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:]

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:]

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:]


  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?