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#26 Re: Help Me ! » Two-blocks number » 2016-05-13 02:18:07
I see.
Maybe this is not the right approach to the problem, so I don't want to mislead you, but, if the final equation lies as in my previous message, we must determine the (possible) boundaries for k, n and therefore a, so as to start testing several values.
Just a few ideas:
Since the denominator is (k+1)^2, (k+1) must be a (double) factor of 10^n + k.
Can we deduce also a relationship between n, k and a, based on the fact that n = log[(c-b)/a]?
#27 Help Me ! » Two-blocks number » 2016-05-12 19:24:08
- chen.aavaz
- Replies: 27
Hi guys,
I need help with this puzzle please:
Find a natural number, which, if appended to the leftmost of another natural number, multiple of the first one, the new number that is formed equals to the square of the sum of the two initial numbers (the first one and its multiple).
The second number cannot be zero.
Just an attempt, but up to a certain point. Then I will leave it to you!
Suppose the numbers are a and b, with b=k*a, k>=1.
Let c be the new number which is formed by the other two.
c=(10^n)*a+b,
again n>=1 (n is the number of digits of a)
We therefore have: c=10^n*a+k*a = a*(10^n+k) = (a+b)^2
So: (a+k*a)^2=a*(10^n+k)-->[a(k+1)]^2=a*(10^n+k)-->a*(k^2+2*k+1)=10^n+k-->a*k^2+2*a*k+a-k=10^n
So a=(10^n+k)/(k^2+2*k+1) and a must be an integer. Is this possible?
#28 Re: Help Me ! » Movement of flea » 2016-05-09 02:20:48
Guys,
Although I have absolutely no clue on how to solve it, I suspect the problem refers to 1-meter "jumps" that the flea does, so maybe during the 1-minute observation by each student, one jump may have "started" (so the flea jumped upwards) but maybe he "landed" at the next student's minute - if you see what I mean. So this 1-meter was not accounted, neither to the previous student's observation, nor to the next one's, because the first one did not see the flea "landing", while the other one did not see him "taking-off" 
I will leave the rest to you
#29 Re: Help Me ! » Movement of flea » 2016-05-06 04:10:33
This is how a friend gave it to me. I think it was originally posted in the Kvant magazine.
#30 Re: Help Me ! » Movement of flea » 2016-05-06 02:19:57
I don't know exactly but I guess it has to do with the series of "jumps" the flea is doing. That is, if he starts a second jump after the completion of the first, but does not "land" within the minute of observation, but in the next one, and then he jumps again... I don't know - just guessing!!
#31 Re: Help Me ! » Movement of flea » 2016-05-05 08:55:46
Let's assume he (or she??) makes jumps of 1m each. On a plane.
On a plane or in 3D?
#32 Re: Help Me ! » Movement of flea » 2016-05-05 08:39:21
In any direction, but we are not really interested in the direction, only the absolute distance he has "traveled".
Hi;
In what directions is he walking? Any direction? How many dimensions, 1,2, 3... is he moving in?
#33 Help Me ! » Movement of flea » 2016-05-05 04:51:15
- chen.aavaz
- Replies: 17
A research team of students in the Physics department of University of South Carolina has been watching the movement of a flea for 60 minutes without interruption.
Each student continuously watched the movement for exactly one minute and during this time, each of them saw the flea move by 1 meter during his 1-minute observation. What is the minimum and maximum distance which the flea could have moved during the 60 minutes of observation? (distance, not displacement)