### 1

```
boundedRatio
You are given an array of integers a and two integers l and r. You task is to calculate a boolean array b, where b[i] = true if there exists an integer x, such that a[i] = (i + 1) * x and l ≤ x ≤ r. Otherwise, b[i] should be set to false.
Example
For a = [8, 5, 6, 16, 5], l = 1, and r = 3, the output should be boundedRatio(a, l, r) = [false, false, true, false, true].
For a[0] = 8, we need to find a value of x such that 1 * x = 8, but the only value that would work is x = 8 which doesn't satisfy the boundaries 1 ≤ x ≤ 3, so b[0] = false.
For a[1] = 5, we need to find a value of x such that 2 * x = 5, but there is no integer value that would satisfy this equation, so b[1] = false.
For a[2] = 6, we can choose x = 2 because 3 * 2 = 6 and 1 ≤ 2 ≤ 3, so b[2] = true.
For a[3] = 16, there is no an integer 1 ≤ x ≤ 3, such that 4 * x = 16, so b[3] = false.
For a[4] = 5, we can choose x = 1 because 5 * 1 = 5 and 1 ≤ 1 ≤ 3, so b[4] = true.
Input/Output
[execution time limit] 4 seconds (php)
[input] array.integer a
An array of integers.
Guaranteed constraints:
1 ≤ a.length ≤ 100,
1 ≤ a[i] ≤ 106.
[input] integer l
An integer representing the lower bound for x.
Guaranteed constraints:
1 ≤ l ≤ 104.
[input] integer r
An integer representing the upper bound for x.
Guaranteed constraints:
1 ≤ r ≤ 104,
l ≤ r.
[output] array.boolean
A boolean array.
```

### 2

```
You are given an array of integers a. A new array b is generated by rearranging the elements of a in the following way:
b[0] is equal to a[0];
b[1] is equal to the last element of a;
b[2] is equal to a[1];
b[3] is equal to the second-last element of a;
b[4] is equal to a[2];
b[5] is equal to the third-last element of a;
and so on.
Your task is to determine whether the new array b is sorted in strictly ascending order or not.
Here is how the process of generating the new array b works:
Example
For a = [1, 3, 5, 6, 4, 2], the output should be alternatingSort(a) = true.
The new array b will look like [1, 2, 3, 4, 5, 6], which is in strictly ascending order, so the answer is true.
For a = [1, 4, 5, 6, 3], the output should be alternatingSort(a) = false.
The new array b will look like [1, 3, 4, 6, 5], which is not in strictly ascending order, so the answer is false.
Input/Output
[execution time limit] 4 seconds (php)
[input] array.integer a
The given array of integers.
Guaranteed constraints:
1 ≤ a.length ≤ 105,
-109 ≤ a[i] ≤ 109.
[output] boolean
A boolean representing whether the new array b will be sorted in strictly ascending order or not.
```

### 3

```
You are given an array of arrays a. Your task is to group the arrays a[i] by their mean values, so that arrays with equal mean values are in the same group, and arrays with different mean values are in different groups.
Each group should contain a set of indices (i, j, etc), such that the corresponding arrays (a[i], a[j], etc) all have the same mean. Return the set of groups as an array of arrays, where the indices within each group are sorted in ascending order, and the groups are sorted in ascending order of their minimum element.
Example
For
a = [[3, 3, 4, 2],
[4, 4],
[4, 0, 3, 3],
[2, 3],
[3, 3, 3]]
the output should be
meanGroups(a) = [[0, 4],
[1],
[2, 3]]
mean(a[0]) = (3 + 3 + 4 + 2) / 4 = 3;
mean(a[1]) = (4 + 4) / 2 = 4;
mean(a[2]) = (4 + 0 + 3 + 3) / 4 = 2.5;
mean(a[3]) = (2 + 3) / 2 = 2.5;
mean(a[4]) = (3 + 3 + 3) / 3 = 3.
There are three groups of means: those with mean 2.5, 3, and 4. And they form the following groups:
Arrays with indices 0 and 4 form a group with mean 3;
Array with index 1 forms a group with mean 4;
Arrays with indices 2 and 3 form a group with mean 2.5.
Note that neither
meanGroups(a) = [[0, 4],
[2, 3],
[1]]
nor
meanGroups(a) = [[0, 4],
[1],
[3, 2]]
will be considered as a correct answer:
In the first case, the minimal element in the array at index 2 is 1, and it is less then the minimal element in the array at index 1, which is 2.
In the second case, the array at index 2 is not sorted in ascending order.
For
a = [[-5, 2, 3],
[0, 0],
[0],
[-100, 100]]
the output should be
meanGroups(a) = [[0, 1, 2, 3]]
The mean values of all of the arrays are 0, so all of them are in the same group.
Input/Output
[execution time limit] 4 seconds (php)
[input] array.array.integer a
An array of arrays of integers.
Guaranteed constraints:
1 ≤ a.length ≤ 100,
1 ≤ a[i].length ≤ 100,
-100 ≤ a[i][j] ≤ 100.
[output] array.array.integer
An array of arrays, representing the groups of indices.
```

### 4

```
You are given an array of integers a. A new array b is generated by rearranging the elements of a in the following way:
b[0] is equal to a[0];
b[1] is equal to the last element of a;
b[2] is equal to a[1];
b[3] is equal to the second-last element of a;
b[4] is equal to a[2];
b[5] is equal to the third-last element of a;
and so on.
Your task is to determine whether the new array b is sorted in strictly ascending order or not.
Here is how the process of generating the new array b works:
Example
For a = [1, 3, 5, 6, 4, 2], the output should be alternatingSort(a) = true.
The new array b will look like [1, 2, 3, 4, 5, 6], which is in strictly ascending order, so the answer is true.
For a = [1, 4, 5, 6, 3], the output should be alternatingSort(a) = false.
The new array b will look like [1, 3, 4, 6, 5], which is not in strictly ascending order, so the answer is false.
Input/Output
[execution time limit] 4 seconds (php)
[input] array.integer a
The given array of integers.
Guaranteed constraints:
1 ≤ a.length ≤ 105,
-109 ≤ a[i] ≤ 109.
[output] boolean
A boolean representing whether the new array b will be sorted in strictly ascending order or not.
```