Subcontests
(5)Area of Pentagon
A given convex pentagon ABCDE has the property that the area of each of five triangles ABC,BCD,CDE,DEA, and EAB is unity (equal to 1). Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property. Approximation of Cube Root of 2
Let R denote a non-negative rational number. Determine a fixed set of integers a,b,c,d,e,f, such that for every choice of R, \left| \frac{aR^2\plus{}bR\plus{}c}{dR^2\plus{}eR\plus{}f}\minus{}\sqrt[3]{2}\right| < \left|R\minus{}\sqrt[3]{2}\right|. Random Selector
A random selector can only select one of the nine integers 1,2,…,9, and it makes these selections with equal probability. Determine the probability that after n selections (n>1), the product of the n numbers selected will be divisible by 10. Ratios of LCM and GCD
The symbols (a,b,…,g) and [a,b,…,g] denote the greatest common divisor and least common multiple, respectively, of the positive integers a,b,…,g. For example, (3,6,18)\equal{}3 and [6,15]\equal{}30. Prove that \frac{[a,b,c]^2}{[a,b][b,c][c,a]}\equal{}\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}.