## 代数100题

## 山大附中:王永喜

这个代数100题是本人近年来给一些机构讲的试题一个汇集(注:当然是讲义挑出一些不太常见的题汇集在此),题源来源比较广,这里就不一一例举了,这些题一般都比较难,不太适合联赛训练,感兴趣的同学可以挑着做做玩玩.

1. 设 $a, b, c > 0$ , 求证

$$
\sqrt {\frac {a}{a + 2 b + c}} + \sqrt {\frac {b}{a + b + 2 c}} + \sqrt {\frac {c}{2 a + b + c}} \leq \frac {3}{2}
$$

2. 设a,b,c,d,e,f是实数,且

$$
a d - b c = 1, \text {且} e, f \geq {\frac {1}{2}}
$$

证明

$$
\sqrt {e ^ {2} (a ^ {2} + b ^ {2} + c ^ {2} + d ^ {2}) + e (a c + b d)} + \sqrt {f ^ {2} (a ^ {2} + b ^ {2} + c ^ {2} + d ^ {2}) - f (a c + b d)} \geq (e + f) \sqrt {2}
$$

3. 求证:

$$
\begin{array}{r} (a _ {1} ^ {2} + b _ {2} ^ {2} + b _ {3} ^ {2}) (a _ {2} ^ {2} + b _ {3} ^ {2} + b _ {1} ^ {2}) (a _ {3} ^ {2} + b _ {1} ^ {2} + b _ {2} ^ {2}) \geq (b _ {1} ^ {2} + b _ {2} ^ {2} + b _ {3} ^ {2}) (a _ {1} b _ {1} + a _ {2} b _ {2} + a _ {3} b _ {3}) ^ {2} \\ + \frac {1}{2} (b _ {1} a _ {2} b _ {3} - b _ {1} b _ {2} a _ {3}) ^ {2} + \frac {1}{2} (b _ {1} b _ {2} a _ {3} - a _ {1} b _ {2} b _ {3}) ^ {2} + \frac {1}{2} (a _ {1} b _ {2} b _ {3} - b _ {1} a _ {2} b _ {3}) ^ {2} \end{array}
$$

其中 $a_{i},b_{i}\in R,i=1,2,3$

4. 设 $a, b, c > 0$ ，且满足 $a + b + c = 3$ ，求证

$$
\sqrt {3 a + \frac {1}{b}} + \sqrt {3 b + \frac {1}{c}} + \sqrt {3 c + \frac {1}{a}} \geq 6
$$

5. 设正整数m,n是给定的,假设 $a_{1},a_{2},\cdots,a_{m}$ 是任意的m个实数,且 $v_{1},v_{2},\cdots,v_{m}$ 是m个不同的n维向量,若

$$
\sum_ {j \neq i} a _ {j} \frac {v _ {j} - v _ {i}}{| | v _ {j} - v _ {i} | | ^ {3}} = 0, i = 1, 2, \dots , m
$$

则

$$
\sum_ {1 \leq i <   j \leq m} \frac {a _ {i} a _ {j}}{| | v _ {j} - v _ {i} | |}
$$

是否能大于0?说明理由.

其中

$$
| | x | | = \sqrt {x _ {1} ^ {2} + x _ {2} ^ {2} + \cdots + x _ {n} ^ {2}}, \qquad x = (x _ {1}, x _ {2}, \dots , x _ {n})
$$

6. : 设 $a_{1}, a_{2}, \cdots, a_{n}$ 是任意实数, 求证

$$
\sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} \frac {a _ {i} a _ {j}}{1 + | i - j |} \geq 0
$$

7. 复系数多项式 $f(z) = z^n + a_1z^{n-1} + \cdots + a_{n-1}z + a_n$ 的 $n$ 个根为 $z_1, z_2, \cdots, z_n$ ，且满足 $\sum_{k=1}^{n} |a_k|^2 \leq 1$ ，求证： $\sum_{k=1}^{n} |z_k|^2 \leq n$ 8. 设 $p$ 是奇素数。对任意正整数 $m < n$ ，证明：

$$
\left| \left(\frac {m}{p}\right) + \left(\frac {m + 1}{p}\right) + \dots + \left(\frac {n}{p}\right) \right| <   \sqrt {p} \ln p
$$

这里 $\left(\frac{\cdot}{-\cdot}\right)$ 表示勒让德符号

9. 设 $a, b, c, x, y, z \in \mathbb{R}$ 满足

$$
\left\{ \begin{array}{l} \{a, b, c, x, y, z \} \subset [ - 1, 1 ] \\ 1 + 2 a b c \geq a ^ {2} + b ^ {2} + c ^ {2} \\ 1 + 2 x y z \geq x ^ {2} + y ^ {2} + z ^ {2} \end{array} \right.
$$

求证:

$$
1 + 2 a b c x y z \geq a ^ {2} x ^ {2} + b ^ {2} y ^ {2} + c ^ {2} z ^ {2}
$$

10. 设 $x_{1}, x_{2}, \cdots, x_{n} \geq 0$ , 证明

$$
\frac {x _ {1} + \sqrt {x _ {1} x _ {2}} + \sqrt [ 3 ]{x _ {1} x _ {2} x _ {3}} + \cdots + \sqrt [ n ]{x _ {1} x _ {2} \cdots x _ {n}}}{n} \leq \sqrt [ n ]{x _ {1} \cdot \frac {x _ {1} + x _ {2}}{2} \cdots \frac {x _ {1} + x _ {2} + \cdots + x _ {n}}{n}}
$$

11. 设数列 $\{x_{n}\}$ 满足 $x_0 = 0, x_1 = x$ ，且满足递推

$$
x _ {n + 1} (x _ {n + 2} + x _ {n + 1} + x _ {n}) = 3, n \in N
$$

求所有实数x的取值可能值,使得 $x_{n}>0,\forall n\in N^{+}$

12. 设 $x_{1}, x_{2}, \cdots, x_{n}$ 是实数, 且满足 $\prod_{k=1}^{n} x_{k} = 1$ , 求证

$$
\sum_ {k = 1} ^ {n} \frac {x _ {k} ^ {2}}{x _ {k} ^ {2} - 2 x _ {k} \cos {\frac {2 \pi}{n}} + 1} \geq 1
$$

