Find the number of cubic polynomials where and are integers in such that there is a unique integer with

绿树教育中心独家解析

是一个有两个整数根的三次方程，因此它有三个整数根。所以， 或 ，其中 。

「Case 1」，则 可以取 ， 可以取从 到 的整数， 个。

「Case 2」，则 可以取 ， 可以取从 到 的整数， 个。

综上，总和为

原题目

There exists a unique positive integer for which the sum

is an integer strictly between and . For that unique , find .

(Note that denotes the greatest integer that is less than or equal to .)

绿树教育中心独家解析

令 ，易知

有

如果 ，则 ，因此

易知 ，则 ，令 ，有

因此，

则

原题目

Find the number of subsets of that contain exactly one pair of consecutive integers. Examples of such subsets are and

官方解析

Define to be the number of subsets of that have consecutive element pairs, and to be the number of subsets that have consecutive pair.

Using casework on where the consecutive element pair is, it is easy to see that

We see that , , and . This is because if the element is included in our subset, then there are possibilities for the rest of the elements (because cannot be used), and otherwise there are possibilities. Thus, by induction, is the th Fibonacci number.

这表明 .

原题目

Let be an equilateral triangle with side length Points and lie on and respectively, with and Point inside has the property that

Find

绿树教育中心独家解析

如下图，易知

计算得 .

原题目

Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths and . The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is , where and are relatively prime positive integers. Find . A parallelepiped is a solid with six parallelogram faces such as the one shown below.

绿树教育中心独家解析

Let one of the vertices be at the origin and the three adjacent vertices be , , and . For one of the parallelepipeds, the three diagonals involving the origin have length . Hence, and . Since all of , , and have equal length, , , and . Symmetrically, , , and . Hence the volume of the parallelepiped is given by

For the other parallelepiped, the three diagonals involving the origin are of length and the volume is

Consequently, the answer is , giving .

原题目

The following analog clock has two hands that can move independently of each other. Initially, both hands point to the number . The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock face while the other hand does not move.

Let be the number of sequences of hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the movements, the hands have returned to their initial position. Find the remainder when is divided by .

绿树教育中心独家解析

不难得到，答案为 .

原题目

Find the largest prime number for which there exists a complex number satisfying

the real and imaginary part of are both integers；

；

there exists a triangle whose three side lengths are the real part of and the imaginary part of