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#1 2010-06-23 06:24:45

ZHero
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Registered: 2008-06-08
Posts: 1,889

Selected Olympiad Questions!

1. A 50-digit natural number is completely divisible by 13. If all the digits, except the 26th digit, are 1 then find the 26th digit.


If two or more thoughts intersect with each other, then there has to be a point.

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#2 2010-06-23 07:43:23

mathsyperson
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Re: Selected Olympiad Questions!


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

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#3 2010-06-23 18:36:13

ZHero
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Re: Selected Olympiad Questions!

mathsyperson: WOW!! Catchy!

2.


If two or more thoughts intersect with each other, then there has to be a point.

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#4 2010-06-23 21:08:36

mathsyperson
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Re: Selected Olympiad Questions!


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

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#5 2010-06-23 22:00:12

phrontister
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From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 3,848

Re: Selected Olympiad Questions!

1.

ZHero wrote:

mathsyperson: WOW!! Catchy!

up

I could only come up with

.

Last edited by phrontister (2010-06-24 15:12:19)


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#6 2010-06-24 17:03:13

ZHero
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Posts: 1,889

Re: Selected Olympiad Questions!

phrontister: UNIQUE! It took me a while understanding why you divided the power of 10 by a 9 but got it now! Comps must be your Favorite Toys!! wink


If two or more thoughts intersect with each other, then there has to be a point.

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#7 2010-06-24 17:04:40

ZHero
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Posts: 1,889

Re: Selected Olympiad Questions!

mathsyperson: you saw it through again!


If two or more thoughts intersect with each other, then there has to be a point.

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#8 2010-06-25 00:23:27

phrontister
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From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 3,848

Re: Selected Olympiad Questions!

ZHero wrote:

Comps must be your Favorite Toys!! wink

Yes...my scientific calculator and my pocket BASIC computer. I know where the buttons are...still trying to suss out what they do! smile


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#9 2010-06-25 07:35:22

ZHero
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Re: Selected Olympiad Questions!

Seems like one day you'll be able to work upon them blindfolded! smile

3. Which is greater?


If two or more thoughts intersect with each other, then there has to be a point.

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#10 2010-06-26 00:08:03

ZHero
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Re: Selected Olympiad Questions!

4. Suppose N is an n-digit positive integer such that
(a) all the n-digits are distinct.
(b) the sum of any three consecutive digits of N is divisible by 5.
How many such N's exist? What is the maximum value of 'n'?


If two or more thoughts intersect with each other, then there has to be a point.

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#11 2010-06-26 12:33:32

phrontister
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From: The Land of Tomorrow
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Re: Selected Olympiad Questions!

Is 4. (a) correct, Zhero? I don't understand it as is, and you might have to explain it to me. I would have thought that "n" should read "N" there. Or maybe, "all of N's digits are distinct".

Also, I'm having trouble understanding Q3.

dizzy


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#12 2010-06-26 17:46:49

ZHero
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Re: Selected Olympiad Questions!

You got it! In 4 (a), the actual statement "all the n-digits are different" actually means that "all digits of N are different".

You might have considered it as "1-digit N, 2-digit N, 7-digit N etc." where all "n"s are different. Instead, consider it as "4-digit N: 1234, 3456, 4567" where "all the 4-digits" of N are distinct.

For 3, its a Power Tower. 2 is raised by 1001 two's and 3 is raised by 1000 three's. You don't start counting from the "base" of the exponent in counting 1001/1000.
Thus, 2² has 2 raised to 2, 1 times.

Is the explanation good?


If two or more thoughts intersect with each other, then there has to be a point.

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#13 2010-06-27 01:20:51

phrontister
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Re: Selected Olympiad Questions!

Thanks for the clarifications, ZHero. I think I get 'em - I think (sic).

4. "How many such N's exist?"

EDIT:

a different output of the same solution. I saw a pattern in the original output and altered the program (BASIC), which now completes in a tiny fraction of the original time. smile

"What is the maximum value of 'n'?"

