Rom

There are only 7, 14, 21, 28, 35 that are factors of 35!

Thus there are only 5 powers of 7 in 35! and thus it is not divisible by 7⁶*3

\(\text{The sum of all the real coefficients is simply the real part of }\\ \left . (1+ix)^{2009} \right |_{x=1}\)

\((1+i)^{2009} = \sqrt{2}^{2009}e^{i 2009 \left(\frac \pi 4\right)} = \sqrt{2}^{2009}\left(\cos\left(\dfrac \pi 4\right)+i \sin\left(\dfrac \pi 4 \right)\right)\)

\(Re\left[(1+i)^{2009}\right] = \sqrt{2}^{2009} \cdot \dfrac{\sqrt{2}}{2} = 2^{1004}\\ \log_2\left(2^{1004}\right) = 1004\)

90 is divisible by 4? 

\(55\% \text{ off means you pay }(100\%-55\%) = 45\%\\ 45\% \text{ of }\$65 = (0.45)65 = \$29.25\)

Fill the 5 cup container full, (5,0)

Fill the 3 cup container full using the 5 cup container, which afterward contains 2 cups, (2,3)

Pour out the 3 cup container and fill it with the 2 cups remaining in the 5 cup container, (0,2)

Fill the 5 cup container, (5,2)

Top off the 3 cup container using the 5 cup container.  This takes 1 cup from the 5 into the 3, (4,3)

You now have 4 cups in the 5 cup container.

\(x+y=a\\ y = a-x\\ x y = x(a-x) =\\ -(x^2 - a x)= \\ -\left(x^2 - ax +\dfrac{a^2}{4}-\dfrac{a^2}{4}\right) = \\ -\left(x-\dfrac a 2\right)^2 + \dfrac{a^2}{4}\)

\(xy \text{ will be a maximum of }\dfrac{a^2}{2} \\ \text{when }x = \dfrac a 2 \text{ i.e. when}\\ x = y = \dfrac a 2\)

\(\text{If you draw the force diagram you see that the gravitational force on the rock}\\ \text{can be broken up into a component normal to the inclined ground and a component }\\ \text{along the direction of potential motion}. \text{The magnitude of the component along the direction of potential motion is}\\ |f_m| = 400 \sin(20^\circ) \approx 136.8~ lbs\\ \text{This is the force that must be applied to the rock in the direction of the incline}\)

\(T(x,y) = x^2 -4x + y^2 + 2y = \\ (x-2)^2 -4 + (y+1)^2 - 1 = \\ (x-2)^2 + (y+1)^2 - 5\)

\(\text{Clearly the first two terms are non-negative and are zero at }x=2,~y=-1\\ T(2,-1) = -5\\ \text{and this is as cold as it gets}\)