13. 设 $a_{1}, a_{2}, \cdots, a_{n} > 0, n \geq 3$ , 求证

$$
\frac {a _ {1}}{a _ {2} + a _ {3}} + \frac {a _ {2}}{a _ {3} + a _ {4}} + \dots + \frac {a _ {n}}{a _ {1} + a _ {2}} > 0. 4 5 n
$$

14. 设 $k$ 是一个正整数, $p(x)$ 是 $n$ 次多项式, 其各项系数分别为 $-1, 1$ 或 $0$ , 满足 $(x - 1)^{k} | p(x)$ , 且

$$
{\frac {q}{\ln q}} <   {\frac {k}{\ln (n + 1)}}
$$

证明:q次单位根是多项式 $p(x)$ 的根(q为奇素数)

15. 设 $a_{1}, a_{2}, \cdots, a_{n}$ 是正实数, 且 $S = a_{1} + a_{2} + \cdots + a_{n}$ , 求证

$$
\frac {1}{n} \sum_ {i = 1} ^ {n} \frac {1}{a _ {i}} + \frac {n (n - 2)}{S} \geq \sum_ {i \neq j} \frac {1}{S + a _ {i} - a _ {j}}
$$

16. 假设 $a_{0}, a_{1}, \cdots, a_{n-1}$ 是n个模都不超过1的n个复数,证明:多项式

$$
P (x) = a _ {0} + a _ {1} x + a _ {2} x ^ {2} + \dots + a _ {n} x ^ {n}, \quad a _ {n} = 1
$$

以1为零点的重数至多为 $5\sqrt{n}$

17. 设 f 是次数不超过 N 的复系数多项式, 满足 $f(1) = 0$ , 且

$$
\left| f \left(e ^ {\frac {i (2 k + 1) \pi}{N}}\right) \right| \leq 1, \forall k = 0, 1, 2, \dots , N - 1
$$

求证:对任意复数 $|z|=1$ ,恒有

$$
\left| \frac {f (z)}{z - 1} \right| \leq \frac {N}{2}, z \neq 1
$$

并说明 $\frac{N}{2}$ 是最佳的

18. 设 $n \geq 2$ 的正整数, 求证:

$$
\sum_ {i = 0} ^ {n} \binom {n} {i} i ^ {n - i} (n - i) ^ {i} \leq \frac {1}{2} n ^ {n}
$$

19. 设正整数 $k, n \geq 2$ 均为给定的,且 $0 \leq a_{1} \leq a_{2} \leq \cdots \leq a_{2017n}$ ,若有

$$
\left(\sum_ {i = 1} ^ {2 0 1 7 n} a _ {i}\right) ^ {2 0 1 7} \geq k _ {n} \sum_ {i = 1} ^ {n} \left(\prod_ {j = 0} ^ {2 0 1 6} a _ {n j + i}\right)
$$

恒成立,求 $k_{n}$ 的最大值

20. 设 $z_{1}, z_{2}, z_{3}$ 是三个复数, 且满足 $|z_{i}| < 1, i = 1, 2, 3$ , 求下列式子最大可能值

$$
f (z _ {1}, z _ {2}, z _ {3}) = \sum_ {1 \leq j <   k \leq 3} \ln | z _ {j} - z _ {k} | + \sum_ {j, k = 1} ^ {3} \ln \left(\sqrt {| 1 - z _ {j} \overline {{z _ {k}}} |}\right)
$$

21. 设 $\{y_{n}\}$ 是实数列, 且满足

$$
\sum_ {i = 1} ^ {n} y _ {i} ^ {3} = 0
$$

证明:必存在正整数 $k\in\{1,2,3,\cdots,n\}$ 使得

$$
\left| y _ {k} \right| \geq \sqrt [ 4 ]{\frac {2 7}{4}} \cdot \frac {\left| y _ {1} + y _ {2} + \cdots + y _ {n} \right|}{n}
$$

22. 设 $a_{i}, b_{i} \in R, n \geq 2$ , 求证

$$
\left(\sum_ {i = 1} ^ {n} a _ {i} ^ {2}\right) \left(\sum_ {i = 1} ^ {n} b _ {i} ^ {2}\right) + \left(\sum_ {i = 1} ^ {n} a _ {i} b _ {i}\right) ^ {2} \geq \sqrt {\left(\sum_ {i = 1} ^ {n} a _ {i} ^ {4}\right) \left(\sum_ {i = 1} ^ {n} b _ {i} ^ {4}\right)} + \sum_ {i = 1} ^ {n} a _ {i} ^ {2} b _ {i} ^ {2}
$$

23. 对于任意非负整数 $n, N, k_i$ , 证明:

$$
\sum_ {k _ {1} + k _ {2} + \dots + k _ {N} = n} \left(\frac {1}{\prod_ {j = 1} ^ {N} [ (N - 1) k _ {j} + 1 ]}\right) \leq 1
$$

24. 设 $x_{i} \geq 0$ ，且

$$
\sum_ {k = 1} ^ {n - 1} \sqrt [ 5 ]{x _ {k} ^ {6}} = \frac {2}{n (n - 1)}, \quad S = \sum_ {k = 1} ^ {n - 1} x _ {k}, \quad , x _ {0} = 0, (n \geq 3)
$$

求证

$$
\sum_ {k = 1} ^ {n - 1} \left(\prod_ {l = 1} ^ {k} \left(S - \sum_ {i = 0} ^ {l - 1} x _ {i}\right)\right) ^ {\frac {2}{k}} <   e (\binom {n} {2}) ^ {- 1}
$$

25. 设 $n \geq 3$ 的正整数, 求证

$$
\sum_ {1 \leq i <   j \leq n} \left| \cos \left(\frac {(i - j) \pi}{n}\right) \right| \leq \frac {1}{2} n (n - 2)
$$

26. 设 $a, b, c, d$ 都不为0, 设 $f: R \to R$ . 且 $f(x) = a + b \cos(2x) + c \sin(5x) + d \cos(8x)$ , 假设存在实数 $t$ . 使得 $f(t) = 4a$ , 证明: 存在实数 $s$ , 使得 $f(s) < 0$

27. :设 $P(x)$ 是多项式且次数不到2n,求证:

$$
| P (n) | \leq (2 \sqrt {n} - 1) \max \left(| P (0) |, | P (1) |, | P (2) |, \dots , | P (n - 1) |, | P (n + 1) |, | P (n + 2) |, \dots , | P (2 n) |\right)
$$

