Some answers:
1. Solve for positive x:
x (8 sqrt(1 - x) + sqrt(1 + x))
The upper bound is clearly x=1 because the square roots cease to exist for x > 1 and and substituting in x = 1 - epsilon^2 gives to first order (8 e + 1) The inverse of y = sqrt(1 + x)/sqrt(1 - x) is with some work x = (y^2 - 1)/(y^2 + 1) so when we divide through by sqrt(1-x) which we know now to be positive, and replace the above, we find:
(y^2 - 1)/(y^2 + 1) * (8 + y) = 11 y - 16
Expanding out we're looking for a root of
-10 y^3 + 24*y^2 - 12 y + 8 = 0
Dividing by -2 to normalize somewhat:
5 y^3 - 12 y^2 + 6 y - 4 = 0
At this point I almost gave up (since it's a cubic and I had no guarantee that x was rational) but got lucky, I started trying positive integers to see where this transitions from negative (y=0) to positive (5y^3) and accidentally found that y=2 solves the equation. Sending it back through I find 3/5, so assuming that it doesn't double back somewhere in the interval [3/5, 1] that's the interval that we're looking for. Phew!
2. Find all functions F(x) : R -> R having the propery that for any x1 and x2,
F(x1) - F(x2)
Yeah, if you're just entering a university you're probably not going to get this one. Rewriting x1 as x + dx, x2 as x, then this says F(x + dx) - F(x)
3. Given a triangle ABC construct with a straightedge and compass a point K on
AB and a point M on BC such that AK = KM = MC.
This seems in general impossible except for some very specific triangles -- is that true? For example if |AB| = 2 and |BC| = 1, it seems that the only point on AB which could possibly be a candidate for K is the midpoint, with M being B. But the only way that the distance from K to M is the same is if BCK is an equilateral triangle, which requires furthermore that the angle BAC is 60 degrees, no?
4. Solve 2 cubert(2y - 1) = y^3 + 1 for real y.
So y = 1 is an obvious solution. Cubing both sides we get
y^9 + 3y^6 + 3y^3 - 16 y + 9 = 0
Then doing polynomial division by y-1 I get
y^8 + y^7 + y^6 + 4 y^5 + 4 y^4 + 4 y^3 + 7 y^2 + 7 y - 9.
At this point I got stuck and turned to automated tools, which say that this is factorizable as:
(y^2 + y - 1)(y^6 + 2 y^4 + 2 y^3 + 4 y^2 + 2 y + 9)
Graphing the right hand side it seems to be consistently positive, so that just leaves the left hand side, which is -1/2 +/- sqrt(5)/2. That's pretty difficult.