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#301 2005-12-23 18:15:05

Ricky
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Registered: 2005-12-04
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Re: Problems and Solutions

Last edited by Ricky (2005-12-23 18:15:23)


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#302 2005-12-23 20:44:00

Jai Ganesh
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Registered: 2005-06-28
Posts: 45,968

Re: Problems and Solutions

Correct, Ricky!

krassi_holmz, the solution to problem # k + 47 iis:-

1 - (8/9)^n - (5/9)^n + (4/9)^n.

A number is divisible by 10 if it has an even number and a 5 amongst its digits. The probability of no 5 is (8/9)^n. The probability of no even number is (5/9)^n. The probability of no 5 and no even number is (4/9)^n. Hence the result is as given above.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#303 2005-12-23 22:34:54

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: Problems and Solutions

With nothing but intuition, I guess

.


Why did the vector cross the road?
It wanted to be normal.

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#304 2005-12-23 23:59:15

Jai Ganesh
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Re: Problems and Solutions

Guessed it right, mathsyperson smile
Either A throws more heads than B, or A throws more tails than B, but (since A has only one extra coin) not both.
By symmetry, these two mutually exclusive possibilities occur with equal probability.

Therefore the probability that A obtains more heads than B is ½.  It is perhaps surprising that this probability is independent of the number of coins held by the players.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#305 2005-12-24 00:43:28

mathsyperson
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Registered: 2005-06-22
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Re: Problems and Solutions

Yes, I was thinking along the same lines, I just hadn't formed it into an expanation. smile


Why did the vector cross the road?
It wanted to be normal.

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#306 2005-12-25 16:18:33

Jai Ganesh
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Re: Problems and Solutions

Right, Mathsy!

Problem # k + 73

Is it possible that the three sides of a right-angled triangle are in Geomtric Progression? If so, what would the lenght of the hypotenuse be?


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#307 2005-12-26 05:24:44

krassi_holmz
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Registered: 2005-12-02
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Re: Problems and Solutions

Let the sides be

,
and
. Then


Let
. Then



, p>0 so

Then the hypotenuse is

Last edited by krassi_holmz (2005-12-26 05:38:21)


IPBLE:  Increasing Performance By Lowering Expectations.

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#308 2005-12-26 05:57:25

mathsyperson
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Re: Problems and Solutions

The other case is when

, which happens when b<1.

In this case,

, but the hypotenuse is just
.

Last edited by mathsyperson (2005-12-26 07:08:21)


Why did the vector cross the road?
It wanted to be normal.

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#309 2005-12-26 06:46:56

krassi_holmz
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Re: Problems and Solutions


IPBLE:  Increasing Performance By Lowering Expectations.

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#310 2005-12-26 06:53:54

krassi_holmz
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Re: Problems and Solutions

In mathsy's case b = sqrt((sqrt(5)-1)/2)q=0.7861513778...q


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#311 2005-12-26 07:09:10

mathsyperson
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Registered: 2005-06-22
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Re: Problems and Solutions

Whoops, you're right. I was so busy trying to get the LaTeX right that I got the maths wrong.


Why did the vector cross the road?
It wanted to be normal.

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#312 2005-12-26 07:43:56

krassi_holmz
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Registered: 2005-12-02
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Re: Problems and Solutions

#k+38-bertrand's postulate. I think there isn't a simple proof. I wrote something with the chebishev's  theta function.


IPBLE:  Increasing Performance By Lowering Expectations.

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#313 2005-12-26 15:18:08

Jai Ganesh
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Re: Problems and Solutions

Solution to problem # k + 73 is correct! Well done, krassi_holmz!
I'll check the solution to Problem # k + 28 and Problem # k + 38 later and post.

Problem # k + 74

The Greatest Common Divisor (or Highest Common Factor) and Least Common Multiple of two numbers are 3 and 84 respectively. If one of these numbers is 9 greater than the other, what's the smaller number?


