Math problems thread

[Kent]

All solution must be shown, explain or suggest ( I will post more problems if i feel like it)

 

1) suppost you are a detective, and you are in a island with three inhabitant. The "Knight" that always tell the truth; the "knave" that always lies , and the "normal" that something tell the truth, and sometimes lie. suppost you already know that one of the three is the murder and he is a knight.

 

Person A said " I am innocent". B said "what A said is true" and C said "B is not normal".

 

Who is the knight?

 

2) Suppose you are one of the prisoners in a prison. The director of the prison tell all of you that starting of the next day at 6 p.m, he would randomly choose one of you to a room where there is a light switch; each hour afterward, he would ramdonly choose( with replacement) a prisoner from the prison population to go into the room. The ligth in the room can be on or off, but you have the option to turn off or on the light as you wish. This procedure will continue when one of you prisoner( any prisoner in the initial population) declear with probability 1 certainty that all prisoners had got into the room at least ones. you and your inmates must devise a plan before the percedure begin, becuase all informations afterward are will be terminated.

 

What is the plan?

 

3)

 

( 100A+10B+C)( A+B+C)=2005

 

What is A, B, C?

 

4)

 

What is the sum to this sequence

 

1,-2, 3,-4, 5, -6.............

 

What is the average of the first 200 terms of the sequence?

 

5)

 

a) How many integer are there from 10 to 99?

:o How many odd interger are there from 10 to 99 have distinct intergers?

 

6)

 

How many sundays and mondays are there in a year?

 

7) How many ways can 982 people sit around a table?

 

8 ) How many zeros end in this number

 

(2^300)( 5^600)(4^400)

 

this number

 

( 200!)

 

This number

 

(200!!!)

 

9) Given a cirle of radus 2. There many line segment of length two that are tengent to the circle. find the area that consisting of all such line segments.

 

10)

 

Given equations

 

(xy/( x+y))=a

 

(xz/(x+z))=b

 

(yz/(y+z))=c

 

Is x rational ?

 

11) Suppose two dies are throw. What is the probability that the sum of the two dies after they hit the ground is at least 8?

[maddog]

1. Knight is A. B or C can be either Knave or normal. Does not matter. Were Knight

anything else B & C would not work.

 

2. This paragraph is to unclear with all the mispellings to be understood.

 

3. Since a polynomial as a product of two factors with three terms. Best to find what is

common. Thus divide 2005 by 5 and get 401. So what

100A + 10B + C = 401 and A + B + C = 5 ?

If C = 1 then

100A + 10B = 400, A + B = 4

A = 4 - B => 100(4 - :o + 10B = 400

400 - 100B + 10B = 400

This is only true if B = 0

Thus A = 4, B = 0, C = 1

 

4. The sequence is n * (-1)^n where n is any integer. This can be rewritten as the

difference of two sums

sum(n=1,infinity) (2n+1) - sum(n=1,infinity)(2n) + 1

To infinity this sum diverges. To 200 the sum is -100. The average is -1/2.

 

5. What is sqrt(i) where i = sqrt(-1) ? :o

 

Maddog

[pgrmdave]

9) Given a cirle of radus 2. There many line segment of length two that are tengent to the circle. find the area that consisting of all such line segments.

 

The line segments would form a solid ring around the outside of the circle. Each single segment would form a triangle, with a base of 2, and a height of two, and unknown sides. If you split it into two triangles, you can get a right triangle, with sides 1, 2, and sqrt5. So, the outer circle would have an area of 5*pi. If you want only the ring, it would be pi (5pi - 4pi).

[Kent]

1. Knight is A. B or C can be either Knave or normal. Does not matter. Were Knight

anything else B & C would not work.

 

Hmm... I would assume person A would be the first to be elimiated from the possibilities, since we know two facts: 1) knight always tell the truth. 2) The knight is the one that commited the crime.

 

 

2. This paragraph is to unclear with all the mispellings to be understood.

 

Well....

 

 

 

4. The sequence is n * (-1)^n where n is any integer. This can be rewritten as the

difference of two sums

sum(n=1,infinity) (2n+1) - sum(n=1,infinity)(2n) + 1

To infinity this sum diverges. To 200 the sum is -100. The average is -1/2.

 

 

You can conceive it by grouping them in pairs with out equations at all.

 

 

What is sqrt(i) where i = sqrt(-1) ?

 

go to chapter on permutations.

[Kent]

The line segments would form a solid ring around the outside of the circle. Each single segment would form a triangle, with a base of 2, and a height of two, and unknown sides. If you split it into two triangles, you can get a right triangle, with sides 1, 2, and sqrt5. So, the outer circle would have an area of 5*pi. If you want only the ring, it would be pi (5pi - 4pi).

 

 

 

pi (5pi - 4pi)

 

 

?

 

Perhaps you mean just Pi, yes?

[pgrmdave]

Yes, the area is pi. I was simply showing how I arrived at that conclusion: area of larger circle (5pi) minus the area of the smaller circle(4pi).

[Kent]

This set of problems should be more easier then my original sets of probelms. i need the practice anyways...

