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## 家里蹲大学数学杂志期刊模式目录

Problem: Let $f$ be a continuous function from $[0, 1]$ into $\bbR^+$. Prove that $$\bex \int_0^1 f(x)\rd x -\exp\sez{\int_0^1 \ln f(x)\rd x} \leq \max{0\leq x, y\leq 1} \sez{\sqrt{f(x)}-\sqrt{f(y)}}^2. \eex$$

Problem: Let $f$ be a continuous,  nonnegative function on $[0, 1]$. Show that $$\bex \int_0^1 f^3(x)\rd x\geq 4\sex{\int_0^1 x^2f(x)\rd x} \sex{\int_0^1 xf^2(x)\rd x}. \eex$$

Problem: Let $f:[0, 1]\to (0, \infty)$ be a continuous function. Calculate $$\bex \vlm{n} \sqrt[n]{\int_0^1 (1+x^n)^n f(x)\rd x}. \eex$$

Problem: Let $a, b, c$ be positive real numbers such that $a+b+c=1$. Prove that $$\bex (1-2a)^5+(1-2b)^5+(1-2c)^5+\frac{80abc}{3}\leq 1. \eex$$

Problem: Show that if $f:[-1, 1]\to \bbR$ is convex and $C^2$ function such that $f(1), f(-1)\geq 0$,  then $$\bex \min{x\in [-1, 1]} f(x)\geq -\int_{-1}^1 f''(x)\rd x. \eex$$

(1). $Ai^2=Ai$,  $i=1, \cdots, k$;

(2). $AiAj=0$,  $i\neq j$.

Problem: For a continuous and nonnegative function $f$ on $[0, 1]$,  let $$\bex \mu_n=\int_0^1 x^nf(x)\rd x. \eex$$ Show that $$\bex \mu_{n+1}\mu_0\geq \mu_n\mu_1, \quad \forall\ n\in\bbN. \eex$$

(2). Prove the following result: Let $g:[a,b]\to\bbR$ be bounded and integrable. Show that its graph $$\bex graph(g)=\sed{(x,g(x));x\in[a,b]} \eex$$ has zero content.

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