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Zielflug [23.3K]
3 years ago
6

Prove that given any 17 integers, there exist nine of them whose sum is divisible by 9.

Mathematics
2 answers:
vaieri [72.5K]3 years ago
7 0

This is a simple consequence of the Zero-sum problem: https://en.wikipedia.org/wiki/Zero-sum_problem

yaroslaw [1]3 years ago
7 0

Answer:

Step-by-step explanation:

let x,x+1,x+2,x+3,x+4,x+5,x+6,x+7,x+8

be 9 integers

x+x+1+x+2+x+3+x+4+x+5+x+6+x+7+x+8=9x+36=9(x+4)

which is divisible by 9

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Given: PRST is a square
xxTIMURxx [149]

Answer:

(1-\sqrt{2})a^2

Step-by-step explanation:

Consider irght triangle PRS. By the Pythagorean theorem,

PS^2=PR^2+RS^2\\ \\PS^2=a^2+a^2\\ \\PS^2=2a^2\\ \\PS=\sqrt{2}a

Thus,

MS=PS-PM=\sqrt{2}a-a=(\sqrt{2}-1)a

Consider isosceles triangle MSC. In this triangle

MS=MC=(\sqrt{2}-1)a.

The area of this triangle is

A_{MSC}=\dfrac{1}{2}MS\cdot MC=\dfrac{1}{2}\cdot (\sqrt{2}-1)a\cdot (\sqrt{2}-1)a=\dfrac{(\sqrt{2}-1)^2a^2}{2}=\dfrac{(3-2\sqrt{2})a^2}{2}

Consider right triangle PTS. The area of this triangle is

A_{PTS}=\dfrac{1}{2}PT\cdot TS=\dfrac{1}{2}a\cdot a=\dfrac{a^2}{2}

The area of the quadrilateral PMCT is the difference in area of triangles PTS and MSC:

A_{PMCT}=\dfrac{(3-2\sqrt{2})a^2}{2}-\dfrac{a^2}{2}=\dfrac{(2-2\sqrt{2})a^2}{2}=(1-\sqrt{2})a^2

5 0
3 years ago
PLEASE HELP! IF YOU DON'T KNOW DON'T ANSWER! 60 POINTS!!
postnew [5]

Answer:

an = 1/2 (4)^ (n-1)

a6 = 512

Step-by-step explanation:

The formula for a geometric sequence is

an = a1 (r)^(n-1)

where an is the term of the sequence

a1 is the initial term of the sequence

r is the ratio

and n is the term number

We know a1 = 1/2 and r =4

I will assume that x=6 means we want to know the 6th term

an = 1/2 (4)^ (n-1)

We want to find the 6th term

a6 = 1/2 * 4^(6-1)

a6 = 1/2 * 4^5

a6 = 512

3 0
3 years ago
Solve using substitution<br>x-2y =2<br>3x+4y=3​
lorasvet [3.4K]

Answer:

7/5, y = -3/10

Step-by-step explanation:

6 0
2 years ago
Read 2 more answers
Can someone help me with this please?
AVprozaik [17]

Answer:

y=70°

z= 60°

w= 70°

Step-by-step explanation:

Since the two triangles are similar, we know that all of the corresponding angles must be the same.  On the triangle on the left, we can see that the angle formed between sides AB and AC is 60°.  This means that the same angle is formed at Z, which is in the same location on the triangle to the right.  From there, we can find the missing angle, W, from the difference of 180°.

180°-50°-60°=70°.

The same way we found Z, we can also find Y, since it has to be the same as W.

8 0
2 years ago
Twice the difference between a number and 4 is twenty. Find the number.
kogti [31]

Answer:

D 14

Step-by-step explanation:

Let x be the number

2(x-4) = 20

Divide each side by 2

2/2(x-4) = 20/2

x-4 = 10

Add 4 to each side

x-4+4 = 10+4

x = 14

7 0
3 years ago
Read 2 more answers
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