MathDB

5

Part of 1994 IMO Shortlist

Problems(3)

Recursively defined function

Source: IMO Shortlist 1994, A5

8/10/2008
Let f(x) \equal{} \frac{x^2\plus{}1}{2x} for x≠0. x \neq 0. Define f^{(0)}(x) \equal{} x and f^{(n)}(x) \equal{} f(f^{(n\minus{}1)}(x)) for all positive integers n n and x≠0. x \neq 0. Prove that for all non-negative integers n n and x \neq \{\minus{}1,0,1\} \frac{f^{(n)}(x)}{f^{(n\plus{}1)}(x)} \equal{} 1 \plus{} \frac{1}{f \left( \left( \frac{x\plus{}1}{x\minus{}1} \right)^{2n} \right)}.
functioninductionalgebrafunctional equationIMO Shortlist
circle with tangent lines

Source: IMO Shortlist 1994, G5

4/15/2004
A circle C C with center O. O. and a line L L which does not touch circle C. C. OQ OQ is perpendicular to L, L, Q Q is on L. L. P P is on L, L, draw two tangents L1,L2 L_1, L_2 to circle C. C. QA,QB QA, QB are perpendicular to L1,L2 L_1, L_2 respectively. (A A on L1, L_1, B B on L2 L_2). Prove that, line AB AB intersect QO QO at a fixed point. Original formulation: A line l l does not meet a circle ω \omega with center O. O. E E is the point on l l such that OE OE is perpendicular to l. l. M M is any point on l l other than E. E. The tangents from M M to ω \omega touch it at A A and B. B. C C is the point on MA MA such that EC EC is perpendicular to MA. MA. D D is the point on MB MB such that ED ED is perpendicular to MB. MB. The line CD CD cuts OE OE at F. F. Prove that the location of F F is independent of that of M. M.
geometrycircumcirclereflectiontrigonometryrotationIMO Shortlist
The game cannot terminate

Source: IMO Shortlist 1994, C5

10/22/2005
1994 1994 girls are seated at a round table. Initially one girl holds n n tokens. Each turn a girl who is holding more than one token passes one token to each of her neighbours. a.) Show that if n<1994 n < 1994, the game must terminate. b.) Show that if n \equal{} 1994 it cannot terminate.
combinatoricsinvariantIMO Shortlistgametermination