iirc, this is very similar to a previous mathcounts nationals problem. here is the solution to that from my document
CFD is a 90, 45, 45 triangle, so CF and DF is $\sqrt2$ $BE = \sqrt(2(2+\sqrt(2))^2) = \sqrt{2} \cdot (2 + \sqrt{2})= 2\sqrt{2} + 2$ $AE^2 = AB^2 + BE^2 = 2 \times (2\sqrt{2} + 2)^2$ $AE = \sqrt{2} \times (2\sqrt{2} + 2) = 4 + 2\sqrt{2}$ $a + b + c = 2 + 4 + 2= \boxed{8}$
You are essentially choosing three items from six
$\dbinom{6}{3} = \frac{6 \cdot 5 \cdot 4}{3 \cdot 2 \cdot 1} = \boxed{20}$
This is , $(x - 4)(2y + 1) = 1 + xy.$ Solving, give $x = 9, y = 6.$ So, the product is $9 \cdot 6 = \boxed{54}$.
The graph is of $f(x).$ $f(2)$ is equal to the $y$ value when $x = 2,$ so $f(2) = \boxed{-2.}$
$5^8 + 5^7 + 5^6 + 5^5 + 5^4 + 5^3 + 5^2 + 5^1 = \boxed{488280}$ words, as there can be words with 1-7 letters as well.
Indeed, the answer is between the values 8080/190 and 8080/199. try guessing which one!
doing it
WHOA LOOK AT THIS DETAILED SOLUTION
Angle of the vertex of the combined shapes so far (without n sided) = $360 - (90 + 120 + 60) = 90.$ so we need a shape that has angle side 90 to finish the 360 degrees. therefore, it is a square: with 4 sides. it is an n-sided polygon, so n = 4
THANK YOU SO MUCH!!!!!!!!!
I had no idea it was that simple... ELEGANT solution
BIG BRAIN
OOPS i didnt see this solution when i posted mine! great they are the same though :)
We have P = k(14,7) where P = pressure and k = the rate
at 0 feet is 14.7 lb
calculate, the pressure increases at a rate of 0.441 psi for each foot you descend.
mile is 5280 feet
so 14.7 + 5280(0.441) = ~2343.2 psi
The minimum possible value is 70 - 50 = 20%, and MAXIMUM is 30 + 50 = 80%
difference = |80 - 20| = 60
there's three UNDEFINED terms we have: point, plane, and line.
point: no size, but has location
plane: infinite size, extends infinitely in two dimensions.
line: no width, extends infinitely in one dimension.
answer is plane
GOOD solution!
Minimizing, to obtain t^2-9t-36 = -225/4. The value of t is 9/2
For the first, we find n^2 = 28, n < 0 so it is n = -2√7.
For the second, we have 6 - 2n = 0, n >= 0 so it is n = 3
|-2√7 - 3| = 3 + 2√7
