proyaop

Yes there is an easier way. Assuming that triangles are still distinct even through rotations/reflections, etc, then the total number of points you could choose is 16 choose 3. However that would get some straight lined triangles, so we have to eliminate those. There are 16 + 16 + 4 + 4 = 40 of these cases counting three consecutive, 2 consecutive and 1 two apart, then considering the diagonal cases. Thus there are 16 choose 3 - 40 = 520 total triangles that are non-degenerate. (assuming that rotating the grid is not going to eliminate cases)

The door is most likely rectangular, meaning that the angles are all 90 degrees. The distance between the opposite corners are the diagonals of the rectangle, which has base length 39 and height 80 (can be used interchangebly). By the pythagorean theorem, the hypotenuse/diagonal squared = width^2 + length^2. So diagonal^2 = 80^2 + 39^2 = 7921. Square root both sides and you get 89.

The distance between the opposite corners of the door is 89 inches.

Open up paranthesis by FOILing: 9x^2 + 36x + 12x + 48 = (4x^2 + 8x + 18x^2)/2x

Combine like terms: 9x^2 + 48x + 48 = (22x^2 + 8x)/2x

Factor out top and bottom by 2x, then cancel terms: (22x^2 + 8x)/2x = 2x(11x + 4)/2x = 11x + 4

9x^2 + 48x + 48 = 11x + 4

Combine like terms: 9x^2 + 37x + 44 = 0

Use quadratic formula: \({-37\pm\sqrt{37^2 - (4)(9)(44)}}\over18\) = \({-37 \pm\sqrt{1369 - 1584}}\over18\) = \(-37\pm i\sqrt{215}\over18\) = x

The middle number is divisible by 5 so it must end in 0 or 5. If it ends in 0, then the next number ends in a 1, however the problem stated it was divisible by 4, and we know that the number must be even, so the last number ends in a 6. The middle number ends in a 5, and the first number ends in a 4. If the first number ends in a 4, we can consider 2 digit numbers, so 24 is div by 3, 25 is div by 5, but 26 is not div by 4. I can say 24 is div by 3 because of the divisibility rule for 3 (the sum of the digits of the number is div by 3), Lets try (2+3) and 4, so 54, 55, and 56. We can see those are divisible by 3, 5, then 4. 54 + 56 + 55 = 110 + 55 = 165

First you look at the x and y intercepts, then you can make a list and find the # of terms in it.

The y-intercept is 18, when y = 0, x = 18, so the x-intercept is also 18.

That means that there is a lattice point at x = 0, x = 1, x = 2,..., x = 18. Altogether 19 lattice points.

Expanded version of \((3n+4)(3n+5) = 9n^2 + 27n + 20\)

If (3n + 4)(3n + 5) = 0, then \(n ={-4\over3}, n = {-5\over3}\)

If ABC is an isosceles triangle, then that means the altituted AD will form a right angle with line BC.

IF BD = BC/2 = 18/2 = 9, and AB = 25, you can use the pythagorean theorem to obtain the length of AD

25^2 - 9^2 = 544. Square root of 544 = \(4\sqrt{34}\)

0 < n < 100

\(5^n + 8^{n + 1} + 11^{n + 2} + 5^{n+2} \)

and the expression above is divisible by 6.

5^n will always end in 5.

11^m will always end in 1.

That means 5^n + 5^(n + 2) will end in 0.

8^(n + 1) will end in an even number, meaning that the expression will be odd because of 11^(n+2). 

If a number has to be even to be divisible by 6, then that means:

The expression will never be divisible by 6 for 0 < n < 100.

Exponent property \((a^m)^n = a^{mn}\)

So (a^9)^5 = a^(9 * 5) = a^45