If $$f\in C^1([a,b])$$ is increasing and nonconstant, then

$\int_a^b\sqrt{1+{f^\prime}^2(x)}\, \mathrm dx \lt b-a+f(b)-f(a).$

For $$\alpha$$, $$\beta\geqslant0$$, $$\sqrt{\alpha^2+\beta^2}\leqslant\alpha+\beta$$, with equality iff one or both of $$\alpha$$, $$\beta$$ equals $$0$$.

Now $$f ^\prime\geqslant 0$$, it follows that

$\sqrt{1+{f^\prime}^2(x)}\leqslant 1+f^\prime(x), x\in [a,b].$

Because $$f(x)$$ is nonconstant, $$f^\prime\gt0$$ in a subinterval. In that subinterval we have strict inequality between these two functions. Integrating both sides then gives the result.

1.令 $$f(x)=\prod\limits_{i=1}^{2013} (x-i)^2+2014$$, $$f(x)$$ 在有理域内可约吗? 证明你的结论.

2. $$M$$, $$N$$ 都是 $$n$$ 阶矩阵, $$n\geq2$$. 如果 $$MNMN$$ 为零矩阵, 那么 $$NMNM$$ 是否也一定是零矩阵? 证明你的结论.

3. $$n\geq2$$. 除了单位矩阵, 还有别的埃尔米特矩阵 $$M$$ 满足下面的条件吗？

$4M^5+2M^3+M=7E_n,$

4. $$\mathbf V$$ 是 $$n$$ 维线性空间. 线性变换 $$\mathcal A$$ 的最小多项式是 $$n$$ 次.
(1) 证明存在向量 $$\alpha$$, 使得 $$\alpha$$, $$\mathcal A\alpha$$, $$\dotsc$$, $$\mathcal A^{n-1}\alpha$$ 是 $$\mathbf V$$  的一组基;
(2) 任何与 $$\mathcal A$$ 可交换的线性变换, 可表示为 $$\mathcal A$$ 的多项式.

5. $$\mathbf V=\Bbb C_{n\times n}$$ 是所有 $$n$$  阶复矩阵组成的向量空间. 求所有形如  $$MN-NM$$ 的矩阵组成的向量空间的维数并给出证明.

6. 欧式空间 $$\mathbf V$$ 中, 对称线性变换 $$\mathcal{A}$$ 称为“正的”, 若对 $$\forall \alpha \in \mathbf V$$, 都有$$(\alpha, \mathcal A(\alpha))\geq 0$$ 成立, 且等号当且仅当 $$\alpha =\mathbf 0$$ 时成立.
(a)证明若线性变换 $$\mathcal A$$ 是正的，则 $$\mathcal A$$ 可逆;
(b)证明若线性变换 $$\mathcal B$$ 是正的, $$\mathcal A-\mathcal B$$ 也是正的，则 $$\mathcal B^{-1}-\mathcal A^{-1}$$ 是正的;
(c)证明对于正的线性变换 $$\mathcal A$$, 总存在正的线性变换 $$\mathcal B$$ 使得 $$\mathcal A=\mathcal B^2$$.

7. 求单叶双曲面

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1$

8.保距变换

$\begin{split} x’& = a_{11}x+a_{12}y+a_{13}z\\ y’& = a_{21}x+a_{22}y+a_{23}z\\ z’& = a_{31}x+a_{32}y+a_{33}z \end{split}$

(a)求不动直线的方向向量;
(b)求旋转角 $$\theta$$.
(原题$$a_{11},\cdots,a_{33}$$皆为具体数字, 现已记不清, 用字母代替之)

9.点 $$A(a_{1},a_{2},a_{3})$$, $$B(b_{1},b_{2},b_{3})$$ 在直线

$\frac{x+a}{2}=\frac{y+b}{2}=\frac{z}{3}$

(原题$$a_{1},\dotsc,b_{3},a,b$$皆为具体数字，现已记不清, 用字母代替之)

1. 叙述实数序列 $$\{x_n\}$$ 的 Cauchy 收敛原理, 并且使用 Bolzano-Weierstrass(波尔查诺-威尔斯特拉斯)定理证明.

