explanation:

55%-40% = 15% cocoa.

Answer is 15% as the chocolate is 40% cocoa before mixture

IM GIVING 43 POINTS PLS ANSWER ASAP I MARK BRAINLIEST

Answers

Answer:

m = 3

Step-by-step explanation:

Get two points and use slope formula

Points

(50, 40) and (60, 70)

Find the slope:

m = (y₂ - y₁)/(x₂ - x₁) m = (70 - 40)/(60 - 50) = 30/10 = 3

Help with #10 please I’m having trouble understanding

Answers

Answer:

u=66(oppodite angles of parallrlogram are equal)

66+3v=180( cointerior angle)

3v=114

V=38

Step-by-step explanation:

plz mske me brainliest

9x + 5 = 4(x + 2) + 5x
solve for X

Answers

Answer:

no solution

Step-by-step explanation:

9x + 5 = 4(x+2) + 5x

9x + 5 = 4x + 8 + 5x

9x + 5 = 9x + 8

9x + 5 can never be equal to 9x + 8, so the answer is no solution

Answer:

x=3

Step-by-step explanation:

9x + 5 =4(x+2) + 5x

9x + 5= 4x + 8+ 5x ( do the parenthesis first- mulitiply  4 and x and get 4x and 4 times 2 = 8)

9x + 5 = 9x + 8 ( add the x's together 4x +5x = 9x )

9x = 9x + 3 ( subtract 5 both sides and you get 3 )

=3 ( subtract 9x from both sides

Brainliest if right i need a couple more to move on :)

and if there is a choice no solution :))

What is the measure of angle AEB? Enter your answer as a whole number.

Answers

Answer: 89 or 90 degrees.

Step-by-step explanation:


Ryan scored 12 more points than Alex on a video game. If together they scored 214 points, how many points
does each person score?

Answers

Answer:

Ryan 113, Alex 101

Step-by-step explanation:

Ryan beats Alex by 12 points

101+101=202

202+12=214

113+101=214

Consider the line - 4x – 3y = 5.
What is the slope of a line perpendicular to this line?
What is the slope of a line parallel to this line?

Answers

Answer:

The slope of the perpendicular line is 1/4

A line parallel to this will have the same gradient of -4.

Need help fast as possible!!!! Pleaseeeee!!!!!!!!

Answers

ita 3 fiths x plus three eigths

plz plz plz help ASAP

Answers

Answer

B. number five is right

Step-by-step explanation:

Answer:

$5

Step-by-step explanation:

First, subtract the $6 toy from the total of $71.

$71 - $6 = $65

The total of the shirts and pants is $65.

She bought 9 shirts and 4 pants, which would be 13 shirts and pants.

We then divide the total cost of the shirts and pants, $65 by the total shirts and pants, 13.

65 / 13 = 5

So, each shirt or pants costs $5.

I would appreciate brainliest, if not that's ok!

Luego de dormir 6 horas al día a usted le quedan 18, que puede repartir entre estudio y ocio. X representa
las horas dedicadas al ocio y la variable Y las horas de estudio. Represente la relación lineal existente entre
las distintas combinaciones posibles de X e Y.​

Answers

¿puede por favor hablar inglés

The linear relationship between the different possible combinations of X and Y is Y= -X + 18.

What is an equation?

An equation contains one or more terms with variables connected by an equal sign.

Example:

2x + 4y = 9 is an equation.

2x = 8 is an equation.

We have,

We know that after sleeping for 6 hours, there are 18 hours left in a day. These 18 hours can be split between leisure time and study time.

Now,

Let's let X be the number of hours of leisure time, and Y be the number of hours of study time.

Since we know that there are a total of 18 hours,

we can set up the following equation.

X + Y = 18

This equation represents all the possible combinations of X and Y that add up to 18 hours.

Now,

To represent a linear relationship between X and Y.

Y = -X + 18

This equation represents a linear relationship between X and Y.

Where X is the independent variable and Y is the dependent variable.

This is in the form of a slope-intercept form.

y = mx + c

Where m = -1 and c = 18

So,

The slope = -1 and the y-intercept is 18.

