MathDB

1

Part of 1994 IMO Shortlist

Problems(4)

Sequence 1994

Source: IMO Shortlist 1994, A1

12/26/2006
Let a_{0} \equal{} 1994 and a_{n \plus{} 1} \equal{} \frac {a_{n}^{2}}{a_{n} \plus{} 1} for each nonnegative integer n n. Prove that 1994 \minus{} n is the greatest integer less than or equal to an a_{n}, 0≤n≤998 0 \leq n \leq 998
Sequencerecurrence relationalgebraIMO Shortlist
Show that EF bisects angle CFD

Source: IMO Shortlist 1994, G1

10/22/2005
C C and D D are points on a semicircle. The tangent at C C meets the extended diameter of the semicircle at B B, and the tangent at D D meets it at A A, so that A A and B B are on opposite sides of the center. The lines AC AC and BD BD meet at E E. F F is the foot of the perpendicular from E E to AB AB. Show that EF EF bisects angle CFD CFD
trigonometrygeometrysymmetryangle bisectorIMO Shortlist
Determine the maximum number of elements in M

Source: IMO Shortlist 1994, N1

8/10/2008
M M is a subset of {1,2,3,…,15} \{1, 2, 3, \ldots, 15\} such that the product of any three distinct elements of M M is not a square. Determine the maximum number of elements in M. M.
number theoryExtremal combinatoricsPerfect SquaresSubsetIMO Shortlist
What is the largest score

Source: IMO Shortlist 1994, C1

10/22/2005
Two players play alternately on a 5×5 5 \times 5 board. The first player always enters a 1 1 into an empty square and the second player always enters a 0 0 into an empty square. When the board is full, the sum of the numbers in each of the nine 3×3 3 \times 3 squares is calculated and the first player's score is the largest such sum. What is the largest score the first player can make, regardless of the responses of the second player?
algorithmcombinatoricsgamegame strategyIMO Shortlist