Voldemort

1 + x + x^2 + x^3 + x^4 = 6

Use geometric sequences!

(x^5-1)/(x-1) = 6

Roots of Unity! One of my favorite topics! We can use root transformataion for this problem!

w^{1997} - 1 = 0 

In order to achieve a polynomial with w + 1 as the roots, we plug in w - 1.

(w-1)^{1997} - 1 = 0

If we want a polynomial with the reciprocal of the roots, we have to flip the coefficients. 

(w-1)^{1997} - 1 = w^{1997} - 1997w^{1996}+....+1997w -1 - 1 = 0. 

Flip the coefficients of the right hand side gives us:

-2w^{1997} + 1997w^{1996}+.... =  0

Can you solve the sum of the roots of the equatoin above? :)

You may also find the avlue of x by setting 2/(x+1) = 1/5 and solve x from there. 

https://artofproblemsolving.com/texer/xjsiitim

Question properly displayed here. Click "With bbcode"

Don't list them all out. That's an unwritten rule for solving a counting math problem. 

For the first segment, you have 7 colors to choose from.

For the second and third segment, you also have 7 choices for both of them. Therefore, there would be 7^3 = 343 ways to color all of them. 

Yikes, if you listed them all out, you would have 343 ways! 

In order for a quadratic equation to have two rational roots, the discriminant has to be greater than 0. Can you find some restrictions for c from here and solve it?

Call the point, F, the foot of the perpendicular line from E to AC. Because triangles AFE and AME are congruent by AAS, FE = 10, and FA = 22. This gives an area of 110. Because AC = 30, FC = 30 - 22 = 8. The area of triangle EFC is therefore 40. 

Adding the two triangle areas up gives us 150 units squared. 

No need for any programs like geogebra. Impressive how you could do that though :D

Because opposite angles of a parallelogram are equal, angles ADF and ABE are congruent. Suppose that \(a\) represents the length of AB and b represents the length of AD.  Using AA similarity, we can find that triangles ABE and ADF are similar. By the fact that the sides of similar triangles are in proportion, we can form the equation, 7/20 = a/b. In addition, because we know that the perimeter of the parallelogram is equal to 124, a + b = 62.

We have a system of equations!

a + b = 62

7b = 20a

b = 20/7a

(20/7)a + a = 62

(27/7)a = 62

a = 434/27

Multiplying this by 20 gives: 8680/27 as the area of the parallelogram.  

Using python codes is a great way to efficiently solve them but we are focused on problem solving our own minds here.

1. 15 = 5 *3

Since the three digit numbers have a unit digits of 5, we just need the number in the form of AB5, to be divisible by 3. And in order for a number to be divisible by 3, its digits need to add up to a multiple of 3. 

A + B + 5 = 0 (mod 3)

A + B = 1 (mod 3)

Can you solve the question from here and find all digits A and B that make the congruence true? (Hint, casework!)

2. One simple way to solve this is by using the loss of generality, plug in any numbers a and b, where their product leaves a remainder of 17 when divided by 20. After finding those numbers, plug it into (a + 10)(b + 10) and see the remainder.