Last edited by phrontister (2010-06-29 06:39:41)


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#14 2010-06-27 02:33:48

ZHero
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Registered: 2008-06-08
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Re: Selected Olympiad Questions!

Awesome job!
I'll need to verify the answer to first part and for the second part perhaps "n" is standing upon its head?


If two or more thoughts intersect with each other, then there has to be a point.

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#15 2010-06-27 02:43:33

phrontister
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From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 3,848

Re: Selected Olympiad Questions!

ZHero wrote:

...for the second part perhaps "n" is standing upon its head?

Oops! Tricky blighter...doing handstands while I wasn't looking! I've stood him back up on his feet again!

Last edited by phrontister (2010-06-27 02:46:12)


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#16 2010-06-30 13:53:43

phrontister
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From: The Land of Tomorrow
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Re: Selected Olympiad Questions!

Hi ZHero!

Have you had a chance yet to verify my answer to the first part of Q4? Just curious as to whether or not I got it right. Ta.


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#17 2010-07-02 03:25:03

ZHero
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Re: Selected Olympiad Questions!

Hi phrontister!
Sorry for the delay! I did not forget bout it but just was a little bit messed up on somethings. I gave the problem to two of my workmates and they came up with

as the total number of numbers possible. The latter one had missed on a few numbers!!
Your solution is flawless!
You Rock!


If two or more thoughts intersect with each other, then there has to be a point.

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#18 2010-07-02 20:41:59

phrontister
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From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 3,848

Re: Selected Olympiad Questions!

Hi ZHero,

Thanks for that...I'd been hoping I'd got it right. smile

Do you know how your workmates came up with their results? Is there a good logical method I missed?

Last edited by phrontister (2010-07-02 20:44:42)


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#19 2010-07-02 21:05:42

ZHero
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Registered: 2008-06-08
Posts: 1,889

Re: Selected Olympiad Questions!

phrontister wrote:

I'd been hoping I'd got it right. smile

By all means! When I asked my colleagues this questions and found a mismatch, I told them that the correct answer has to be

for the reasons you can understand! wink

The method they used was....


If two or more thoughts intersect with each other, then there has to be a point.

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#20 2010-07-03 23:57:58

phrontister
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From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 3,848

Re: Selected Olympiad Questions!

Hi, ZHero! Thanks.

Finding n's maximum value is neat! smile Good logic...I'd never have thought of that. My method was to record the length of the highest-value number by comparing the results during the calcs.

Re finding all Ns...from your info I can't see how the initial 3/6 and 4/5 numbers are established before going cycling with them, but unless there's an astounding logical solution I guess I don't really have to know.

Nice problem, btw! smile

Last edited by phrontister (2010-07-03 23:58:28)


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#21 2010-07-04 03:00:14

ZHero
Real Member
Registered: 2008-06-08
Posts: 1,889

Re: Selected Olympiad Questions!

Hi phrontister!
There's no logical method known to me to find those 3, 4, 5 and 6 digit Ns. sad
Your computer program is certainly better and faster too!


If two or more thoughts intersect with each other, then there has to be a point.

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#22 2010-11-08 21:45:46

nerissa22
Member
Registered: 2010-11-08
Posts: 2

Re: Selected Olympiad Questions!

i need a math tutor..please help me...

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#23 2010-11-08 21:55:40

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,676

Re: Selected Olympiad Questions!

Hi nerissa22;

Welcome to the forum. Please post your math problems in the Help me section. You will get all the help you want.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#24 2012-04-24 17:48:08

manpreet singh saluja
Member
Registered: 2012-04-23
Posts: 1

Re: Selected Olympiad Questions!

ZHero wrote:

Seems like one day you'll be able to work upon them blindfolded! smile

3. Which is greater?

please give a solution to this problem.

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#25 2012-04-24 18:31:37

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,676

Re: Selected Olympiad Questions!

Hi manpreet singh saluja;

So call 3^3 = a and 2^2^2 = b. Now,

call 3^a = a1 and 2^b = b1, now we can continue

Call 3^a1 = a2 and 2^b1 = b2. We can continue recursively like this until we work our way down the tower. So

Welcome to the forum!


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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