2. 序列 $$\{x_n\}$$ 满足 $$x_1=1$$, $$x_{n+1}=\sqrt{4+3x_n}$$, $$n=1$$, $$2$$,$$\dotsc$$. 证明此序列收敛并求极限.

3. 计算 $$\iiint_{\Omega}\sqrt{x^2+y^2}\, \mathrm dx\mathrm dy\mathrm dz$$, 其中 $$\Omega$$ 是曲面 $$z=\sqrt{x^2+y^2}$$ 与 $$z=1$$ 围成的有界区域.

4. 证明函数项级数 $$\sum\limits_{n=1}^{+\infty}x^3e^{-nx^2}$$ 在 $$[0,+\infty)$$ 一致收敛.

5. 讨论级数 $$\sum\limits_{n=3}^{+\infty}\ln \cos\dfrac\pi n$$ 的敛散性.

6. 设函数 $$f\colon\Bbb R^n\to\Bbb R$$ 在 $$\Bbb R^n\setminus\mathbf0$$ 可微, 在 $$\mathbf0$$ 点连续, 且 $$\lim\limits_{\mathbf p\to \mathbf0} \dfrac{\partial f(\mathbf{p})}{\partial x_i}=0$$, $$i=1$$, $$2$$, $$\dotsc$$, $$n$$. 证明 $$f$$ 在 $$\mathbf0$$ 处可微.

7.  设 $$f(x)$$, $$g(x)$$ 是 $$[0,1]$$ 上的连续函数, 且 $$\sup\limits_{x\in [0,1]}f(x)=\sup\limits_{x\in [0,1]}g(x)$$. 证明存在 $$x_0\in[0,1]$$, 使得 $$e^{f(x_0)}+3f(x_0)=e^{g(x_0)}+3g(x_0)$$.

8. 记 $$\Omega=\{\mathbf p\in\Bbb R^3| |\mathbf p|\leq1 \}$$, 设 $$V\colon\Bbb R^3\to\Bbb R^3$$, $$V=(V_1, V_2, V_3)$$ 是 $$C^1$$ 向量场, $$V$$ 在 $$\Bbb R^3\setminus\Omega$$ 恒为 $$0$$, $$\dfrac{\partial V_1}{\partial x}+\dfrac{\partial V_2}{\partial y}+ \dfrac{\partial V_3}{\partial z}$$在 $$\Bbb R^3$$ 恒为 $$0$$.
(1) 若 $$f\colon\Bbb R^3\to\Bbb R$$ 是 $$C^1$$ 函数, 求 $$\iiint_{\Omega}\bigtriangledown f\cdot V\,\mathrm dx\mathrm dy\mathrm dz$$.
(2) 求 $$\iiint_{\Omega}V_1\, \mathrm dx\mathrm dy\mathrm dz$$.

9. 设 $$f\colon\Bbb R\to\Bbb R$$ 是有界连续函数, 求 $$\lim\limits_{t\to0^+}\int_{-\infty}^{+\infty} f(x) \frac{t}{t^2 + x^2}\,\mathrm dx$$.

10. 设 $$f \colon [0,1] \to [0,1]$$ 是 $$C^2$$ 函数, $$f(0)=f(1)=0$$, 且 $$f^{\prime\prime}(x)\lt0$$, $$\forall x\in[0,1]$$. 记曲线 $$\{(x,f(x))|x\in[0,1]\}$$ 的弧长是 $$L$$. 证明 $$L\lt3$$.

“霍比特人2: 史矛革之战(The Hobbit: Desolation of Smaug)” 一鼓作气席卷扑面而来! 翘首以盼快一年了!! 嗯, 嗯! 挪威(Norway)的朋友有幸今天 12 月 10 日, 最先观赏到这部充满期待的影片. 喜爱魔戒的影迷赶快重温下 “霍比特人 1: 意外之旅(The Hobbit: An Unexpected Journey)” 吧! 俺第一次看”霍比特人 1″, 是今年初 3 月 25 日在北京清河的 CGV 希界维国际影城. 影院在五彩城购物中心的最顶 7-8 两层, 环境还不错. 这电影院当时刚开张, 做了一些活动.

