Search found 3 matches

by Durjoy Sarkar
Tue Mar 06, 2018 1:02 pm
Forum: Higher Secondary Level
Topic: USAMO #1
Replies: 1
Views: 3696

USAMO #1

Prove that there are infinitely many distinct pairs $(a, b)$ of relatively prime integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$.
by Durjoy Sarkar
Mon Mar 05, 2018 10:06 pm
Forum: National Math Olympiad (BdMO)
Topic: BDMO 2017 National round Secondary 5
Replies: 15
Views: 5801

Re: BDMO 2017 National round Secondary 5

radius of two big circle are same. let X be the point where little circle is tangent.
it is well known the center of little circle, tangent point are lies on $OB$.
$MO=OX$
$BX= BO+OX=BO+MO $
by Durjoy Sarkar
Sun Mar 04, 2018 10:37 pm
Forum: National Math Olympiad (BdMO)
Topic: BDMO 2017 National round Secondary 5
Replies: 15
Views: 5801

Re: BDMO 2017 National round Secondary 5

Ooh I got a cool solution to this. So, $OM+OB = AB$ and $MB=\dfrac{AB}{2}$. Also by pythagorean theorem, $MB^2+OM^2=OB^2$. Substituting, $\dfrac{AB^2}{4} + (AB-OB)^2 = OB^2$ Or, $\dfrac{AB^2}{4} + AB^2 - 2\times AB \times OB + OB^2 = OB^2$ Or, $\dfrac{AB^2}{4} + AB^2 - 2\times AB \times OB = 0$ Or,...