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[namcpnh]

Bài toán 32 : Tìm $f:\mathbb{R}\rightarrow \mathbb{R}$ thỏa $\left | \sum_{k=1}^{n}[3^k(f(x+ky)-f(x-ky))] \right |\leq 1$.

[hoangtrunghieu22101997]

Bài toán 32 : Tìm $f:\mathbb{R}\rightarrow \mathbb{R}$ thỏa $\left | \sum_{k=1}^{n}[3^k(f(x+ky)-f(x-ky))] \right |\leq 1$.

Ta có:
$-1 \le \sum_{k=1}^{n}[3^k(f(x+ky)-f(x-ky))] \le 1$
$\to -1 \le \sum_{k=1}^{n-1}[3^k(f(x+ky)-f(x-ky))] \le 1$
Trừ vế cho vế ta có:
$-2 \le 3^n(f(x+ny)-f(x-ny)) \le 2$
Đặt $u=x+ny; v=x-ny$
Thì $|f(u)-f(v)| \le 2/3^n$
$\to f(x)=$const

Edited by hoangtrunghieu22101997, 30-05-2013 - 16:09.