Voldemort

From the fact that f(2) = 0 and f(0) = 0, we know that 0 and 2 are both roots/x-intercepts of this graph. Suppose that a is the other root, this gives us:

f(x) = (x - 0)(x - 2)(x - a) 

Plugging in 1 for x gives us

f(1) = -1(1 - a)

5 = -1 + a

a = 6

Therefore, the x intercepts, are, 0, 2, and 6. 

I bet many people are falling for this:

sqrt(x^2) does not necessarily equal x.

I repeat, sqrt(x^2) does NOT Necessasrliy equal x. 

What if x is negative? :) 

We'll have to take 2 cases:
1. x >= 0; 

This gives us sqrt(120 - x) is an integer. 

This means that 120-x is a perfect square. Since x is nonnegative, the amount of values of x is the amount of perfect squares less than 120.
This ranges from 1^2.... to 10^2.. Therefore there are 10 values where x is nonnegative. 

2. x < 0 

This gives us that sqrt(120 + x)  is an integer. However, note that x is negative and there will be the same values of x before because x in our first case is like -x in our case here. Hence, there are 10 values where x is negative.

Adding up these two cases, gives us 20 values. 

Suppose that the roots are r, s

P(x) = ax^2 + bx + c = (x - r)(x-s)

Since we know what r and s is, we know that P(10) is equal to (10 + 0.18859048)(10 + 0.58919729)

Yikes, this doens't seem like the right path. 

We may want to find the coefficients instead of the exact value of each of these decimals. If we use vieta's formula, you DONT need a calculator(I will show this magic to you :) )

We know that the product of these two roots is equal to c/a and that it's a very tiny value. Any value greater than 1/9 will surely not equal the product of these two numbers. Therefore, c = 1, a = 9..

By vieta's,  b/a is approximately 0.77(You don't hae to expliclity sum up ALL those values).. Therefore, since a = 9, b is equal to 7.

Therefore, our polynomial is:

P(x) = 9x^2 + 7x + 1

We will plug in 10 and this gives us \(\boxed{971}\)

I believe the problem is asking for odd-digit numbers and not odd numbers. So we have to consider three cases:

1. 1 digit numbers
There are 5 1 digit numbers that may be formed. 

2. 3 digit numbers

There are 5 x 4 x 3 = 60 3 digit numbers.

3. 5 digit numbers. 

There are 5 x 4 x 3 x 2 x 1 = 120 5 digit numbers.

Adding all of these up gives us 185. 

Probability = (Number of successful outcomes)/(Number of Total Outcomes)

The number of total outcomes is how many diagonals there are in total. 20(17)/2 is the number of diagonals which is equal to 170.

The shortest diagonal comes from connecting one vertex to another that is only two "sides" away. The amount of these diagonals is  20. 

Therefore, the answer is 20/170 = 2/17. 

Edit: Below whoops, didn't see that they also needed second shortest and third shortest... In that case, we'll have to multiply 20 by 3 for the other cases as well giving us 

6/17. 

Sorry with my previous answer, I accidentally multiplied by 5 fo rsome reason.

There are 120 ways to arrange these 5-digit positive integers. That means that the digits will be added 24 times in each digit of the sum. Therefore, our sum would be equal to:
24(10000 + 20000 + 30000 + 50000+80000) + 24(1000 + 2000 + 3000 + 5000 + 8000) + ... + 24(1 + 2 + 3 + 5 + 8))

We know that this is equal to:

24(11111 + 22222 + 33333 + 55555 + 88888) 

= 5,066,616