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1.(希腊)设a_1=1,a_2=3,且对子所有的正整数n,a_(n+2)=(n+3)a_(n+1)-(n+2)a_n 。试求所有使a_n可被11整除的n的值。 2.(保加利亚)考虑下式定义的一个多项式:a_0+a_1x+a_2x~2+…十a_((2)_n)x~(2n)=(x+2x~2+…+nx~2)~2。求证: 3.(南斯拉夫)设A_1B_1C_1是不等边锐角△ABC的垂足三角形,A_2、B_2、c_2是内切于△A_1B_1C_1的圆与它的边的切点。求证:△A_2B_2c_2和△ABC的欧拉直线重合。注1.已知三角形的垂足三角形以原三角形高线的足为顶点。注2.已知三角形的欧拉直线由它的垂心(三条高的交点)和它的外接圆心确定。
1. (Greece) Let a_1=1, a_2=3, and all positive integers of the pair, a_(n+2)=(n+3)a_(n+1)-(n+2)a_n. Try to find all the values of n that make a_n divisible by 11. 2. (Bulgaria) Consider a polynomial defined by: a_0+a_1x+a_2x~2+...a_((2)_n)x~(2n)=(x+2x~2+...+nx~2)~ 2. Proof: 3. (Yugoslavia) Let A_1B_1C_1 be the equilateral acute triangle angle △ABC’s foot triangle, A_2, B_2, c_2 is the cut point of the circle △A_1B_1C_1 and its edge. Proof: The Euler linear coincidence of △A_2B_2c_2 and △ABC. Note 1. The foot triangle of the known triangle is the vertex of the foot of the original triangle high line. Note 2. The Euler line of the known triangle is determined by its vertical center (three high points of intersection) and its circumscribed circle.