竞赛习题精选

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:\[\sum_{k=1}^{n} a_{k}^{\frac{k}{k+1}} \leq \sum_{k=1}^{n} a_{k}+\sqrt{\frac{2\left(\pi^{2}-3\right)}{9} \sum_{k=1}^{n} a_{k}}.\]

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$\frac{1}{p}+\frac{1}{q}=1, p>1$.求证:

\[\left(\sum_{i=1}^{n} a_{i}^{p}\right)^{1 / p}\left(\sum_{i=1}^{n} b_{i}^{q}\right)^{1 / q} \leq \frac{1}{p^{1 / p}} \frac{1}{q^{1 / q}} \frac{A^{p} B^{q}-1}{\left(B A^{p}-A\right)^{1 / q}\left(A B^{q}-B\right)^{1 / p}} \sum_{i=1}^{n} a_{i} b_{i}.\]

4.设$a_i>0$,求证:

\[\sum_{k=1}^{n} \frac{k}{a_{1}+a_{2}+\cdots+a_{k}} \leq\left(2-\frac{7 \ln 2}{8 \ln n}\right) \sum_{k=1}^{n} \frac{1}{a_{k}}.\]

参考:王永喜《2017年集训队讲义》

5.设函数$f(x)$是以$2\pi$为周期且具有$2$阶连续导数的函数.定义

\[b_{n}=\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin n x d x,\quad b_{n}^{\ast}=\frac{1}{\pi} \int_{-\pi}^{\pi} f''(x) \sin n x d x.\]

求证:若级数$\sum_{n=1}^\infty b_n^\ast$绝对收敛,则\[\sum_{n=1}^{\infty} \sqrt{\left|b_{n}\right|} \leq \frac{1}{2}\left(2+\sum_{n=1}^{\infty}\left|b_{n}^\ast\right|\right).\]

6.设$0\leq p_i\leq 1,\sum_{i=1}^kp_i=1$,求$-\sum_{i=1}^kp_i\log_2p_i$的最大值.