I'm really scared

Find the sum of all the integers a , b , c a,b,c with 1 < a < b < c 1<a<b<c such that ( a 1 ) ( b 1 ) ( c 1 ) (a-1)(b-1)(c-1) divides a b c 1 abc-1 .


Source : IMO.


The answer is 37.

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1 solution

Ayush Kumar
Aug 23, 2018

Put a-1=x,b-1=y,c-1=z. Now to find 1<= x<y<z for which 1/x +1/y +1/z +1/xy + 1/yz+1/zx=p is also a integer. we observed that x>=1,y>=2 and z>=3 P<=1+1/2+1/3+1/2+1/3+1/6=17/6<3 Thus p=1 or 2. Taking two cases each of p=1 and p=2 we get triples (a,b,c)={3,5,15},{2,4,8} So total sum of all values =3+5+15+2+4+8=37. Hope u enjoyed this problem of IMO1992.

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