在数学竞赛中经常会碰到一些涉及两数(式)和与两数(式)积的问题,这类问题一般难度较大,不易解答。解答这类问题需要掌握一定的策略。本文举例说明解答这类问题常见的策略,供同学们参考。1 利用完全平方式转化和积 例1 已知x,y,z为实数,且x+y+z=5,xy+yz+zx=3,试求z的最大值与最小值。(加拿大第10届数学竞赛题) 解由题意有x+y=5-z①,xy+(x+y)z=3,所以xy=3-(x+y)z=3-(5-z)z=z2-5z+3②,由①②利用公式(x+y)2-4xy=(x-y)2≥0得(5-z)2-4(z2-5z+3)≥0,即3z2-10z-13≤0,解之得-1≤z≤13/3,故z In mathematics competitions, there are often problems involving two numbers (formulas) and two numbers (formulas). Such problems are generally difficult and difficult to answer. Answering such questions requires a certain strategy. This article gives examples of common strategies for answering such questions for students’ reference. 1 Using the Completely Flat Transformation and Example 1 Knowing that x, y, and z are real numbers, and x+y+z=5 and xy+yz+zx=3, try to find the maximum and minimum values ​​of z. (Canada’s 10th Mathematical Contest Question) The solution meaning is x+y=5-z1, xy+(x+y)z=3, so xy=3-(x+y)z=3-(5-z ) z = z2-5z + 32, from 12 using the formula (x + y) 2-4xy = (xy) 2 ≥ 0 to (5-z) 2-4 (z2-5z + 3) ≥ 0, that is, 3z2- 10z-13 ≤ 0, the solution is -1 ≤ z ≤ 13/3, so z