Difference between revisions of "1998 IMO Problems/Problem 4"
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Determine all pairs <math>(a, b)</math> of positive integers such that <math>ab^{2} + b + 7</math> divides | Determine all pairs <math>(a, b)</math> of positive integers such that <math>ab^{2} + b + 7</math> divides | ||
<math>a^{2}b + a + b</math>. | <math>a^{2}b + a + b</math>. | ||
| − | + | ===Video Solution(In Chinese)=== | |
| + | https://youtu.be/TAfXdhndY5M | ||
===Solution=== | ===Solution=== | ||
We use the division algorithm to obtain <math>ab^2+b+7 \mid 7a-b^2</math> | We use the division algorithm to obtain <math>ab^2+b+7 \mid 7a-b^2</math> | ||
| Line 20: | Line 21: | ||
Written by dabab_kebab | Written by dabab_kebab | ||
| − | == | + | ==Video Solution by Flammable Maths== |
| − | + | https://www.youtube.com/watch?v=MjyJzQeX6ec | |
{{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 | ||
