竞赛4-数学组卷-自助组卷

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竞赛4

数学

一、填空题（共 1 题），请把答案直接填写在答题纸上

1. 正实数

x, y, z

满足

x + 2 y^{2} + 4 x^{2} y^{2} z^{2} = 8

, 则

\log_{4} x + \log_{2} y + \log_{8} z

的最大值为

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二、解答题（共 20 题），解答过程应写出必要的文字说明、证明过程或演算步骤

2. 设

d \geq 0

是整数,

V

是

2 d + 1

维复线性空间, 有一组基

{v_{1}, v_{2}, \dots, v_{2 d + 1}} .

对任一整数

j (0 \leq j \leq \frac{d}{2})

, 记

U_{j}

是

v_{2 j + 1}, v_{2 j + 3}, \dots, v_{2 d - 2 j + 1}

生成的子空间. 定义线性变换

f : V \to V

为

f (v_{i}) = \frac{(i - 1) (2 d + 2 - i)}{2} v_{i - 1} + \frac{1}{2} v_{i + 1}, 1 \leq i \leq 2 d + 1 .

这里我们约定

v_{0} = v_{2 d + 2} = 0

.
(1) 证明:

f

的全部特征值为

- d, - d + 1, \dots, d

.
(2) 记

W

是从属于特征值

- d + 2 k (0 \leq k \leq d)

的

f

的特征子空间的和. 求

W \cap U_{0}

的维数.
(3) 对任一整数

j (1 \leq j \leq \frac{d}{2})

, 求

W \cap U_{j}

的维数.

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3. 双曲线

Γ : \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

的左右顶点

A, B

的距离为 4 ，

M, N

是

Γ

右支上不重合的两动点且满足

k_{B N} + 2 k_{A M} = 0

(

k_{A M}, k_{B N}

是相应直线的斜率). 求动直线

M N

经过的定点的坐标.

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4. Let

G

be a finite group.
(1) Let

K

be a field. Show that

G

has a finite-dimensional fait hful

K

-linear representation.
(2) Show that

G

has a faithful one-dimensional complex repr esentation if and only if

G

is cyclic.
(3) Assume moreover that

G

is commutative. Let

n \geq 1

be an integer. Show that

G

has a faithful

n

-dimensional complex re presentation if and only if

G

can be generated by

n

elements.
(4) Classify all finite groups having a faithful 2-dimensional re al representation.

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5. Let

n \geq 1

be an integer. Let

A

be a discrete valuation ring wit h

K

its field of fractions and

π \in A

a uniformizer. For

λ = (λ_{1}, \dots, λ_{n}) \in Z^{n}

write

D_{λ} = diag (π^{λ_{1}}, \dots, π^{λ_{n}}) = (\begin{array}{cccc} π^{λ_{1}} & 0 & \dots & 0 \\ 0 & π^{λ_{2}} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & π^{λ_{n}} \end{array}) \in {GL}_{n} (K) .

Show that, for

λ, μ \in Z^{n}

, the following intersection inside

{GL}_{n} (K)

{GL}_{n} (A) \cdot D_{μ} \cdot {GL}_{n} (A) ⋂ U (K) \cdot D_{λ}

is non-empty if and only if

λ_{dom} \leq μ_{dom}

. Here
-

{GL}_{n} (K)

(resp.

{GL}_{n} (A)

) is the group of invertible

n \times n sq

uare matrices with coefficients in

K

(resp.in

A

), and

U (K) \subset {GL}_{n} (K)

is the standard unipotent subgroup, that

i

s

, the subgroup of upper triangular matrices with coefficients 1 on the diagonal.
- for

α = (a_{1}, \dots, a_{n})

and

β = (b_{1}, \dots, b_{n})

two elements

i

n Z^{n}

, we write

α \leq β

\sum_{i = 1}^{k} a_{i} \leq \sum_{i = 1}^{k} b_{i}, for any 1 \leq k \leq n

and if

\sum_{i = 1}^{n} a_{i} = \sum_{i = 1}^{n} b_{i}

. Write also

α_{dom} := (a_{1}^{'}, \dots, a_{n}^{'})

with

a_{1}^{'}, \dots, a_{n}^{'}

an arrangement of

a_{1}, \dots, a_{n}

such that

a_{1}^{'} \geq a_{2}^{'} \geq \dots \geq a_{n}^{'}

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6. Let

k

be an imperfect field of characteristic

p > 0

. Let

a \in k ∖ k^{p}

.
(1) Show that the polynomial

X^{p} - a \in k [X]

is irreducible.
(2) Let

A = k [X] / (X^{p^{2}} - a X^{p})

. Compute

A_{red}

, the quotien

t

A

by its nilpotent radical.