 

explain your solutions

 

11 )

 

suppose you have this pattern

 

abcdefg

bcdefg

cdefg

defg

efg

fg

g

 

moving from the upper left hand corrner of the pattern, starting with "a". You can more either right or down. How many ways can you spell abcdefg?

 

12) what is the value of this

 

if x rise to the power of x rise to the power x rise to the power of x ..............etc

 

The value is equal to 2. What is the value of x?

 

13)

 

((x^2)-9x+19)^( (2x^3)-(x^2)-10x)

 

^ the total value equal to 1

 

How many solutions do the equation have?

 

14)

 

Find the least positive integer that is divisible by the first 15 natural numbers

 

15) How many handshake is it posible for "n" number of people in a room?

 

16) How many "breaks" do you need to divid up a piece of chocolate(spell?) in to "n" number of pieces?

 

17) every person alive had shake hands with a certain number of people. prove to me that the people who skake hands an odd number of times must yield an even number?

 

18)

 

prove that (2222^5555) + (5555^2222) is divisible by 7

 

 

19) If the coefficient of

 

a(x^2) + bx + c =0

 

are odd. Can the root be rational?

 

20) proof to me that if the sum of the digits of a number is divisible by 9, then that number must be divisible by 9

[Kent]

19) If the coefficient of

 

a(x^2) + bx + c =0

 

are odd. Can the root be rational?

 

This a old math olymiad problem from the ussr . do not know which year

 

You can see instantly that this is a quadratic equation.

 

The only way for the solution x to be rational, is for the discriminant to be a perfact square, yes.

 

Since we know a, b, c is odd, let me say

 

a=2k+1

 

b=2q+1

 

c=2r+1

 

Substitude a, b and c into the discriminate (b^2)- 4ac

 

We would yield

 

8( some number ) +5 ---------Important.

 

Since 8( some number) +5 must be a perfect squre for x to be rational, and that 8( some number)+8 is odd. we must find a general form of a odd number rise to the power of 2 to see if it match 8(some number)+5.

 

all interger can be express in mod4, yes?

 

4k, 4k+1, 4k+2, 4k+3.

 

since we want to limited ourself to only odd numbers.

 

We must have 4k+1 and 4k+3

 

or in other word. 4k plus or minus 1.

 

( 4k plus or minus 1 ) ^2

 

produce an answer with the form 8( some number ) +1 --------important

 

which does not match the form 8(some numer) +5

 

the discrimiate can not be a perfect square, and thus x can not be rational.

20) proof to me that if the sum of the digits of a number is divisible by 9, then that number must be divisible by 9

 

That is easy my dear.

 

Why? suppose there is a number n with digits abcde:

 

n=10,000a + 1,000b+ 100c+10d+ e

 

Then

 

n=( 9999+1)a + ( 999+1)b +( 99+1)c+ (9+1)d+ e

 

n=( 9999a + 999b +99c +9d) +( a+ b +c+d+e )

 

( a+ b +c+d+e ) is divisible by 9.

 

n is dividsible by 9

[Kent]

11 )

 

suppose you have this pattern

 

abcdefg

bcdefg

cdefg

defg

efg

fg

g

 

moving from the upper left hand corrner of the pattern, starting with "a". You can more either right or down. How many ways can you spell abcdefg?

 

the answer must be 2^ 6

 

Why? every step, there are two possibility. You can either move rither right or down

 

There are 7 letters in abcdefg. You can continue until the last letter( g) where there is zero possibility. That mean 2*2*2*2*2*2 =2^6

 

A simple extention of the multiplication principle in pre-calculus. if you can do one thing is K ways , and another thing in P ways, then you can do both things in kp ways. the keyword for the day? Ways

 

15) How many handshake is it posible for "n" number of people in a room?

 

Hmm..... there are one person in a room!

 

Another person come in to the room. I skate his hand. That is one handskake, for two person(n=2)

 

Another person come in to the room, there are two hand skate. ( N=3)

 

Another person come into the room, there are three hand skate ( N=4)

 

In other word, for any n person in a room, there are 1+2+3+.......+(n-1) hand skate possible

 

it is a familiar series S=k(k+1)/2

 

substitude k for n-1

 

the sum is (n-1)n/2 which is the least number of handskate for n number of people.

[Tim_Lou]

12) what is the value of this

If x rise to the power of x rise to the power x rise to the power of x ..............etc

The value is equal to 2. What is the value of x?

 

I assume you mean: x^(x^(x^(x^(......x^x)))).. thats the only way that the problem would converge...

let A= the whole thing,

A=2 (equation1)

Alnx=ln2 (equation2)

equation2/equation1

lnx=(ln2)/2

x= sqrt (2)

[Tim_Lou]

13)

((x^2)-9x+19)^( (2x^3)-(x^2)-10x)

^ the total value equal to 1

How many solutions do the equation have?

 

let A be ((x^2)-9x+19)^( (2x^3)-(x^2)-10x)

so, A^A = 1 (is that what you mean?)

the only possibility is A=1

((x^2)-9x+19)^( (2x^3)-(x^2)-10x)=1

there are 2 possibilies that this equation would work,

either ((x^2)-9x+19)=1

or (2x^3)-(x^2)-10x=0 (*while ((x^2)-9x+19) cant be 0)

solution set: {6,3,0,2.5,-2}

there are 5 solutions