Contest Center

Due to a prolonged spell of wet weather in the UK, and the demise of New Scientist Enigma, I have recently turned my attention to the Contest Center, which has a lot of interesting maths puzzles.

I concentrated on problems that did not yet have any solvers, and managed to solve a few of them. (hints under plain cover to genuine puzzle enthusiasts only)

Point on Circumcircle (contributed by Fotos Fotiadis) 
 ABC is a triangle whose smallest angle is A. K is a point on the arc BC of the circumcircle. The perpendicular bisectors of AB and AC intersect the line AK at L and M, respectively. The lines BL and CM intersect at T. Prove that BT+CT=AK. 

Greatest Divisor (contributed by Paul Cleary) 
When n=8, the expression ab(aⁿ-bⁿ) is divisible by 30 for all positive integers a and b, and 30 is the greatest such divisor. Find a positive integer n such that the greatest common divisor of ab(aⁿ-bⁿ) for all positive integers a and b is n. 

26 Numbers (Contributed by Paul Cleary) 
There are infinitely many sets of 26 real numbers where their sum is 200 and the sum of their squares is S. The difference between the smallest and largest of the numbers in these sets is 60/13. What is the value of S? 

Sums of Polynomials 
Let p_i(x) = x²+m_ix+n_i for i=1,2,3,4 be four given polynomials with m_i and n_i integers, and m₁ through m₄ not all odd. Show that there is an integer N such that any integer n > N can be expressed as the sum p₁(a₁)+p₂(a₂)+p₃(a₃)+p₄(a₄) for some integers a₁ through a₄. 

Rational Powers (contributing the criterion gcd(a,b,c)=1)
 Let x be a rational number x=p/q with q>1 and gcd(p,q)=1. Let a, b and c be positive integers, with a^x+b^x=c^x and gcd(a,b,c)=1. Prove that a, b and c must be q-th powers, or find a counterexample. 


Repeated Digits  (not a first to solve, but most solutions found to date)
The square of 88 is 7744, where each digit of the square is repeated. Find additional squares (not ending with 0) where every digit is part of a repeated sequence, such as 11000555544. We will list the number of squares found by each solver. The notation +F indicates that the solver also found an infinite family of solutions. 

Update June 2014:
Square Heronian (contributed by Lee Morgenstern) 
A Heronian triangle has integer sides and integer area. Find a Heronian triangles where all 3 sides are squares. [Only one solution is known. Extra credit for anyone who finds a second solution.] 

Update September 2014:
** Square Rearranger 
Find the smallest three distinct whole numbers A, B and C such that you can rearrange the digits of A and B to get C², the digits of A and C to get B², and the digits of B and C to get A². [Leading zeroes are not allowed.]