1.The lengths of BC, CA and AB of triangle ABC are a, b and c respectively, with b < c. D is a point on BC such that AD bisects angle A.
(i) Find the necessary and sufficient condition for ensuring there are
points E, F on segments AB, AC, other than the vertices, which satisfy BE = CF and . (Express the condition in terms of angles A, B and C);
(ii) Under this condition, express the length of BE in terms of a, b and c.
2. The polynomial sequence is defined as:
Let be the sum of the absolute values of the coefficients of . For any positive integer n, find the non-negative integer so that just divides .
3. 18 football teams plays a tournament. In each round, the teams are divided into 9 groups and each group plays a game. They play a total of 17 rounds so that each team can play a game with every other team. Find the maximum possible value of n, so that after n rounds, there always exists 4 teams among which only one game is played.
4. are any 4 distinct points on a plane , find the minimum value of the following:.
4.对平面上任意4个不同的点 ,求 的最小值。
5. A rational point is a point with rational numbers as x and y coordinates.
Prove that all rational points on the plane can be divided into 3 disjoint sets satisfying:
(i) In any circle with a rational point as the center there are points from every one of the 3 sets;
(ii) On any straight line there does not exist 3 points each of which
belongs to each of the 3 sets.
6. Given , find the minimum value of constant M, so that for any positive integer n >= 2 and positive real numbers ，satisfying ,we have ,where m is the greatest integer not exceeding cn.
6.给定 ,及实数 ，只要满足 ,总有 , 其中m为不超过cn的最大整数