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

设 $d \geq 0$ 是整数, $V$ 是 $2 d+1$ 维复线性空间, 有一组基
$$
\left\{v_1, v_2, \cdots, v_{2 d+1}\right\} \text {. }
$$

对任一整数 $j\left(0 \leq j \leq \frac{d}{2}\right)$, 记 $U_j$ 是
$$
v_{2 j+1}, v_{2 j+3}, \cdots, v_{2 d-2 j+1}
$$

生成的子空间. 定义线性变换 $f: V \rightarrow V$ 为
$$
f\left(v_i\right)=\frac{(i-1)(2 d+2-i)}{2} v_{i-1}+\frac{1}{2} v_{i+1}, 1 \leq i \leq 2 d+1 .
$$

这里我们约定 $v_0=v_{2 d+2}=0$.
(1) 证明: $f$ 的全部特征值为 $-d,-d+1, \cdots, d$.
(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 $j\left(1 \leq j \leq \frac{d}{2}\right)$, 求 $W \cap U_j$ 的维数.

双曲线 $\Gamma: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ 的左右顶点 $A, B$ 的距离为 4 ，
$M, N$ 是 $\Gamma$ 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.

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.

Let $n \geq 1$ be an integer. Let $A$ be a discrete valuation ring wit h $K$ its field of fractions and $\pi \in A$ a uniformizer. For $\lambda=\left(\lambda_1, \cdots, \lambda_n\right) \in \mathbb{Z}^n$ write
$$
D_\lambda=\operatorname{diag}\left(\pi^{\lambda_1}, \cdots, \pi^{\lambda_n}\right)=\left(\begin{array}{cccc}
\pi^{\lambda_1} & 0 & \cdots & 0 \\
0 & \pi^{\lambda_2} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \pi^{\lambda_n}
\end{array}\right) \in \mathrm{GL}_n(K) .
$$

Show that, for $\lambda, \mu \in \mathbb{Z}^n$, the following intersection inside $\mathrm{GL}_n(K)$
$$
\mathrm{GL}_n(A) \cdot D_\mu \cdot \mathrm{GL}_n(A) \bigcap U(K) \cdot D_\lambda
$$
is non-empty if and only if $\lambda_{\text {dom }} \leq \mu_{\text {dom }}$. Here
- $\mathrm{GL}_n(K)$ (resp. $\mathrm{GL}_n(A)$ ) is the group of invertible $n \times n \mathrm{sq}$ uare matrices with coefficients in $K$ (resp.in $A$ ), and $U(K) \subset \mathrm{GL}_n(K)$ is the standard unipotent subgroup, that $\mathrm{i}$ $\mathrm{s}$, the subgroup of upper triangular matrices with coefficients 1 on the diagonal.
- for $\alpha=\left(a_1, \cdots, a_n\right)$ and $\beta=\left(b_1, \cdots, b_n\right)$ two elements $\mathrm{i}$ $\mathrm{n} \mathbb{Z}^n$, we write $\alpha \leq \beta$ if
$$
\sum_{i=1}^k a_i \leq \sum_{i=1}^k b_i, \quad \text { for any } 1 \leq k \leq n
$$
and if $\sum_{i=1}^n a_i=\sum_{i=1}^n b_i$. Write also $\alpha_{\text {dom }}:=\left(a_1^{\prime}, \cdots, a_n^{\prime}\right)$ with $a_1^{\prime}, \cdots, a_n^{\prime}$ an arrangement of $a_1, \cdots, a_n$ such that
$$
a_1^{\prime} \geq a_2^{\prime} \geq \cdots \geq a_n^{\prime}
$$

Let $k$ be an imperfect field of characteristic $p>0$. Let $a \in k \backslash k^p$.
(1) Show that the polynomial $X^p-a \in k[X]$ is irreducible.
(2) Let $A=k[X] /\left(X^{p^2}-a X^p\right)$. Compute $A_{\text {red }}$, the quotien $\mathrm{t}$ of $A$ by its nilpotent radical.