28. 设 $n$ 是正整数, 且

$$
A _ {r} = \sum_ {k = 0} ^ {r} \binom {r} {k} p _ {k}, B _ {r} = \sum_ {k = 0} ^ {r} \binom {r} {k} w _ {k}, C _ {r} = \sum_ {k = 0} ^ {r} \binom {r} {k} q _ {k}, p _ {k}, w _ {k}, q _ {k} \geq 0
$$

求证:

$$
T (A, B, C) = \frac {1}{n} \sum_ {r = 0} ^ {n - 1} A _ {r} B _ {r} C _ {r} - \frac {1}{n ^ {3}} \sum_ {r = 0} ^ {n} A _ {r} \sum_ {r = 0} ^ {n} B _ {r} \sum_ {r = 0} ^ {n} C _ {r} \leq \frac {1}{8} A _ {n} B _ {n} C _ {n}
$$

29. 设 $x_{1}, x_{2}, \cdots, x_{n} \geq 0, x_{1} + x_{2} + \cdots + x_{n} = 1, n \geq 5$ , 求

$$
x _ {1} x _ {2} x _ {3} x _ {4} + x _ {2} x _ {3} x _ {4} x _ {5} + x _ {3} x _ {4} x _ {5} x _ {6} + \dots + x _ {n} x _ {1} x _ {2} x _ {3}
$$

的最大值?

30. 设多项式 $f(x) = \sum_{i=1}^{n} a_i x^i (a_i \in R) = 0$ 只有实数根, 求证:

$$
g (x) = \sum_ {i = 1} ^ {n} {\binom {n} {i}} a _ {i} x ^ {i} = 0
$$

只有实数根.

31. 设 $\langle x\rangle$ 表示最接近实数x的整数,求证: $\langle f(n)\rangle$ 能整除n-1,但不能整除n

其中

$$
f (n) = \left(\frac {n !}{e} \left(\tan \frac {\pi}{1 6} \cdot \frac {\sum_ {k = 1} ^ {n ^ {2} + 2 n} \sqrt {\sqrt {2 n + 2} + \sqrt {n + 1 + \sqrt {k}}}}{\sum_ {k = 1} ^ {n ^ {2} + 2 n} \sqrt {\sqrt {2 n + 2} - \sqrt {n + 1 + \sqrt {k}}}}\right)\right)
$$

32. 设 $m, n$ 是两个自然数, 且 $S_{m,n}$ 表示不定方程

$$
\frac {1}{x _ {1}} + \frac {1}{x _ {2}} + \dots + \frac {1}{x _ {m}} = \frac {1}{n}
$$

正整数解 $x=(x_{1},x_{2},\cdots,x_{m})$ 解的个数,定义

$$
k _ {m, n} = \max _ {x \in S _ {m, n}} \left(\max _ {1 \leq j \leq m} x _ {j}\right)
$$

对于所有 $m\geq2,n\in N^{+}$ ,求证: $\frac{k_{m,n}}{k_{m-1,n}(k_{m-1,n}+1)}$ 恒是定值.

33. 对给定的正整数 $a_1, a_2, a_3, a_1 \geq a_2 \geq a_3$ , 若不定方程

$$
\frac {a _ {1}}{x _ {1}} + \frac {a _ {2}}{x _ {2}} + \frac {a _ {3}}{x _ {3}} = 1
$$

的正整数解 $(x_{1},x_{2},x_{3})$ 的组数为 $N(a_{1},a_{2},a_{3})$ .证明

$$
N (a _ {1}, a _ {2}, a _ {3}) \leq 6 a _ {1} a _ {2} a _ {3} (3 + \ln (2 a _ {1}))
$$

34. 设 $a, b, c, n$ 是正整数, 且 $b, n \geq 2$ 求证: 对于 $n \equiv 4 \pmod{8}$ , 不定方程

$$
\left(\sum_ {k = 1} ^ {n} k ^ {a}\right) ^ {b} = \sum_ {k = 1} ^ {n} k ^ {c}
$$

的解是有限的,并求出其所有解?

35. 设 $\pi_{n}(x)$ 是次数不超过 n 的实系数多项式, 且满足

$$
| \pi_ {n} (x) | \leq \sqrt {1 - x ^ {2}}, - 1 \leq x \leq 1
$$

求证:

$$
| \pi_ {n} ^ {\prime} (x) | \leq 2 (n - 1)
$$

36. 给定正整数 $m, n, p$ , 满足 $m \leq n \leq p$ , 和正实数 $a, x_1, x_2, \cdots, x_p$ , 求证

$$
n ^ {m a} \binom {p} {m} \sum_ {I \subseteq \{1, 2, \dots , p \}, | I | = n} \left(\sum_ {i \in I} x _ {i}\right) ^ {- m a} \leq \binom {p} {n} \sum_ {I \subseteq \{1, 2, \dots , p \}, | I | = m} \left(\prod_ {i \in I} (x _ {i}) ^ {- a}\right)
$$

37. : 设 $\varphi(z)=\sum_{n=1}^{\infty}a_{n}z^{n}$ 是在 z=0 附近具有导数的, 令 $\psi(z)=e^{\varphi(z)}=\sum_{n=0}^{\infty}\beta_{n}z^{n}$

则对 $n \geq 1$ ,有

$$
\sum_ {k = 0} ^ {n} | \beta_ {k} | ^ {2} <   (n + 1) \exp \left[ \frac {1}{n + 1} \sum_ {k = 1} ^ {n} (n + 1 - k) \left(k | a _ {k} | ^ {2} - \frac {1}{k}\right) \right]
$$

38. :设x,y,z是三个复数,定义

$$
d (x, y) = \frac {| x - y |}{\sqrt {(1 + | x | ^ {2}) (1 + | y | ^ {2})}}
$$

求证

$$
d (x, y) \leq d (x, z) + d (z, y)
$$

39. 如果非实数 $z$ 满足 $Re(z) \geq 1$ , 那么对任意正整数 $n$ , 有

$$
\left| z ^ {n + 1} - 1 \right| > \left| z ^ {n} \right| \cdot \left| z - 1 \right|
$$

40. 设 $a, b, c > 0$ , 且一元二次方程 $ax^2 + bx + c = 0$ 有两个不相等的实数根. 数列 $\{a_n\}, \{b_n\}$ 满足

$$
b _ {0} = c a _ {0} > 0, b _ {1} = c a _ {1} - b a _ {0} > 0, b _ {n} = a a _ {n - 2} - b a _ {n - 1} + c a _ {n}, n \geq 2
$$

若 $b_{n}>0$ ,是否有 $a_{n}>0$ ,并证明你的结论.

