Selected Olympiad Questions!

ZHero · 2010-06-23 06:24:45

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.

mathsyperson · 2010-06-23 07:43:23

ZHero · 2010-06-23 18:36:13

mathsyperson: WOW!! Catchy!

2.

mathsyperson · 2010-06-23 21:08:36

phrontister · 2010-06-23 22:00:12

1.

ZHero wrote:

mathsyperson: WOW!! Catchy!

I could only come up with

.

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

ZHero · 2010-06-24 17:03:13

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!!

ZHero · 2010-06-24 17:04:40

mathsyperson: you saw it through again!

phrontister · 2010-06-25 00:23:27

ZHero wrote:

Comps must be your Favorite Toys!!

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

ZHero · 2010-06-25 07:35:22

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

3. Which is greater?

ZHero · 2010-06-26 00:08:03

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'?

phrontister · 2010-06-26 12:33:32

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.

ZHero · 2010-06-26 17:46:49

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?

phrontister · 2010-06-27 01:20:51

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.

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

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

ZHero · 2010-06-27 02:33:48

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

phrontister · 2010-06-27 02:43:33

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)

phrontister · 2010-06-30 13:53:43

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.

ZHero · 2010-07-02 03:25:03

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!

phrontister · 2010-07-02 20:41:59

Hi ZHero,

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

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)

ZHero · 2010-07-02 21:05:42

phrontister wrote:

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

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!

The method they used was....

phrontister · 2010-07-03 23:57:58

Hi, ZHero! Thanks.

Finding n's maximum value is neat! 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!

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

ZHero · 2010-07-04 03:00:14

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

nerissa22 · 2010-11-08 21:45:46

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

bobbym · 2010-11-08 21:55:40

Hi nerissa22;

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

manpreet singh saluja · 2012-04-24 17:48:08

ZHero wrote:

Seems like one day you'll be able to work upon them blindfolded!
3. Which is greater?

please give a solution to this problem.

bobbym · 2012-04-24 18:31:37

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!

Math Is Fun Forum

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

Selected Olympiad Questions!

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

Re: Selected Olympiad Questions!

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

Re: Selected Olympiad Questions!

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

Re: Selected Olympiad Questions!

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

Re: Selected Olympiad Questions!

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

Re: Selected Olympiad Questions!

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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