MathDB

3

Part of 1996 IMO Shortlist

Problems(4)

Recursive sequence satisfies inequality

Source: IMO Shortlist 1996, A3

8/9/2008
Let a>2 a > 2 be given, and starting a_0 \equal{} 1, a_1 \equal{} a define recursively: a_{n\plus{}1} \equal{} \left(\frac{a^2_n}{a^2_{n\minus{}1}} \minus{} 2 \right) \cdot a_n. Show that for all integers k>0, k > 0, we have: \sum^k_{i \equal{} 0} \frac{1}{a_i} < \frac12 \cdot (2 \plus{} a \minus{} \sqrt{a^2\minus{}4}).
inequalitiesalgebraSequenceRecurrenceIMO Shortlist
I need some pure geometry :))

Source: IMO Shortlist 1996 problem G3

10/4/2003
Let OO be the circumcenter and HH the orthocenter of an acute-angled triangle ABCABC such that BC>CABC>CA. Let FF be the foot of the altitude CHCH of triangle ABCABC. The perpendicular to the line OFOF at the point FF intersects the line ACAC at PP. Prove that ∡FHP=∡BAC\measuredangle FHP=\measuredangle BAC.
geometrycircumcirclesymmetrytrigonometryorthocenterIMO Shortlist
Maximum size of a subset S

Source: IMO Shortlist 1996, C3, Iran PPCE 1997, E2, P1

8/9/2008
Let k,m,n k,m,n be integers such that 1 < n \leq m \minus{} 1 \leq k. Determine the maximum size of a subset S S of the set \{1,2,3, \ldots, k\minus{}1,k\} such that no n n distinct elements of S S add up to m. m.
combinatoricsAdditive combinatoricsAdditive Number TheoryExtremal combinatoricsSubsetsIMO Shortlist
There exists a quadratic sequence with a_0 = 0, a_n = 1996

Source: IMO Shortlist 1996, N3

8/9/2008
A finite sequence of integers a0,a1,…,an a_0, a_1, \ldots, a_n is called quadratic if for each i i in the set {1,2…,n} \{1,2 \ldots, n\} we have the equality |a_i \minus{} a_{i\minus{}1}| \equal{} i^2. a.) Prove that any two integers b b and c, c, there exists a natural number n n and a quadratic sequence with a_0 \equal{} b and a_n \equal{} c. b.) Find the smallest natural number n n for which there exists a quadratic sequence with a_0 \equal{} 0 and a_n \equal{} 1996.
number theoryInteger sequencePerfect SquaresExtremal combinatoricsrecurrence relationIMO Shortlist