2016数学全国竞赛班寒假课程讲义

1、初中数学竞赛班讲义:代数式求值问题主讲教师:Update on 2016 年 1 月 20 日目录1常用公式2例题选讲233练习题511常用公式a2 + b2 + c2 + 2(ab + bc + ca) = (a + b + c)21.a2 + b2 + c2 ab bc ca =(a b)2 + (b c)2 + (c a)212.2a3 + b3 + c3 3abc = (a + b + c)(a2 + b2 + c2 ab bc ca)abc + ab + bc + ca + a + b + c + 1 = (a + 1)(b + 1)(c + 1) a2b + b2c + c2a a

2、b2 bc2 ca2 = (a b)(b c)(c a)3.4.5.22例题选讲已知实数 a, b, c 满足1.abc = 1, a + b + c = 4,abca2 3a 1 + b2 3b 1 + c2 3c 1 = 1.求 a2 + b2 + c2 的值.若正数 a, b, c 满足2.()()()222222222222b + c ac + a ba + b c+= 3,2bc2ca2ab求代数式b2 + c2 a2c2 + a2 b2a2 + b2 c2+2bc2ca2ab的值.设 abc = 0, a + b + c = a2 + b2 + c2 = 2.求代数式3.(1 a)

3、2(1 b)2(1 c)2+bccaab的值.4.已知实数 x, y, z, u 满足xyzuy + z + u = z + u + x = u + x + y = x + y + z .求代数式x + y + y + z + z + u + u + xz + uu + xx + yy + z的值.5.已知正实数 x, y, z 满足x3 xyz = 5,y 3 xyz = 2,z3 xyz = 21.求 x + y + z 的值.已知实数 x, y, z 满足6.x4 + y 2 + 4 = 5yz,y 4 + z2 + 4 = 5zx,z4 + x2 + 4 = 5xy.3求 x + y

4、+ z 的值.已知正实数 x, y, z 满足7.x2 + xy + y 2 = 9,y 2 + yz + z2 = 16,z2 + zx + x2 = 25.求 xy + yz + zx 的值.若8.a2b2c2= 0,b + c a + c + a b + a + b c求代数式abcb + c a + c + a b + a + b c的值.9.已知 abc = 0, 且abcab + a + 1 + bc + b + 1 + ca + c + 1 = 1.求 abc 的值.43练习题已知1.1 1 1=,+xy + z2111=,+yz + x31 1 1+=.zx + y4234求的值.x + y + z2.已知实数 x, y, z 满足1x += 4,y1.y + z = 1,17z + x = 3求 xyz 的值.已知互不相等的实数 a, b, c 满足3.111a + b = b + c = c + a = t .求 t 的值.4.已知 x, y, z 满足x2y 2z2y + z + z + x + x + y= 0.求代数式 x + y + z 的值.y + zz + xx + y设不全相等的