1992年全国高中数学联赛竞赛一试第6题
设$$f\left( x \right)$$是定义在实数集$$R$$上的函数，且满足下列关系：$$f\left( 10+x \right)=f\left( 10-x \right)$$．$$f\left( 20-x \right)=-f\left( 20+x \right)$$．则$$f\left( x \right)$$是（~ ）．
竞赛->知识点->函数->函数的图像与性质
由所给第一式得  $$f\left[ 10+\left( 10-x \right) \right]=f\left[ 10-\left( 10-x \right) \right]$$，  ∴~~~~  $$f\left( x \right)=f\left( 20-x \right)$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ①  又由所给第二式得  $$f\left( x \right)=-f\left( 20-x \right)$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ②  ∴~~~~  $$\begin{matrix}f\left( 40+x \right)=f\left[ 20+\left( 20+x \right) \right] =-f\left( 20+x \right)=f\left( x \right)   \end{matrix}$$  可见$$f(x)$$是周期函数．  由①，②得  $$f\left( -x \right)=f\left( 20+x \right)=-f\left( x \right)$$，  ∴$$f\left( x \right)$$是奇函数．因此答案是（$$\text{C}$$）． 