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7. Let

k

be a field of character

p > 0

and consider

k (t) \subset k ((t))

.
(1) Show that the field extension

k ((t)) / k (t)

is transcendent al, i.e.,

k ((t))

contains at least one transcendental element ov er

k (t)

. (Hint: don't spend too much time on this question.)
(2) Let

α \in k [[t]]

which is transcendental over

k (t)

, and write

β = α^{p} \in k [[t]]

. Let

K = k (t, β)

and

L = k (t, α)

: both are subfields of

k ((t))

. Let

A = k [[t]] \cap k (t, β)

. Determine the i ntegral closure

B

A

L

, and show that

A

and

B

are two discrete valuation rings.
(3) Is

B

finitely generated over

A

as a module? Justify your as sertion.

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8. Let

p

be a prime number. Consider

Z_{p}

(resp.

Q_{p}

) the ring of

p

adic integers (resp. field of

p

-adic numbers). Clearly

Z \subset Z_{p}

.
(1) Show that the set

Z

is dense in

Z_{p}

, and deduce that a ma

p f : Z \to Q_{p}

can be extended to a continuous function on

Z_{p}

if and only if

f

is uniformly continuous, i.e., for any

ε > 0

. there exists some integer

N > 0

so that

| f (n) - f (m) | < ε

for any integers

n, m \in Z

with

m \equiv n (mod p^{N})

.
(2) Let

a \in Q_{p} ∖ {0}

. Under what condition on

a

, the map

f : Z \to Q_{p}, n \mapsto a^{n}

can be extended to a continuous function over

Z_{p}

? Justify yo ur assertion.
(3) Assume that the condition in (2) is fulfilled. Can we even

e

xtend the function

f

in (2) to a continuous map

a^{x} : Q_{p} \to Q_{p},

so that

a^{x + y} = a^{x} a^{y}

for any

x, y \in Q_{p}

? Justify your assertio n.

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9. Let

n \geq 1

be an integer and write

Φ_{n} (X)

the

n

-th cyclotomi c polynomial, that is, the minimal polynomial of a primitive

n

th root of unity in

C

over

Q

. Write also

φ (n) = \deg (Φ_{n} (X))

(1) Let

q

be a power of a prime number such that

(q, n) = 1

. Show that

Φ_{n}

, viewed as an element in

F_{q} [X]

, can be decom posed as a product of

φ (n) / d

irreducible polynomials of deg ree

d

, with

d

the order of

q

in the multiplicative group

(Z / n Z)^{\times}

.
(2) From now on, assume

n = 2^{r + 1}

for some integer

r \geq 1

. L et

ζ = ζ_{n}

be a primitive

n

-th root of unity and

K = Q [ζ]

. Let

p

be a prime with

p \equiv - 3 (mod 8)

.
(a) For

x, y \in K = Q [ζ]

, define

(x, y) := \sum_{τ} τ (x) \cdot \overset{―}{τ (y)}

where

τ

runs through all the embeddings

K ↪ C

K

into the field

C

of complex numbers. Write

K_{R} = K \otimes_{Q} R

, and we use the same notation to denote the (à priori

C

-valued) bi linear form on

K_{R}

obtained by extension of scalars. Show tha

t (\cdot, \cdot)

gives an inner product on

K_{R}

and for

0 \leq i, j < 2^{r}

(ζ^{i}, ζ^{j}) = {\begin{cases} 2^{r}, & if i = j \\ 0, & otherwise \end{cases}

In particular, we obtain an Euclidean space

K_{R}

and

{(ζ^{i} / \sqrt{2^{r}})}_{0 \leq i < 2^{r}}

is an orthonormal basis.
(b) Decompose

p O_{K}

into a product of prime ideals.
(c) Let

p \subset O_{K}

be a prime ideal of

O_{K}

containing

p

. Show th at for every

α \in p, | α |^{2} \in 2^{r} p Z

, and compute the length of

t

he shortest non-zero vector in the prime ideal

p \subset K_{R}

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10. Let

A \in R^{n \times n}

be a non-singular matrix. Let

u, v \in R^{n}

be col umn vectors. Define the rank 1 perturbation

\hat{A} = A + u v^{T}

.
(a) Derive a necessary and sufficient condition for

\hat{A}

to be inv ertible.
(b) Let

x, z

and

b

be column vectors in

R^{n}

. Suppose one can solve

A z = b

with

O (n)

floating-point operations (flops). Un der conditions derived in (a), design an algorithm to solve

