本文应用“三元齐次线性方程有非零解的充要条件是它的系数行列式为零的定理”,通过构造三阶行列式,对一部分数学竞赛题进行巧思妙证,现分类举例说明如下:1解代数题例1若实数a,b,x,y满足ax+by=3,ax2+by2=7,ax3+by3=16,ax4+by4=42,求ax5+by5的值.(第8届美国数学邀请赛试题)(AI M In this paper, the “The necessary and sufficient condition for the nonzero solution of the ternary homogeneous linear equation is the theorem of its coefficient determinant is zero”. By constructing the third-order determinant, some mathematical contest problems are inferred. The explanation is as follows: 1 Solving algebraic problem example 1 If the real numbers a, b, x, y satisfy ax+by=3, ax2+by2=7, ax3+by3=16, ax4+by4=42, find the value of ax5+by5. (The 8th American Mathematical Invitational Questions) (AI M