竞赛习题精选

Posted 2022-03-27 eufisky

1.设$x,y,z$是三个复数,定义\[d(x, y)=\frac|x-y|\sqrt\left(1+|x|^2\right)\left(1+|y|^2\right).\]求证:\[d(x, y) \leq d(x, z)+d(z, y).\]

2.设$a_1, a_2, \cdots, a_n$是正实数,求证:\[\sum_k=1^n a_k^\frackk+1 \leq \sum_k=1^n a_k+\sqrt\frac2\left(\pi^2-3\right)9 \sum_k=1^n a_k.\]

3.设$a_i>0,b_i>0$,其中$i=1,\cdots,n$.记$A=\frac\max a_k\min a_k, B=\frac\max b_k\min b_k$和$\frac1p+\frac1q=1, p>1$.求证:

\[\left(\sum_i=1^n a_i^p\right)^1 / p\left(\sum_i=1^n b_i^q\right)^1 / q \leq \frac1p^1 / p \frac1q^1 / q \fracA^p B^q-1\left(B A^p-A\right)^1 / q\left(A B^q-B\right)^1 / p \sum_i=1^n a_i b_i.\]

4.设$a_i>0$,求证:

\[\sum_k=1^n \fracka_1+a_2+\cdots+a_k \leq\left(2-\frac7 \ln 28 \ln n\right) \sum_k=1^n \frac1a_k.\]

参考:王永喜《2017年集训队讲义》