Difference between revisions of "1998 IMO Problems/Problem 4"
Bobwang001 (talk | contribs) (Ph.D degree, IMO coach,https://www.youtube.com/@math000) |
Robindabank (talk | contribs) (→Solution) |
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− | == | + | ==Video Solution by Flammable Maths== |
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{{IMO box|year=1998|num-b=3|num-a=5}} | {{IMO box|year=1998|num-b=3|num-a=5}} |
Latest revision as of 00:50, 28 August 2024
Determine all pairs of positive integers such that divides .
Video Solution(In Chinese)
Solution
We use the division algorithm to obtain Here is a solution of the original statement, possible when and where is any natural number. This is easily verified.
Otherwise we obtain the inequality (by basic properties of divisiblity): So
Testing for we find that Therefore, , and we can easily check these.
Testing for and applying the division algorithm we find that , having no solutions in natural .
Hence, the only solutions are: for all natural .
Written by dabab_kebab
Video Solution by Flammable Maths
https://www.youtube.com/watch?v=MjyJzQeX6ec
1998 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |