Ukrainian Maths Olympiad 2005 | Maths Olympiad Question #olympiad #imo #algebra #ukraine

Ukrainian math Olympiad question of 2005
#olympiad #imo #algebra #premath #maths #mindyourdecision #ukraine

Find all m and n for all positive integer to st. this equation.

[anisderouich]

very very very very very good
thnx so much 🥰❤❤

[shrutikumari]

Evaluative, good question and explanation 👏

[prabhudasmandal]

I have given a thought over this problem for logical soluution .RHS is 2006 means 2005+1. Now LHS is√m +2005/√n i.e √m must be 1 and 2005/√n =2005 or vice versa to make 2006. Now, to make 2005/√n a +ve integer value of √n must be 2005 or its multiples i.e 1, 2005, 5, , 401.putting these values of √n, it is easy to get value of √m or m, n.

[zplusacademy]

Extremely beautiful effort and execution

[peludogameplaysmachopeludo]

m + 2005/n = 2006.

mn + 2005 = 2006n.

2005 = n(2006 - m).

5 . 401 = n(2006 - m)

401 is prime, therefore only possible values for n and m, respectively, are:

5², 1605².
1, 1
401², 2001²
2005², 2005²

[holyshit]

First guess m=n=1
but there are more solutions
If you want a try look for divisors of 2005

[ivankaznacheyeu]

Author has lost half of the solution, where he proves that there are no other solutions.

[Alemdar-]

M=1 n=1 i write don't watch before

[GrigSV]

Сначала разложим 2005 на 5 и 401 плюс 1, т.к. должно получится целое число 1, 5, 401) - это SQL(N) => N это (1, 25, 160801),
тогда остальное просто.
Очень простая задача. Странно, что она на олимпиаде. Слишком просто.

Very, Very simple.