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#314 2005-12-26 18:37:47

Jai Ganesh
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Re: Problems and Solutions

krassi_holmz, I am not fully convinced by your proof (solution to Problem # k + 28), although you seem to be on the right track. The reason, I think, is I am unable to follow what you intend conveying. Maybe, you have got the full proof in your mind but had not posted it. I shall wait (for a week) for amendment from you, or proof by someone else before I post the solution.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#315 2005-12-26 18:46:54

krassi_holmz
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Re: Problems and Solutions

Last edited by krassi_holmz (2005-12-26 18:49:07)


IPBLE:  Increasing Performance By Lowering Expectations.

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#316 2005-12-26 19:16:16

Jai Ganesh
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Re: Problems and Solutions

krassi_holmz, you get this smile
job_well_done_award.jpg


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#317 2005-12-27 16:36:38

Jai Ganesh
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Re: Problems and Solutions

Problem # k + 75

X,Y,and Z take 20, 30 and 60 days respectively to complete a job independently.
They set out to complete a job together . However Y leaves after 4 days and Z leaves after another 6 days .
How many more days will it take for X alone to complete the job now?


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#318 2005-12-27 22:40:59

krassi_holmz
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Re: Problems and Solutions


IPBLE:  Increasing Performance By Lowering Expectations.

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#319 2005-12-27 22:42:03

krassi_holmz
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Registered: 2005-12-02
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Re: Problems and Solutions

Yesterday I solved two of the unsolved problems. When i have time, i'll post the solutions.


IPBLE:  Increasing Performance By Lowering Expectations.

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#320 2005-12-27 22:47:58

krassi_holmz
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Registered: 2005-12-02
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Re: Problems and Solutions

k+28
My last proof wasn't correct at all.
In it i proved only that
LCM(1^n,2^n, ... ,n^n)/n^2!
But LCM(1^n,2^n, ...n^n) {!=} n!^n

Then I proved that
(n-k)^(n+k)/n^2!
but this wasn't enough.

I found a beautiful number theory proof with prime numbers, floors and an numberic theory theorem:
Let write n! as

, where p_1 , p_2 ... ,p_k ∈ P(prime numbers) and p_k = max(p ∈ P : p <= n). Then
.
Let write n^2! ia the same way:
.
To proof that n!^n/n^2! it is enough to proof that


...

Now we will use a theorem from the numeric theory:
The biggest power of the prime number p that divides n! is exactly

,
where [x] means Floor(x).
Now we are using it:

Now, at last, we will prove that


We use the following:
If n ∈ N and x ∈ R then
[nx]=[n([x]+{x})]=[n[x]+n{x}]=n[x]+[n{x}] >= n[x];

so
=>
<=>
=>
.

Last edited by krassi_holmz (2005-12-28 00:12:15)


IPBLE:  Increasing Performance By Lowering Expectations.

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#321 2005-12-27 23:35:46

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Problems and Solutions

And for k+47
Not I understood that the product of the numbers must be divisible by 10. When i wrote the my post i though that the SUM must be divisible by 10. My mistake.


IPBLE:  Increasing Performance By Lowering Expectations.

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#322 2005-12-27 23:40:17

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Problems and Solutions

k+48:

?


IPBLE:  Increasing Performance By Lowering Expectations.

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#323 2005-12-27 23:48:36

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Problems and Solutions

here is a picture:


IPBLE:  Increasing Performance By Lowering Expectations.

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#324 2005-12-28 00:17:51

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Problems and Solutions

Triangle MHO is simular to triangle OHB.
Let yr=h. then hr=yr²=x. But
x²=y²+h²
(yr²)²=y²+y²r²
Let f = r². Then
f²=1+f
f²-f-1=0
D=1+4=5
f= (1+sqrt(5))/2
r=x/h=2x/2h=AB/BC=sqrt((1+sqrt(5))/2)

Last edited by krassi_holmz (2005-12-28 00:19:16)


IPBLE:  Increasing Performance By Lowering Expectations.

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#325 2005-12-28 15:53:30

Jai Ganesh
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Registered: 2005-06-28
Posts: 45,968

Re: Problems and Solutions

krassi_holmz, your solution to problem # k + 75 is correct. Well done.
Regarding the solution to problem # k + 48 and the proof, I shall tell you after putting the answers under a microscope. I had a different proof in my mind, similar to yours though.

Problem # k + 76

What is the angle between the two hands of a clock when the time is 02.35 ?


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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