perplexus.info :: Numbers : Algebra busters

Algebra busters (Posted on 2020-04-22)

Find the number of solutions to the system of equations:

k₁+k₂+k₃+...+k_n=5n-4
1/k₁+1/k₂+1/k₃+...+1/k_n=1

where k₁,k₂,k₃,...,k_n and n are positive integers.

No Solution Yet Submitted by Danish Ahmed Khan

Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

Idea (Hasn't taken me anywhere much, yet)

| Comment 3 of 6 |

Let's try a kind of generator function approach. Define:

h(x) = x^(k_1-1) + ... + x^(k_n-1)

then

h'(1) = 5n - 4 - n = 4(n-1)

Integral from 0 to 1 of dx h(x) = 1

h passes through (0,0) and (1,n)

Posted by FrankM on 2020-05-05 21:52:50

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