\hat{A} x = b

with

O (n)

flops, and provide justification for your an swer.

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11. Consider the integral

\int_{0}^{\infty} f (x) d x

where

f

is continuous,

f^{'} (0) \neq 0

, and

f (x)

decays like

x^{- 1 - α}

with

α > 0

in the limit

x \to \infty

.
(a) Suppose you apply the equispaced composite trapezoid

r

ule with

n

subintervals to approximate

\int_{0}^{L} f (x) d x

What is the asymptotic error formula for the error in the limit

n \to \infty

with

L

fixed?
(b) Suppose you consider the quadrature from (a) to be an ap proximation to the full integral from 0 to

\infty

. How should

L

in crease with

n

to optimize the asymptotic rate of total error

d

ecay? What is the rate of error decrease with this choice of

L

?
(c) Make the following change of variable

x = \frac{L (1 + y)}{1 - y}

y = \frac{x - L}{x + L}

in the original integral to obtain

\int_{- 1}^{1} F_{L} (y) d y

Suppose you apply the equispaced composite trapezoid rule; what is the asymptotic error formula for fixed

L

?
(d) Depending on

α

, which method - domain truncation of ch ange-of-variable - is preferable?

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12. Consider the Chebyshev polynomial of the first kind

T_{n} (x) = \cos (n θ), x = \cos (θ), x \in [- 1, 1] .

The Chebyshev polynomials of the second kind are defined a

s

U_{n} (x) = \frac{1}{n + 1} T^{'} (x), n \geq 0 .

(a) Derive a recursive formula for computing

U_{n} (x)

for all

n \geq 0

.
(b) Show that the Chebyshev polynomials of the second kind are orthogonal with respect to the inner product

⟨ f, g ⟩ = \int_{- 1}^{1} f (x) g (x) \sqrt{1 - x^{2}} d x .

\int_{- 1}^{1} f (x) \sqrt{1 - x^{2}} d x = \sum_{j = 1}^{3} w_{j} f (x_{j}) .

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13. Consider the boundary value problem

- \frac{d}{d x} (a (x) \frac{d u}{d x}) = f (x), u (0) = u (1) = 0

where

a (x) > δ \geq 0

is a bounded differentiable function in

[0, 1]

. We assume that, although

a (x)

is available, an expressi on for its derivative,

\frac{d a}{d x}

, is not available.
(a) Using finite differences and an equally spaced gird in

[0, 1], x_{l} = h l, l = 0, \dots, n

and

h = 1 / n

, we discretize the ODE to obtain a linear system of equations, yielding an

O (h^{2})

approximation of the ODE. After the application of the boundary conditions, the resulting coefficient matrix of the li near system is an

(n - 1) \times (n - 1)

tridiagonal matrix.

Provide a derivation and write down the resulting linear syste

m

(by giving the expressions of the elements).
(b) Utilizing all the information provided, find a disc in

C

, the smaller the better, that is guaranteed to contain all the eigenv alues of the linear system constructed in part (a).

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14. (a) Verify that the PDE

u_{t} = u_{x x x}

is well posed as an initial value problem.
(b) Consider solving it numerically using the scheme

\frac{u (t + k, x) - u (t - k, x)}{2 k} = \frac{- \frac{1}{2} u (x - 2 h, t) + u (x - h, t) - u (x + h, t) + \frac{1}{2} u (x + 2 h, t)}{h} .

Determine this scheme's stability condition.

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15. Consider the diffusion equation

\frac{\partial v}{\partial t} = μ \frac{\partial^{2} v}{\partial x^{2}}, v (x, 0) = ϕ (x), \int_{a}^{b} v (x, t) d x = 0

with

x \in [a, b]

and periodic boundary conditions. The solutio

n

is to be approximated using the central difference operator

L

for the 1D Laplacian.