41. 设 $a_{i}>0, b_{i}>0$ ，其中 $i=1,\cdots,n$ ，记 $A=\frac{\max a_{k}}{\min a_{k}}, B=\frac{\max b_{k}}{\min b_{k}}$ 和 $\frac{1}{p}+\frac{1}{q}=1, p>1$ 。求证：

$$
\left(\sum_ {i = 1} ^ {n} a _ {i} ^ {p}\right) ^ {1 / p} \left(\sum_ {i = 1} ^ {n} b _ {i} ^ {q}\right) ^ {1 / q} \leq \frac {1}{p ^ {1 / p}} \frac {1}{q ^ {1 / q}} \frac {A ^ {p} B ^ {q} - 1}{(B A ^ {p} - A) ^ {1 / q} (A B ^ {q} - B) ^ {1 / p}} \sum_ {i = 1} ^ {n} a _ {i} b _ {i}
$$

42. 设 $0 < a_{1} < a_{2} < \cdots < a_{k}$ 是整数, 且 $b_{0}, b_{1}, \cdots, b_{k}$ 是实数满足 $b_{k} = \pm 1$ , 且 $b_{0} + b_{1} x^{a_{1}} + \cdots + b_{k} x^{a_{k}}$ 的所有根的模为 1, 设函数 f 是有界的且满足

$$
b _ {0} f (x) + b _ {1} f (x + a _ {1}) + \dots + b _ {k} f (x + a _ {k}) = 0
$$

求证: f 是周期函数

43. 设 $p \geq 3$ 的素数且 $p \equiv 1 \pmod{4}$ , 若 $F_{n}$ 是第 $\mathfrak{n}$ 个斐波那契数列, 即 $F_{0} = 0, F_{1} = 1, F_{2} = 1, F_{n+1} = F_{n} + F_{n-1}$ 求证:

$$
\sum_ {k = 0} ^ {p - 1} \left(\binom {2 k} {k}\right) ^ {2} \frac {F _ {k}}{1 6 ^ {k}} \equiv 0 \pmod {p ^ {2}}
$$

44. 设 $n$ 是给定的正整数, 定义一种运算: 设 $x, y$ 满足 $x^{2} = y^{2} = 1$ , $xy \neq yx$ , 求

$$
((1 + x) (1 + y)) ^ {n}
$$

的常数项,

注： $(x + y)(x + y) = x^{2} + xy + yx + y^{2} = 1 + xy + yx + 1 = 2 + xy + yx$ 的常数项为2

$$
(1 + x) (1 + y) = 1 + x + y + x y
$$

的常数项为1

$$
((1 + x) (1 + y)) ^ {2} = (1 + x + y + x y) (1 + x + y + x y) = 1 + x + y + x y + x + 1 + x y + y + y + y x + 1 + y x y + x y + x y x + x + x y x y
$$

的常数项为3

45. 设 $a_{i} > 0 (i = 1,2,\cdots,n)$ , 记 $A = \sum_{i=1}^{n} a_{i}, x_{i} = \frac{a_{i}}{A - a_{i}}, i = 1,2,\cdots,n$ 证明: 序列 $\left\{\frac{(n-1)^k \sigma_k}{\binom{n}{k}}\right\}$ 单调不增, 其中 $\sigma_k = \sum_{1 \leq i_1 < i_2 < \cdots < i_k \leq n} x_{i_1} x_{i_2} \cdots x_{i_k}$

46. 设数列 $\{x_{n}\}, \{y_{n}\}$ 满足 $x_0 = 1, y_0 = 2020$ , 且有

$$
\left\{ \begin{array}{l} x _ {n + 1} = x _ {n} - (x _ {n} y _ {n} + x _ {n + 1} y _ {n + 1} - 2) (y _ {n} + y _ {n + 1}) \\ y _ {n + 1} = y _ {n} - (x _ {n} y _ {n} + x _ {n + 1} y _ {n + 1} - 2) (x _ {n} + x _ {n + 1}) \end{array} \right.
$$

求证:

$$
x _ {n} ^ {2} \leq 2 0 2 0, \forall n \in N ^ {+}
$$

47. 数列 $\{x_{n}\}$ 满足 $x_{1}=0,x_{2}=1$ ,且当 $n\geq2$ 时,有

$$
x _ {n + 1} = \left(1 + \frac {1}{n}\right) x _ {n} - x _ {n - 1}
$$

求证:

$$
x _ {n} ^ {2} \leq \frac {8 n}{3 n + 1}
$$

48. 设正整数q且不是立方,设 $\|x\|$ 表示与实数x的最近整数的距离,求证:存在正常数C,使得

$$
| | n q ^ {\frac {1}{3}} | | + | | n q ^ {\frac {2}{3}} | | \geq \frac {C}{\sqrt {n}}
$$

49. 设 $n \geq 2$ , 且实系数多项式 $p(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ 是实系数多项式, 若正整数 $k$ 满足 $(x - 1)^{k+1}|p(x)$ , 求证

$$
\sum_ {l = 0} ^ {n - 1} \left| a _ {l} \right| > 1 + \frac {2 k ^ {2}}{n}
$$

50. 设 $n$ 次复系数多项式 $p(z)$ 的所有根都在复平面的单位圆周上, 证明: 对每个实数 $c \geq 0$ , 多项式 $2z(z - 1)p'(z) + ((c - n)z + (c + n))p(z)$ 的所有根也在单位圆圆周上.

