第八届罗马尼亚大师杯数学竞赛

• 作 者: 赵斌
• 出 处: 中等数学
• 发表日期: 2016年07期
• 关键词: 三线共点 对称点 正整数 数学竞赛 圆半径 平方数 韦达 三角形区域 四点共圆 伯努利不等式

1.在△ABC中,点D在线段BC上,且D与B、C均不重合,△ABD的外接圆与线段AC的另一个交点为E,△ACD的外接圆与线段AB的另一个交点为F.记A′为点A关于直线BC的对称点,直线A′C与DE交于点P,且直线A′B与DF交于点Q.证明:直线AD、BP、CQ三线共点或互相平行2.给定正整数m,且n≥m,在一个m×2n的方格表中最多能放入多少块多米诺骨牌

1. In △ ABC, the point D is on the line segment BC, and D and B, C are not coincident, △ ABD circumcircle and line segment AC of the other intersection is E, △ ACD circumcircle and line segment AB the other The intersection point is F. Let A ’be the symmetry point of point A with respect to line BC, the line A’C intersects DE with point P, and the line A’B with DF intersect with point Q. Proof: The three lines of AD, BP, CQ Point or parallel to each other 2. Given a positive integer m, and n ≥ m, how many pieces of domino can be placed in an m × 2n box