Thus,

Y= -X + 18 represents all the different possible combinations of leisure time and study time that adds up to 18 hours.

Learn more about equations here:

https://brainly.com/question/17194269

#SPJ5

The complete question.

After sleeping 6 hours a day, you have 18 hours left, which you can divide between study and leisure. X represents the hours dedicated to leisure and the variable Y the hours of study. Represent the linear relationship between the different possible combinations of X and Y.​


The rectangle is enlarged by a scale factor of 5. What is the perimeter of
the enlarged rectangle?

Answers

Answer:

I believe its 90

Step-by-step explanation:

You multiply each side by 5

5x5=25

double it for the other side

4x5= 20

double it

50+40=90

CAN I GET HELP PLEASE!

Answers

Answer:

0.30 of a book / hour or 3/10 of a book per hour.

Step-by-step explanation:

Usually I tell people to put the numbers into decimals. Unfortunately for this question, it won't work. So we have any choice but to do it with fractions.

r = amount / time

amount = 1/4

time = 5/6

r = [tex]\frac{\frac{1}{4} }{\frac{5}{6} }[/tex]

Invert the denominator and multiply

r = [tex]\frac{1}{4} * \frac{6}{5}[/tex]

r = 6 * 1 / (5*4) = 6/20 = 3/10

The answer is 3/10 of a book per hour

5/6 of an hour is almost an hour.

1/4 of a book is 0.25 of a book.

3/10 of a book is 0.30 of a book. So it looks like it is just about right.

HELP meeeeee please

Answers

Answer: a = 23, b = 21, c = 136

Step-by-step explanation:

5(wx + yz) =5wx + 5xy

Answers

Answer:

Step-by-step explanation:

5wx+5yz=5wx+5xy

5wx-5wx+5yz=5xy

5yz=5xy

5Y(X-Z)=0
Step-by-step explanation:
5(WX+YZ)= 5WX+5XY
Open bracket l.h.s
5wx+ 5yz
Put like terms together
5wx-5wx= 5xy-5yz
Substitute
0=5y(x-z)

can someone solve this?

Answers

47z²−25

Step-by-step explanation:

3627272727822881818181

Answer:

130z-3

Step-by-step explanation:

Break down the equation:

(9[tex]z^{2}[/tex]-4)=

9z * 9z= 81z

You can't subtract 4, so I left it at (81z- 4)

(7[tex]z^{2}[/tex]+ 1)=

7z * 7z= 49z

You can't add 1, so I left it at (49z+ 1)

Then Solve:

You can add 81z and 49z.

81z+49z= 130z

-4+1= -3

Hope this helps!

HELP PLS!!! WILL GIVE BRAINLIEST!!!

Answers

Answer:

The correct answer is B.2 plz mark as brainliest.

Step-by-step explanation:

is 144 a natural number yes or no​

Answers

Answer:

is 144 a natural number yes or no

Yes

I'm not sure

Spins a fair spinner numbered 1 - 5 and flips a fair coin. What is the probability of obtaining a factor of 15 and a tail?

Answers

Answer:

3/10

Step-by-step explanation:

Probability is the ratio of the number of possible outcome to the number of total outcome.

Given that the fair spinner is numbered 1 - 5, the factors of 15 there are 1, 3 and 5 hence the probability of obtaining a factor of 15

= 3/5

The probability of obtaining a tail from the flip of a coin

= 1/2

Hence the probability of obtaining a factor of 15 and a tail

= 3/5 * 1/2

= 3/10

HELP ASAP ILL GIVE U BRAINLEIST

Answers

Answer:

"A number cubed is a power where the base is the variable n and the exponent is 3"

"Increased by 4 means add 4"

"The expression is written as n^3 + 4"

I agree with the person above me she or he should get brainlyest

On a map with a scale of 1 cm : 2.5 km two towns are 5 cm apart. What is this distance in real life?

Answers

Answer:

12.5 km

Step-by-step explanation:

1cm = 2.5km

2cm = 5km

3cm = 7.5km

4cm = 10km

5cm = 12 5km

The expression logx (ab) is equivalent to

Answers

Answer:

logx a + logx b

Step-by-step explanation:

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