51. 设n是大于2的正整数,且 $x_{i}(i=1,2,\cdots,n)$ 是n个互不相等的实数,求证:

$$
\sum_ {1 \leq i <   j \leq n} \left(\frac {1}{(x _ {i} - x _ {j}) ^ {2}} + (x _ {i} - x _ {j}) ^ {2}\right) \geq \frac {n (n - 1)}{2} \sqrt {2 n}
$$

52. 设 $a_{i}>0, i=1,2,\cdots,n$ , 记 $s_{1},s_{2},\cdots,s_{n}$ 表示 $a_{1},a_{2},\cdots,a_{n}$ 的n个初等对称式, 求证

$$
\sum_ {k = 1} ^ {n} (- 1) ^ {k - 1} k ^ {2} (s _ {1}) ^ {n - k} s _ {k} \geq 0
$$

53. 求所有整系数多项式 $P(x)$ ，使得对所有正整数n，以及素数满足： $p\nmid nP(n)$ 和 $ord_{p}(n)\geq ord_{p}(P(n))$

54. 设 $f(x) = \sum_{k=0}^{n} \binom{n}{k} a_k x^k (1 - x)^{n-k}, 0 \leq a_k \leq 1, 0 < x < 1$ , 求证

$$
[ f ^ {\prime} (x) ] ^ {2} \leq \frac {n f (x) [ 1 - f (x) ]}{x (1 - x)}
$$

55. 设 $\{T_{n}\}$ 满足

$$
T _ {0} = 0, T _ {1} = T _ {2} = 1, T _ {n + 3} = T _ {n + 2} + T _ {n + 1} + T _ {n}
$$

.求证:存在无穷多个正整数n,使得

$$
n | T _ {n}
$$

56. 设 $x_{i} \in [0, 0.5], i = 1, 2, \cdots, n$ , 求证

$$
\frac {x _ {1} ^ {4} + x _ {2} ^ {4} + \cdots + x _ {n} ^ {4}}{x _ {1} x _ {2} \cdots x _ {n}} \geq \frac {(1 - x _ {1}) ^ {4} + (1 - x _ {2}) ^ {4} + \cdots + (1 - x _ {n}) ^ {4}}{(1 - x _ {1}) (1 - x _ {2}) \cdots (1 - x _ {n})}, n \geq 4
$$

57. 求所有函数 $f: R \to R$ , 且对任意实数 $x, y$ , 有

$$
f (2 x y + f (x + y)) = x f (y) + y f (x)
$$

58. 设n是大于2的正整数, $x_{i}(i=1,2,\cdots,n)$ 是非负实数,满足任意两个数不能同时为0,求证

$$
(n - 2) \sum_ {1 \leq i <   j \leq n} \frac {1}{x _ {i} ^ {2} + x _ {j} ^ {2}} + 8 \sum_ {1 \leq i <   j <   k \leq n} \frac {1}{x _ {i} ^ {2} + x _ {j} ^ {2} + x _ {k} ^ {2}} \geq 6 \sum_ {1 \leq i <   j <   k \leq n} \frac {1}{x _ {i} x _ {j} + x _ {j} x _ {k} + x _ {i} x _ {k}}
$$

59. 设K是给定的正整数,设 $(a_{0},a_{1},\cdots)$ 是实数列,满足 $a_{0}=-1$ 且

$$
\sum_ {i _ {0}, i _ {1}, \dots , i _ {K} \geq 0, i _ {0} + i _ {1} + \dots + i _ {K} = n} \frac {a _ {i _ {1}} a _ {i _ {2}} \cdots a _ {i _ {K}}}{i _ {0} + 1} = 0, \forall n \in N ^ {+}
$$

求证 $a_{n} > 0$ $n\geq 1$

60. 设 $n \geq 3$ 是给定的正整数, 且 $x_{i} \geq 0 (i = 1, 2, \cdots, n)$ , 且 $x_{1} + x_{2} + \cdots + x_{n} = 1$ , 试求

$$
F = x _ {1} ^ {3} x _ {2} ^ {2} + x _ {2} ^ {3} x _ {3} ^ {2} + \dots + x _ {n} ^ {3} x _ {1} ^ {2} + n ^ {2 (n - 1)} x _ {1} ^ {3} x _ {2} ^ {3} \dots x _ {n} ^ {3}
$$

的最大可能值

61. 设 $x_{i}, y_{i} > 0, i = 1, 2, \dots, n$ , 满足 $\sum_{i=1}^{n} x_{i}^{2} = \sum_{i=1}^{n} y_{i}^{2} = 1$ , 记 $a = \sum_{i=1}^{n} x_{i} y_{i}$ , 证明

$$
\sum_ {i = 1} ^ {n} x _ {i} ^ {2} \ln x _ {i} + \sum_ {i = 1} ^ {n} y _ {i} ^ {2} \ln y _ {i} \leq a ^ {2} \sum_ {i = 1} ^ {n} \frac {x _ {i} y _ {i}}{a} \ln \frac {x _ {i} y _ {i}}{a}
$$

62. 设 $\{u_n\}$ 是一个单调递增的数列, 满足 $u_0 = 0, p$ 是一个正整数, 求证

$$
\sum_ {n = 1} ^ {N} \frac {(u _ {n} - u _ {n - 1}) ^ {p - 1} (n - 1) ! A _ {n}}{(n + p) !} \geq 0
$$

其中 $A_{n}=(np-n+1+\frac{n}{p})u_{n}-(np-n+1+\frac{n}{p}+p)u_{n-1}.$

63. 设n,m是正整数,且 $n\geq2m$ ,设 $a_{1}\leq a_{2}\leq\cdots\leq a_{n}$ 是正整数,满足

$$
a _ {n} <   m + \frac {1}{2 m} \left(\sum_ {i = 1} ^ {m} \binom {n} {2 i} \binom {2 i} {i}\right)
$$

求证:存在两组不同的数组 $(\xi_{1},\xi_{2},\cdots,\xi_{n}),(\mu_{1},\mu_{2},\cdots,\mu_{n})$ 其元素均为0,1,2且满足

$$
\sum_ {i = 1} ^ {n} \xi_ {i} = \sum_ {i = 1} ^ {n} \mu_ {i} \leq 2 m, \quad \sum_ {i = 1} ^ {n} \xi_ {i} a _ {i} = \sum_ {i = 1} ^ {n} \xi_ {i} a _ {i}
$$

64. 设 $m, n, r$ 是整数, 且满足 $0 \leq r \leq n \leq m - 2$ , 求证

$$
\sum_ {k = 0} ^ {r} (- 1) ^ {k} \binom {m + n - 2 (k + 1)} {n} \binom {r} {k} > 0
$$

65. 设 $z_{1}, z_{2}, \cdots, z_{n}$ 是n个复数,且满足

$$
\sum_ {i = 1} ^ {n} z _ {i} = 0, \quad \sum_ {i = 1} ^ {n} | z _ {i} | ^ {2} = 1
$$

