Points and are given on side of convex quadrilateral (with closer than to ). It is known that and . Prove that .

M. Smurov

Suppose that m,n are in natural numbers. find all m,n that (m^n-n)^m=n!+m

Let a1 be a member of the integers and an+1=an^2-an-1. Show that (an+1,2n+1)=1

Find all pairs of positive integers such that

Find all functions f:R+--R+ such that for all a,b>0, f(af(b)+a)(f(bf(a))+a)=1

Let be nonnegative real numbers such that . Find min

Let and Prove that
Where

a^2+b^2+c^2+x^2+y^2=1. Find the maximum value of the expression (ax+by)^2+(bx+cy)^2

Three distinct circles , , cut three common chords concurrent at . Consider two distinct circles , which are internally tangent to all . Determine, with proof, which of the following two statements is true.

(1) is the insimilicenter of and
(2) is the exsimilicenter of and .

Three turtles are crawling along a straight road heading in the same
direction. "Two other turtles are behind me," says the first turtle. "One turtle is
behind me and one other is ahead," says the second. "Two turtles are ahead of me
and one other is behind," says the third turtle. How can this be possible?