MathDB

6

Part of 2010 Hanoi Open Mathematics Competitions

Problems(2)

greatest integer <= (2 +\sqrt3)^5 (HOMC 2010 J Q6)

Source:

7/25/2019
Find the greatest integer less than (2+3)5(2 +\sqrt3)^5 .
(A): 721721 (B): 722722 (C): 723723 (D): 724724 (E) None of the above.
floor functionalgebra
2(abc+bcd+cda+dab)/(p^2+q^2+r^2+s^2) \in Z (HOMC 2010 Q6)

Source:

7/25/2019
Let a,ba,b be the roots of the equation x2−px+q=0x^2-px+q = 0 and let c,dc, d be the roots of the equation x2−rx+s=0x^2 - rx + s = 0, where p,q,r,sp, q, r,s are some positive real numbers. Suppose that M=2(abc+bcd+cda+dab)p2+q2+r2+s2M =\frac{2(abc + bcd + cda + dab)}{p^2 + q^2 + r^2 + s^2} is an integer. Determine a,b,c,da, b, c, d.
number theoryquadraticsInteger