设 $r_{ij}(i<j)$ 是非负实数,求证

$$
\sum_ {i <   j} | z _ {i} - z _ {j} | ^ {2} r _ {i j} \leq \sum_ {i <   j} r _ {i j} + \max _ {i <   j} (r _ {i j})
$$

66. 设 $\{a_{n}\}$ 整数列, 且满足

$$
0 \leq a _ {n - 1} + \frac {1 - \sqrt {5}}{2} a _ {n} + a _ {n + 1} <   1, \forall n \geq 2
$$

证明: $\{a_{n}\}$ 是周期列

67. 设 $a_{i} > 0$ ，则有

$$
\frac {a _ {1}}{a _ {1} + a _ {2}} + \frac {a _ {2}}{a _ {2} + a _ {3}} + \dots + \frac {a _ {n}}{a _ {n} + a _ {1}} \geq \frac {a _ {1} + a _ {2} + \dots + a _ {n}}{a _ {1} + a _ {2} + \dots + a _ {n} - (n - 2) \sqrt [ n ]{a _ {1} a _ {2} \dots a _ {n}}}
$$

68. 若 $n, k$ 是任意给定的正整数, 假设 $P$ 是以首一系数的且次数为 $\mathbf{n}$ 的复系数多项式构成的集合, 求

$$
\min _ {F \in P} \left(| F (1) | ^ {2} + | F (2) | ^ {2} + \dots + | F (n + k + 1) | ^ {2}\right)
$$

的可能值.

69. 设 $a_{1}, a_{2}, \cdots, a_{n} \neq 0$ 的实数,且满足

$$
a _ {1}, a _ {2}, \dots , a _ {n} \geq n - 1 - \sqrt {n ^ {2} - n + 1}, a _ {1} + a _ {2} + \dots + a _ {n} = n
$$

求证

$$
\frac {1}{a _ {1} ^ {2}} + \frac {1}{a _ {2} ^ {2}} + \dots + \frac {1}{a _ {n} ^ {2}} \geq n
$$

70. 设 $a_{i}>0(i=1,2,\cdots,n)$ ，设 $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$ ，求证

$$
\sum_ {k = 1} ^ {n} \left(\frac {k a _ {k}}{S _ {k}}\right) ^ {\alpha} \cdot \frac {1}{a _ {k}} \leq 2 ^ {\alpha} \sum_ {k = 1} ^ {n} \frac {1}{a _ {k}}
$$

其中 $0 \leq \alpha \leq 2$

71. 设

$$
a _ {1} = 1; a _ {n} = \frac {a _ {[ \frac {n}{2} ]}}{2} + \frac {a _ {[ \frac {n}{3} ]}}{3} + \dots + \frac {a _ {[ \frac {n}{n} ]}}{n}
$$

.求证

$$
a _ {2 n} <   2 a _ {n} (\forall n \in \mathbb {N})
$$

72. 设 $n$ 是正整数, 求证

$$
(1 + 2 n) ^ {n} \geq 1 ^ {n} + 2 ^ {n} + 4 ^ {n} + 6 ^ {n} + \dots + (2 n) ^ {n}
$$

73. 设实系数多项式

$$
p (x) = \sum_ {i = 0} ^ {n} {\binom {n} {i}} a _ {i} x ^ {i} (1 - x) ^ {n - i}
$$

满足

$$
a _ {0} + \sum_ {a _ {i} <   0} (1 - \frac {i}{n}) {\binom {n} {i}} a _ {i} > 0, \quad a _ {n} + \sum_ {a _ {i} <   0} \frac {i}{n} {\binom {n} {i}} a _ {i} > 0
$$

求证:

$$
p (x) > 0, \forall x \in [ 0, 1 ]
$$

74. 设 $k, n$ 是正整数, 且 $v_{1}, v_{2}, \cdots, v_{n}$ 是 $n$ 个单位向量, 满足对任意 $k$ 个中都存在两个是正交的, 求证

$$
| \sum_ {i = 1} ^ {n} v _ {i} | \leq e n ^ {1 - \frac {1}{k}}
$$

这里的 $e=2.718\cdots$ ，正交的定义:若向量a,b满足 $a\cdot b=0$ ,则称a,b为正交的

75. 定义

$$
P _ {n} = \{p: p (x) = \sum_ {k = 0} ^ {n} a _ {k} x ^ {k}, a _ {k} \in R \}, \alpha , \beta \in R
$$

假设 $x^{2}-\alpha x+\beta$ 没有非负实数根,若存在两个多项式 $p(x),q(x)$ ，满足

$$
(x ^ {2} - \alpha x + \beta) q (x) = p (x)
$$

其中 $p,q\in P_{m}$ 且系数都是非负系数.求证:

$$
m \leq 1 0 \left(1 - \frac {\alpha^ {2}}{\beta}\right) ^ {- \frac {1}{2}}
$$

76. 设数列 $\{a_{n}\}$ ，满足 $a_{n}\geq0$ ，且

$$
\frac {a _ {i - 1} + a _ {i + 1}}{2} \leq a _ {i}, i = 1, 2, \dots , n - 1
$$

其中 $a_{0}=0$ 求证:存在一个大于0的正常数c,使得对任意正正整数n有

$$
\left(\sum_ {k = 1} ^ {n} a _ {k}\right) ^ {2} \geq \frac {3 n - c}{4} \sum_ {k = 1} ^ {n} a _ {k} ^ {2}
$$

77. 设 $a_{1}, a_{2}, \cdots, a_{n}$ 是正整数, 求证

$$
\sum_ {k = 1} ^ {n} \frac {\sqrt {a _ {k}}}{1 + a _ {1} + a _ {2} + \cdots + a _ {k}} \leq \sum_ {k = 1} ^ {n} \frac {1}{k}
$$

78. 设 $A_{1}, A_{2}, \ldots, A_{n}$ 是任意n个有限非空集合. 求证:

$$
\sum_ {i = 1} ^ {n} \frac {1}{| A _ {i} |} - \sum_ {1 \leq i <   j \leq n} \frac {1}{| A _ {i} \cup A _ {j} |} + \sum_ {1 \leq i <   j <   k \leq n} \frac {1}{| A _ {i} \cup A _ {j} \cup A _ {k} |} - \dots + \frac {(- 1) ^ {n - 1}}{| A _ {1} \cup \cdots \cup A _ {n} |} > 0
$$

