CF1616D.Keep the Average High

You are given an array of integers $a_1, a_2, \ldots, a_n$ and an integer $x$ .

You need to select the maximum number of elements in the array, such that for every subsegment $a_l, a_{l + 1}, \ldots, a_r$ containing strictly more than one element $(l < r)$ , either:

• At least one element on this subsegment is not selected, or
• $a_l + a_{l+1} + \ldots + a_r \geq x \cdot (r - l + 1)$ .

The first line of input contains one integer $t$ ( $1 \leq t \leq 10$ ): the number of test cases.

The descriptions of $t$ test cases follow, three lines per test case.

In the first line you are given one integer $n$ ( $1 \leq n \leq 50\,000$ ): the number of integers in the array.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $-100\,000 \leq a_i \leq 100\,000$ ).

The third line contains one integer $x$ ( $-100\,000 \leq x \leq 100\,000$ ).

For each test case, print one integer: the maximum number of elements that you can select.

In the first example, one valid way to select the elements is $[\underline{1}, 2, \underline{3}, \underline{4}, \underline{5}]$ . All subsegments satisfy at least one of the criteria. For example, for the subsegment $l = 1$ , $r = 2$ we have that the element $2$ is not selected, satisfying the first criterion. For the subsegment $l = 3$ , $r = 5$ we have $3 + 4 + 5 = 12 \ge 2 \cdot 3$ , satisfying the second criterion.

We can't select all elements, because in this case for $l = 1$ , $r = 2$ all elements are selected and we have $a_1 + a_2 = 3 < 2 \cdot 2$ . Thus, the maximum number of selected elements is $4$ .

In the second example, one valid solution is $[\underline{2}, \underline{4}, 2, \underline{4}, \underline{2}, \underline{4}, 2, \underline{4}, \underline{2}, \underline{4}]$ .

In the third example, one valid solution is $[\underline{-10}, -5, \underline{-10}]$ .

In the fourth example, one valid solution is $[\underline{9}, \underline{9}, -3]$ .