## Sunday, December 4, 2005

### Just Practice. Topic: All. Level: AIME/Olympiad.

Practice Problem #1: (2000 USA TST - #1) Given three non-negative reals $a,b,c$ prove that

$\frac{a+b+c}{3}-\sqrt[3]{abc} \le \max\{(\sqrt{a}-\sqrt{b})^2, (\sqrt{b}-\sqrt{c})^2, (\sqrt{c}-\sqrt{a})^2\}$

Practice Problem #2: (2004 AIME1 - #3) P is a convex polyhedron with $26$ vertices, $60$ edges and $36$ faces. $24$ of the faces are triangular and $12$ are quadrilaterals. A space diagonal is a line segment connecting two vertices which do not belong to the same face. How many space diagonals does P have? [Source: http://www.kalva.demon.co.uk/aime/aime04a.html]

Practice Problem #3: (2001 AIME1 - #12) Find the inradius of the tetrahedron with vertices $(6,0,0)$, $(0,4,0)$, $(0,0,2)$ and $(0,0,0)$. [Source: http://www.kalva.demon.co.uk/aime/aime01a.html]

Practice Problem #4: (2001 AIME1 - #3) Find the sum of the roots of the polynomial $x^{2001} + \left(\frac{1}{2} - x\right)^{2001}$. [Source: http://www.kalva.demon.co.uk/aime/aime01a.html]

1. I know #4, too lazy to post solution, so I"ll give you a question from my mock AIME in progress instead (I'll probably change it later):

Triangle ABC has AB = 13, BC = 37. Point D is on AC such that BD = \sqrt{193}. Point E is on BC such that BE = 10. AE and BD intersect at point F. CF intersects AB at point G. If \frac{AG}{BG} = \frac{70}{81}, find the area of triangle ABC.

(This is #8 so far :) )

2. Well I have an ugly solution, aka Stewart's+Ceva to find the length of AC, then Heron's...

3. Stewart's + Ceva is the solution I intended, but the last step can be done slightly more simply by drawing the altitude from B to AC. :)

4. Haha ic. I assume it turns out much more nicely than it looks, probably thanks to all those ugly numbers given in the problem.

5. QC, do you have an AoPS username?

6. QC = t0rajir0u