2011 USA Mathematical Olympiad - DAY I
3:50 AM
Muhammad Yusuf
, Posted in
USAMO 1. Let a, b, c be positive real numbers such
that a2 + b2 + c2 + (a + b + c)2 ≤ 4. Prove that
USAMO 2. An integer is assigned
to each vertex of a regular pentagon so that the sum of the five integers is
2011. A turn of a solitaire game consists of subtracting an integer m from each of the integers at two
neighboring vertices and adding 2m to the opposite vertex,
which is not adjacent to either of the first two vertices. (The amount m and the vertices chosen can vary
from turn to turn.) The game is won at a certain vertex if, after some number of
turns, that vertex has the number 2011 and the other four vertices have the
number 0. Prove that for any choice of the initial integers, there is exactly
one vertex at which the game can be won.
USAMO 3. In hexagon ABCDEF, which is nonconvex but not
self-intersecting, no pair of opposite sides are parallel. The internal angles
satisfy ∠A = 3∠D, ∠C = 3∠F, and ∠E = 3∠B. Furthermore AB = DE, BC = EF, and CD = FA. Prove that diagonals AD, BE, and CF are concurrent.
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