In a polygon ,the measure of each interior angle is a distinct integer.If the largest interior angle is 145?

Question

1.In a polygon ,the measure of each interior angle is a distinct integer.If the largest interior angle is 145 degrees and the polygon has the maximum number of sides possible,then find its largest exterior angle.Also find the number of sides of the polygon.

2.Find the last two digits of 2238473^982973.

3.Fin the area of the largest semicircle that can be inscribed in a square of unit square.

Answer 1

for #2, the last two digits start repeating at ^21 (ie, x^1 = x^21, x^2 = x^22)

20 goes into 982973, 49148 times evenly with 13 left over, so 3^13 = 1594323, last 2 digits are 23.

srry, on your own with the other 2

Answer 2

1) (CORRECTED)

The exterior angles sum to 360.

The exterior angle corresponding to 145 is 35.

360 - 35 = 325

We need distinct integers adding to 325

The other exterior angles will be larger, 145 is largest interior angle.

To have the maximum number of sides (and therefore

angles too), we want the smallest possible exterior angles.

We could have exterior angles of

36, 37, 38, 39, 40, 41, 42 which sum to 273,

leaving 52

The corresponding interior angles are

144, 143, 142, 141, 140, 139, 138, and 128.

Total of 9 sides.

The largest exterior angle is 52.

It is possible to distribute the remaining degrees

in many other ways.

For example, the last two exterior angles could be

43 + 51, or 44+50, or 45 + 49 or 46 + 48.

Similarly with the others.

The problem as stated does not have a unique solution.

3) Inscribed semi-circle.

I worked hours on this and finally found the solution.  

Post a new question with just this one and I'll answer it.

You should send me notice of the new question,

because I'm not going to search for it.

You should post separate - and unrelated - problems separately anyway.