MathDB

4

Part of 2012 JBMO ShortLists

Problems(3)

Inequality transformed in an equation!

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2/4/2015
Solve the following equation for x,y,z∈Nx , y , z \in \mathbb{N} : (1+xy+z)2+(1+yz+x)2+(1+zx+y)2=274\left (1+ \frac{x}{y+z} \right )^2+\left (1+ \frac{y}{z+x} \right )^2+\left (1+ \frac{z}{x+y} \right )^2=\frac{27}{4}
inequalities
Isogonal conjugates!

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2/4/2015
Let ABCABC be an acute-angled triangle with circumcircle ω\omega , and let OO , HH be the triangle's circumcenter and orthocenter respectively . Let also A′A^{'} be the point where the angle bisector of the angle BACBAC meets ω\omega . If A′H=AHA^{'}H=AH , then find the measure of the angle BACBAC.
geometrycircumcirclerhombusangle bisector
Diophantine equation !

Source:

2/4/2015
Determine all triples (m,n,p)(m , n , p) satisfying : n2p=m2+n2+p+1n^{2p}=m^2+n^2+p+1 where mm and nn are integers and pp is a prime number.
number theory