IMO Number system!

Number Theory Level 4

Find the sum of all natural numbers x x in decimal system such that the product of their digits is = x 2 − 10 x − 22 = x^{2}-10x-22 .

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The answer is 12.

Krishna Ar by Krishna Ar , 21, India

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2 solutions

Dinesh Chavan
Jul 3, 2014

The question is from IMO 1968 1968 . The solution can be found here

3 Helpful 0 Interesting 0 Brilliant 0 Confused

I don't want to look at the solution but I tried this with 12. By using the quadratic formula... and letting k as the product of the digits. where x = (1/2)(10+2 sqrt(47 + k)) finding the possible squares give k = 2, 17, 34, ... (of the form k = m^2 - 47). We prove why 12 is only possible. From the k earlier, 2 only satisfies the conditions giving x = 12. We use the trend of the quadratic function f(x) = x^2 - 10x - 22 and f(x) = product of the digits of x. For each increased number, the quadratic function increases rapidly as x tends to infinity and for the product function, each values increases steadily by different factors (and goes on again after passing a number with digit 0). The intersection between these functions must be of a small value. I wonder if this is enough.

John Ashley Capellan - 6 years, 11 months ago

Yes. How did you know about this @dinesh chavan

Krishna Ar - 6 years, 11 months ago

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In My leasure time, I often solve problems from maths olympiads from AoPS, So I accidently remembered this question :)

Dinesh Chavan - 6 years, 11 months ago

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:) Nice 2 know

Krishna Ar - 6 years, 11 months ago

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@Krishna Ar – Plus, if you google the problem, the site shows up.

Roberto Vázquez - 6 years, 10 months ago

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@Roberto Vázquez – Yes, but isnt that cheating? ;)

Krishna Ar - 6 years, 9 months ago

I don't want to look at the solution below but I tried this with 12. By using the quadratic formula... and letting k as the product of the digits. where x = (1/2)(10+2 sqrt(47 + k)) finding the possible squares give k = 2, 17, 34, ... (of the form k = m^2 - 47). We prove why 12 is only possible. From the k earlier, 2 only satisfies the conditions giving x = 12. We use the trend of the quadratic function f(x) = x^2 - 10x - 22 and f(x) = product of the digits of x. For each increased number, the quadratic function increases rapidly as x tends to infinity and for the product function, each values increases steadily by different factors (and goes on again after passing a number with digit 0). The intersection between these functions must be of a small value. I wonder if this is enough.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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