Solutions to the 74th William Lowell Putnam Mathematical Competition 2013  are now posted

Solutions to the 74th William Lowell Putnam Mathematical Competition

Problems from the 2013 William Lowell Putnam Mathematical Competition, held December 7, are now posted.

The 74th William Lowell Putnam Mathematical Competition

2013 年第四届丘成桐(Shing-Tung Yau)大学生数学竞赛(S.T. Yau College Student Mathematics Contests)已经落下帷幕. 决赛已经于 8 月 11 日和 12 日在北京中国科学院数学与系统科学院思源楼和晨兴中心举行, 颁奖典礼也已于 8 月 12 日在清华大学举行.

Analysis and differential equations 2013 Individual

Geometry and topology 2013 Individual

Algebra and number theory 2013 Individual

Probability and statistics 2013 Individual

Applied Math. and Computational Math. 2013 Individual

Team 2013

# Day 1

Problem 1. Let  $$\rm A$$ and $$\rm B$$ be real symmetric matrixes with all eigenvalues strictly greater than $$1$$. Let $$\lambda$$ be a real eigenvalue of matrix $$\rm {AB}$$. Prove that $$\left| \lambda \right|\gt1$$.

Problem 2.  Let $$f:\Bbb R \to \Bbb R$$ be a twice differentiable function. Suppose  $$f(0) = 0$$. Prove that there exists $$\xi \in\left({-\frac\pi2,\frac\pi2}\right)$$ such that

$f^{\prime\prime}\left( \xi \right) = f\left( \xi \right)\left(1 + 2\tan^2\xi \right).$

Problem 3. There are $$2n$$ students in a school $$\left( {n \in {\Bbb N},n \geqslant 2} \right)$$. Each week $$n$$ students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?

Problem 4. Let $$n\geqslant 3$$ and let $$x_1,x_2,\dotsc,x_n$$ be nonnegative real numbers. Define $$A = \sum\limits_{i = 1}^n x_i,B = \sum\limits_{i = 1}^n x_i^2,C=\sum\limits_{i = 1}^n x_i^3$$. Prove that:

$\left(n+1\right)A^2B+\left(n-2\right)B^2\geqslant A^4+\left(2n-2\right)AC.$

Problem 5. Does there exist a sequence $$(a_n)$$ of complex numbers such that for every positive integer $$p$$ we have that $$\sum\limits_{n=1}^\infty a_n^p$$ converges if and only if $$p$$ is not a prime?

# Day 2

Problem 1. Let $$z$$ be a complex number with $$\left|z+1\right|>2$$. Prove that $$\left|z^3+1\right| > 1$$.

Problem 2. Let $$p$$ and $$q$$ be relatively prime positive integers. Prove that

$\sum_{k=0}^{pq-1}(-1)^{\left[\frac kp\right]+\left[\frac kq\right]}=\begin{cases}0 &\text{if} pq \text{is even},\\1 &\text{if } pq \text{ is odd}.\end{cases}$

(Here $$[x]$$ denotes the integer part of $$x$$.)

Problem 3. Suppose that $$\mathbf v_1,\mathbf v_2,\dotsc,\mathbf v_d$$ are unit vectors in $$\Bbb R^d$$. Prove that there exists a unit vector $$\mathbf u$$ such that

$\left| \mathbf u\cdot\mathbf v_i \right| \leqslant \frac1{\sqrt d}$

for $$i = 1,2,\dotsc,d$$.
( Here $$\cdot$$ denotes the usual scalar product on $$\Bbb R^d$$).

Problem 4.  Does there exist an infinite set $$M$$ consisting of positive integers such that for any $$a,b \in M$$, with $$a\lt b$$, the sum $$a+b$$ is square-free?
( A positive integer is called square-free if no perfect square greater than $$1$$ divides it ).

Problem 5.  Consider a circular necklace with $$2013$$ beads. Each bead can be painted either white or green. A painting of the necklace is called good if among any $$21$$ successive beads there is at least one green bead. Prove that the number of good paintings of the necklace is odd.
(Two paintings that differ on some beads, but can be obtained from each other by rotating or flipping the necklace, are counted as different paintings. )