Jan 062014
 

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?

\[4M^5+2M^3+M=7E_n,\]

其中, \(M\) 是与 \(E_n\) 同阶的矩阵.

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

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

垂直的直母线交点的轨迹.

8.保距变换

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

可以看做绕不动直线旋转一个角度而得到.
(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

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

上的投影为 \(A_{1}, B_{1}\), 求 \(A_{1}, B_{1}\) 坐标以及两点间距离.
(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)

Jan 052014
 

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 \(L\lt3\).

Dec 102013
 

“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.

咕噜(Gollum)无疑是”魔戒”里的悲剧人物之一. 咕噜没有在拥有魔戒的五百年间, 使用魔戒做任何事, 最终还因为魔戒而坠入末日火山的火焰. 他因为魔戒, 心灵扭曲; 他的大半生, 人不像人鬼不像鬼.

由于这个系列有各种不同版本, 片长不一, 这里顺便提一下各部电影最长的时间: 指环王1 魔戒现身加长版 228分钟;  指环王 2 双塔奇兵加长版 235 分钟; 指环王 3 王者归来加长版 263 分钟; 霍比特人1 意外之旅加长版 182 分钟.

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

Solutions to the 74th William Lowell Putnam Mathematical Competition

第 74 届普特南数学竞赛的官方解答

 Posted by at 1:40 am
Oct 112013
 

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.

个人赛试题

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

团体赛试题

Team 2013

感谢博士数学论坛的网友数函的分享

Aug 092013
 

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. )

 Posted by at 6:09 pm
Mar 212013
 

第四届全国大学生数学竞赛决赛试题

2013 年3 月16 日 电子科技大学(成都)

1.(15分) 设 \(A\) 为正常数, 直线 \(L\) 与双曲线 \(x^2-y^2=2(x>0)\) 所围成的面积为 \(A\). 证明:
(i) 上述 \(L\) 被双曲线  \(x^2-y^2=2(x>0)\) 所截线段的中点的轨迹为双曲线;
(ii) \(L\) 总是 (i) 中轨迹曲线的切线.

2.(15分) 设函数 \(f(x)\) 满足条件:
(1) \(-\infty<a \leqslant  f(x) \leqslant b <+\infty, a\leqslant x\leqslant b \);
(2) 对任意 \(x,y\in[a,b]\) 有 \(|f(x)-f(y)|<L|x-y|\),其中 \(L\) 是大于 \(0\) 小于 \(1\) 的常数.
设 \(x_1\in[a,b]\), 令 \(x_{n+1}=\frac12[x_n+f(x_n)],n=1,2,\dotsc\).
证明 \(\lim\limits_{n\to\infty}x_n=x\) 存在, 且 \(f(x)=x\).

3.(15分) 设实 \(n\) 阶方阵 \(A\) 的每个元素的绝对值为 \(2\), 证明: 当 \(n\geqslant 3\) 时, \(|A|\leqslant\frac13\cdot2^{n+1}n!\).

4.(15分) 设函数 \(f(x)\) 为区间 \((a,b)\) 上的可导函数. 对 \(x_0\in (a,b)\), 若存在 \(x_0\) 的领域 \(U\) 使得任意的\(x\in U\backslash\{x_0\}\) 有 \(f(x)>f(x_0)+f^\prime(x_0)(x-x_0)\), 则称 \(x_0\) 为 \(f(x)\) 的凹点. 类似地, 若存在\(x_0\) 的领域 \(U\) 使得任意的\(x\in U\backslash\{x_0\}\) 有 \(f(x)<f(x_0)+f^\prime(x_0)(x-x_0)\), 则称 \(x_0\) 为 \(f(x)\) 的凸点.
求证: 若 \(f(x)\) 是区间 \((a,b)\) 上的可导函数且不是一次函数, 则 \(f(x)\) 一定存在凹点或凸点.

5.(20分) 设 \(A=\pmatrix{
a_{11}&a_{12}&a_{13}\cr
a_{21}&a_{22}&a_{23}\cr
a_{31}&a_{32}&a_{33}\cr
}\)为实对称矩阵, \(A^*\) 为 \(A\) 的伴随矩阵, 记 \(f(x_1,x_2,x_3,x_4)=
\left|\matrix{
x_1^2&x_2&x_3&x_4\cr
-x_2&a_{11}&a_{12}&a_{13}\cr
-x_3&a_{21}&a_{22}&a_{23}\cr
-x_4&a_{31}&a_{32}&a_{33}
}\right|\). 若 \(A\) 的行列式为 \(-12\),  \(A\) 的所有特征值的和为 \(1\), 且 \((1,0,-2)^T\) 为 \((A^*-4I)X=0\) 的一个解. 试给出一正交变换 \(\left(\matrix{x_1\cr x_2\cr x_3\cr x_4}\right)=Q\left(\matrix{y_1\cr y_2\cr y_3\cr y_4}\right)\) 使得 \(f(x_1,x_2,x_3,x_4)\) 化为标准型.

6.(20分) 令 \(\Bbb R\) 为实数集, \(n\) 为给定的正整数, \(A\) 表示所有 \(n\) 次首一实系数多项式组成的集合. 证明
\[\inf_{b\in\Bbb R, c>0, P(x)\in A}\dfrac{\int_b^{b+c}|P(x)|\,\rm dx}{c^{n+1}}>0.\]

 Posted by at 6:12 pm