L v_{m} = \frac{v_{m + 1} - 2 v_{m} + v_{m - 1}}{h^{2}}

and the following two finite different approximations, (i) Forw ard-Euler

v_{n + 1} = v_{n} + μ k L v_{n}, (1)

and (ii) Crank-Nicolson

v_{n + 1} = v_{n} + μ k (L v_{n} + L v_{n + 1})

Throughout, consider

[a, b] = [0, 2 π]

and the finite differenc e stencil to have periodic boundary conditions on the spatial lattice

[0, h, 2 h, \dots, (N - 1) h]

where

h = \frac{2 π}{N}

and

N

is ev en.
(a) Determine the order of accuracy of the central difference operator

L v

is approximating the second derivative

v_{x x}

.
(b) Using

v_{m}^{n} = \sum_{l = 0}^{N - 1} {\hat{v}}_{l}^{n} \exp (- i \frac{2 π l m}{N})

give the updates

{\hat{v}}_{l}^{n + 1}

in terms of

{\hat{v}}_{l}^{n}

for each of the methods, including the ca se

l = 0

.
(c) Give the solution for

v_{m}^{n}

for each method when the initial condition is

ϕ (m Δ x) = (- 1)^{m}

.
(d) What are the stability constraints on the time step

k

for ea ch of the methods, if any, in equation (1) and (2)? Show there are either no constraints or express them in the form

k \leq F (h, μ)

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16. 双曲线

Γ : \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

的左右顶点

A, B

的距离为

4, M, N

是

Γ

右支上不重合的两动点且满足

k_{B N} + 2 k_{A M} = 0

(

k_{A M}, k_{B N}

是相应直线的斜率). 求动直线

M N

经过的定点的坐标.

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17. 实数

a, b, c

满足

a b + b c + c a = 44

, 求

(a^{2} + 4) (b^{2} + 4) (c^{2} + 4)

的最小值.

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18. 设

m, n

是大于 2 的整数. 已知平面上的正

n

边形区域

T

包含某个边长为 1 的正

m

边形区域. 求证: 该平面上的任意一个边长为

\cos \frac{π}{[m, n]}

的正

m

边形区域

S

可以平移嵌入区域

T

, 即存在向量

\vec{α}

, 满足区域

S

中的每个点平移

\vec{α}

后都落在区域

T

中.
注: 多边形区域包含内部和边界.

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19. 对互质的正整数

a, b

, 用

(a^{- 1} mod b)

表示满足

a m \equiv 1 (mod b)

且

0 \leq m <

b

的唯一整数

m

.
(1) 求证: 对任意两两互质的正整数

a, b, c, 1 < a < b < c

, 都有

(a^{- 1} mod b) + (b^{- 1} mod c) + (c^{- 1} mod a) > \sqrt{a} .

(2) 求证: 对任意正整数

M

, 存在两两互质的正整数

a, b, c

, 满足

M < a < b <

c

, 且

(a^{- 1} mod b) + (b^{- 1} mod c) + (c^{- 1} mod a) < 100 \sqrt{a} .

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20. 设

m, n

是自然数,

a_{0}, a_{1}, \dots, a_{m}, b_{0}, b_{1}, \dots, b_{n}

是非负实数. 对

0 \leq k \leq

m + n

, 记

c_{k} = max_{i + j = k} a_{i} b_{j}

. 求证:

\frac{c_{0} + c_{1} + \dots + c_{m + n}}{m + n + 1} \geq \frac{a_{0} + a_{1} + \dots + a_{m}}{m + 1} \cdot \frac{b_{0} + b_{1} + \dots + b_{n}}{n + 1} .

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21. 若实数

τ

满足: 对任意正整数

x, y, z

, 均有

x^{2} + 2 y^{2} + 4 z^{2} + 8 \geq 2 x (y + z + τ),

则称

τ

为 "平生数". 记最大的平生数为

τ_{0}

.
(1) 求

τ_{0}

的值;
(2) 求方程

x^{2} + 2 y^{2} + 4 z^{2} + 8 = 2 x (y + z + τ_{0})

的所有正整数解

(x, y, z)

.
(董秋仙供题)

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