MathDB

1

Part of 1987 China Team Selection Test

Problems(2)

China TST 1987 smallest integer from five sets

Source: China TST 1987, problem 1

5/16/2005
a.) For all positive integer kk find the smallest positive integer f(k)f(k) such that 55 sets s1,s2,…,s5s_1,s_2, \ldots , s_5 exist satisfying: i. each has kk elements; ii. sis_i and si+1s_{i+1} are disjoint for i=1,2,...,5i=1,2,...,5 (s6=s1s_6=s_1) iii. the union of the 55 sets has exactly f(k)f(k) elements.
b.) Generalisation: Consider n≥3n \geq 3 sets instead of 55.
combinatoricsNumber theoretic functionsSetsChinaTeam Selection Test
construct a rectangle A inside it with maximum area

Source: China TST 1987, problem 4

5/16/2005
Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle AA inside it with maximum area (over all posible rectangles). Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure. Find the smallest lamda such that it works for all convex figures.
geometryrectangleratioanalytic geometrysymmetryconicshyperbola