Difference between revisions of "1967 IMO Problems/Problem 3"
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==Solution== | ==Solution== | ||
− | We have that <math>c_1c_2c_3...c_n=n!(n+1)</math> | + | We have that <math>c_1c_2c_3...c_n=n!(n+1)!</math> |
and we have that <math>c_a-c_b=a^2-b^2+a-b=(a-b)(a+b+1)</math> | and we have that <math>c_a-c_b=a^2-b^2+a-b=(a-b)(a+b+1)</math> |
Revision as of 23:07, 12 December 2022
Problem
Let be natural numbers such that
is a prime greater than
Let
Prove that the product
is divisible by the product
.
Solution
We have that
and we have that
So we have that We have to show that:
is an integer
But is an integer and
is an integer because
but does not divide neither
nor
because
is prime and it is greater than
(given in the hypotesis) and
.
The above solution was posted and copyrighted by Simo_the_Wolf. The original thread can be found here: [1]
See Also
1967 IMO (Problems) • Resources | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
All IMO Problems and Solutions |