79. 设 $x_{i} \in R, i = 1, 2, \cdots, n$ , 且 $p_{i} \geq 0, i = 1, 2, \cdots, n$ , 满足

$$
p _ {1} + p _ {2} + \dots + p _ {n} = 1
$$

定义

$$
S _ {k} = \sum_ {i = 1} ^ {n} p _ {i} x _ {i} ^ {k} - \left(\sum_ {i = 1} ^ {n} p _ {i} x _ {i}\right) ^ {k}
$$

证明:

$$
S _ {2 m} S _ {2 m + 2} \geq \left(1 - \frac {1}{m (2 m + 1)}\right) S _ {2 m + 1} ^ {2}, m \in N ^ {+}
$$

80. 设 $x_{i}.y_{i}\in R$ ，且

$$
\sum_ {i = 1} ^ {n} x _ {i} ^ {2} = \sum_ {i = 1} ^ {n} y _ {i} ^ {2} = 1, \sum_ {i = 1} ^ {n} x _ {i} = \sum_ {i = 1} ^ {n} y _ {i} = 0
$$

求

$$
\sum_ {1 \leq i <   j \leq n} | x _ {i} - x _ {j} | | y _ {i} - y _ {j} |
$$

的最小值

81. 设 $y_{1}, y_{2}, y_{3}, a_{1}, a_{2}, a_{3}$ 均是正实数,且满足 $y_{1}y_{2}y_{3}=a_{1}a_{2}a_{3}$ 和

$$
y _ {1} + y _ {2} + y _ {3} \geq a _ {1} + a _ {2} + a _ {3}, y _ {1} y _ {2} + y _ {2} y _ {3} + y _ {1} y _ {3} \geq a _ {1} a _ {2} + a _ {2} a _ {3} + a _ {1} a _ {3}
$$

求证

$$
(\ln y _ {1}) ^ {2} + (\ln y _ {2}) ^ {2} + (\ln y _ {3}) ^ {2} \geq (\ln a _ {1}) ^ {2} + (\ln a _ {2}) ^ {2} + (\ln a _ {3}) ^ {2}
$$

82. 设 $a_{k}, b_{k}, c_{k}, k = 1, 2, \cdots, n$ 是三个实数列, 求证

$$
\left(\sum_ {i = 1} ^ {n} a _ {i} ^ {2}\right) \left(\sum_ {i = 1} ^ {n} b _ {i} c _ {i}\right) ^ {2} + \left(\sum_ {i = 1} ^ {n} b _ {i} ^ {2}\right) \left(\sum_ {i = 1} ^ {n} a _ {i} c _ {i}\right) ^ {2} \leq \left(\sum_ {i = 1} ^ {n} a _ {i} ^ {2}\right) \left(\sum_ {i = 1} ^ {n} b _ {i} ^ {2}\right) \left(\sum_ {i = 1} ^ {n} c _ {i} ^ {2}\right) + \left| \sum_ {i = 1} ^ {n} a _ {i} b _ {i} \right| \sqrt {\sum_ {i = 1} ^ {n} a _ {i} ^ {2} \sum_ {i = 1} ^ {n} b _ {i} ^ {2}} \sum_ {i = 1} ^ {n} c _ {i} ^ {2}
$$

83. 设 $z_{i}(i=1,2,\cdots,2m)$ 为不全为实数的复数,且满足 $|z_{k}|\leq n(k=1,2,\cdots,2m)$ 和 $\sum_{k=0}^{2m}z_{k}=0$ ,我们记

$$
t _ {k} = \sum_ {j = k} ^ {k + m - 1} z _ {j} (k = 1, 2, \dots , 2 m), z _ {2 m + l} = z _ {l} (l = 1, 2, \dots , m - 1)
$$

求证

$$
\sum_ {k = 1} ^ {m} | t _ {k} | ^ {2} \leq m \left(n \csc {\frac {\pi}{2 m}}\right) ^ {2}
$$

84. 设 $x_{i} > 0, i = 1,2,\dots ,n,n\geq 3,x_{n + 1} = x_1,s,m\in R,$ 满足 $s\geq 1,m\geq s + 1$ ，求证

$$
\sum_ {i = 1} ^ {n} \frac {x _ {i} ^ {m}}{x _ {i + 1} ^ {s}} \geq \sum_ {i = 1} ^ {n} x _ {i} ^ {m - s} + \frac {4 (x _ {1} - x _ {2}) ^ {2}}{(m - s) \sum_ {i = 1} ^ {n} x _ {i} ^ {s + 2 - m}}
$$

85. 对任意实数 $x_{1},x_{2},\cdots,x_{n}$ ，则一定存在正整数 $k\in\{1,2,\cdots,n\}$ 满足

$$
\sum_ {i = 1} ^ {n} \left\{k x _ {i} \right\} \left(1 - \left\{k x _ {i} \right\}\right) <   \frac {n + 1}{6}
$$

其中 $\{x\}=x-\lfloor x\rfloor.$

86. 给定整数 $n \geq 2$ , 求最大的实数 $\lambda$ , 使得对任意模不大于1的复数 $z_{i}$ , 都有

$$
1 + | \sigma_ {1} | + | \sigma_ {2} | + \dots + | \sigma_ {n} | \geq \lambda (| z _ {1} | + | z _ {2} | + \dots + | z _ {n} |)
$$

其中对 $1 \leq k \leq n, \sigma_{k} = \sum_{1 \leq i_{1} < i_{2} < \dots < i_{k} \leq n} z_{i_{1}} z_{i_{2}} \cdots z_{i_{k}}$

87. 设m,n为整数, $1\leq m<n$ ,令 $a_{ij},i=1,2,\cdots,m;j=1,2,\cdots,n$ .为mn个不全为零的整数,并记 $a=\max_{1\leq i\leq m,1\leq j\leq n}|a_{ij}|$ .
求证:方程组

$$
\left\{ \begin{array}{l} a _ {1 1} x _ {1} + a _ {1 2} x _ {2} + \dots + a _ {1 n} x _ {n} = 0 \\ a _ {2 1} x _ {1} + a _ {2 2} x _ {2} + \dots + a _ {2 n} x _ {n} = 0 \\ \vdots \\ a _ {m 1} x _ {1} + a _ {m 2} x _ {2} + \dots + a _ {m n} x _ {n} = 0 \end{array} \right.
$$

满足条件: $|x_{j}|\leq[(2na)^{\frac{m}{n-m}}],1\leq j\leq n$

88. 数列 $\{c_{n}\}, \{d_{n}\}$ 满足 $c_{0} = 0, d_{0} = 1$ , 且满足递推关系

$$
\left\{ \begin{array}{l} (n + 1) c _ {n} = n c _ {n - 1} + 2 n d _ {n - 1} \\ d _ {n} = 2 c _ {n - 1} + d _ {n - 1} \end{array} \right.
$$

.求证: $c_{n},d_{n}$ 均是整数

89. 设斐波那契数列 $F_{1}=1, F_{2}=1, F_{n}=F_{n-1}+F_{n-2}$ ，求证

$$
\sqrt {\frac {F _ {n + 3}}{F _ {n}}} + \sqrt {\frac {F _ {n} + F _ {n + 2}}{F _ {n + 1}}} > 1 + 2 \left(\sqrt {\frac {F _ {n}}{F _ {n + 3}}} + \sqrt {\frac {F _ {n + 1}}{F _ {n} + F _ {n + 2}}}\right)
$$

90. 设 $F_{n}$ 表示第n个斐波那契数列, 即 $F_{n}=F_{n-1}+F_{n-2}, F_{0}=0, F_{1}=F_{2}=1$ . 求证

$$
2 (F _ {n} ^ {4} + F _ {n + 1} ^ {4} + F _ {n + 2} ^ {4}) \left(\frac {1}{F _ {n} ^ {2}} + \frac {1}{F _ {n + 1} ^ {2}} + \frac {1}{F _ {n + 2} ^ {2}}\right) ^ {2} > 8 1
$$

91. 设多项式 $f(x) = \sum_{i=0}^{3} a_i x^i (a_i > 0)$ 有三个实数根, 求证: 若实数 $m \geq 1$ , 则

$$
g (x) = \sum_ {i = 0} ^ {3} a _ {i} ^ {m} x ^ {i}
$$

也只有实数根.

92. 设 $f$ 为定义于 $R^{+}$ 上的实值函数, 满足

$$
f \left(\frac {x + y}{2}\right) + f \left(\frac {2 x y}{x + y}\right) = f (x) + f) u), \forall x, y > 0
$$

求证:对任意x,y>0,恒有

$$
2 f (\sqrt {x y}) = f (x) + f (y)
$$

93. 求 $f: R \to R$ , 满足对任意实数 $x, y, z, t$ 都有

$$
(f (x) + f (z)) (f (y) + f (t)) = f (x y - z t) + f (x t + y z)
$$

94. 设 $f: N \to N$ , 且对任意 $p, q, r \geq 2$ , 有

$$
f ^ {p q r - p} (p q r) + f ^ {p q r - q} (p q r) + f ^ {p q r - r} (p q r) = p + q + r
$$

其中 $f^{1}(n)=f(n),f^{k}(n)=f(f^{k-1}(n))$ .求 $f(x),(x\geq3)$

95. 设 f 是实值函数, 若对于任意 x, y > 0, 有

$$
f (x y) = x f (y) + y f (x), \quad f (x + 1) \leq f (x)
$$

且 $f(0.5)=0.5$ ,求证:

$$
f (x) + f (1 - x) \geq - x \log_ {2} x - (1 - x) \log_ {2} (1 - x), 0 <   x <   1
$$

96. 设 $0 < a_{1} \leq a_{2} \leq \cdots \leq a_{n}$ , 求证

$$
\frac {1}{2 n ^ {2} a _ {n}} \sum_ {1 \leq i <   j \leq n} (a _ {i} - a _ {j}) ^ {2} \leq \frac {a _ {1} + a _ {2} + \cdots + a _ {n}}{n} - \sqrt [ n ]{a _ {1} a _ {2} \cdots a _ {n}} \leq \frac {1}{2 n ^ {2} a _ {1}} \sum_ {1 \leq i <   j \leq n} (a _ {i} - a _ {j}) ^ {2}
$$

97. 设x,y,z是任意给定的正数,且在任意三角形 $\Delta ABC$ ,求证:

$$
(\sin A) ^ {x} \cdot (\sin B) ^ {y} \cdot (\sin C) ^ {z} \leq \sqrt {\frac {x ^ {x} y ^ {y} z ^ {z} (x + y + z) ^ {x + y + z}}{(x + y) ^ {x + y} (y + z) ^ {y + z} (z + x) ^ {z + x}}}
$$

98. 设 $a_{i}, b_{j}$ 是正实数,且对任意正整数i, j恒有 $a_{i} < b_{j}$ ,求证

$$
\sqrt {\frac {1}{3 n} \sum_ {k = 1} ^ {n} (a _ {k} ^ {2} + a _ {k} b _ {k} + b _ {k} ^ {2})} \geq \sqrt [ 3 ]{\frac {1}{2 n} \sum_ {k = 1} ^ {n} (a _ {k} + b _ {k}) a _ {k} b _ {k}}
$$

99. 设 $z_{1}, z_{2}, \cdots, z_{n} \in C (n \geq 2)$ , 且 $z_{1} + z_{2} + \cdots + z_{n} = 0$ . 定义

$$
| | Z | | = \sqrt {\sum_ {k = 1} ^ {n} | z _ {k} | ^ {2}}, \quad \min | \Delta Z | = \min (| z _ {2} - z _ {1} |, | z _ {3} - z _ {2} |, \dots , | z _ {n} - z _ {n - 1} |)
$$

求 $\beta(n)$ 的最大值,使得

$$
| | Z | | \geq \beta (n) \min | \Delta Z |
$$

100. 设实系数多项式 $f(x) = a_0 + a_1x + \cdots + a_mx^m$ 的所有根都在(0,1)内,且 $(-1)^m a_m > 0$ . 求证: 对任意非负整数 $n, p (n+p < m)$ , 下面不等式总成立

$$
(- 1) ^ {p + 1} \sum_ {k = n} ^ {m - p - 1} (- 1) ^ {k} \binom {k} {n} \sum_ {h = 0} ^ {m} a _ {h} \sum_ {i + j = k} (- 1) ^ {j} \binom {h} {i} \binom {m - h} {j} > 0
$$