MathDB

3

Part of 1995 IMO Shortlist

Problems(3)

Base values between 2 and 3 satisfy fractional inequality

Source: IMO Shortlist 1995, A3

8/10/2008
Let n n be an integer, n≥3. n \geq 3. Let a1,a2,…,an a_1, a_2, \ldots, a_n be real numbers such that 2≤ai≤3 2 \leq a_i \leq 3 for i \equal{} 1, 2, \ldots, n. If s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n, prove that
\frac{a^2_1 \plus{} a^2_2 \minus{} a^2_3}{a_1 \plus{} a_2 \minus{} a_3} \plus{} \frac{a^2_2 \plus{} a^2_3 \minus{} a^2_4}{a_2 \plus{} a_3 \minus{} a_4} \plus{} \ldots \plus{} \frac{a^2_n \plus{} a^2_1 \minus{} a^2_2}{a_n \plus{} a_1 \minus{} a_2} \leq 2s \minus{} 2n.
inequalitiesalgebran-variable inequalityIMO Shortlist
E, F, Z, Y are concyclic

Source: IMO Shortlist 1995, G3

5/31/2008
The incircle of triangle △ABC \triangle ABC touches the sides BC BC, CA CA, AB AB at D,E,F D, E, F respectively. X X is a point inside triangle of △ABC \triangle ABC such that the incircle of triangle △XBC \triangle XBC touches BC BC at D D, and touches CX CX and XB XB at Y Y and Z Z respectively. Show that E,F,Z,Y E, F, Z, Y are concyclic.
geometrytrigonometryIMO Shortlistharmonic divisionpower of a pointincirclegeometry solved
q(x) to be the product of all primes less than p(x)

Source: IMO Shortlist 1995, S3

8/10/2008
For an integer x≥1x \geq 1, let p(x)p(x) be the least prime that does not divide xx, and define q(x)q(x) to be the product of all primes less than p(x)p(x). In particular, p(1)=2.p(1) = 2. For xx having p(x)=2p(x) = 2, define q(x)=1q(x) = 1. Consider the sequence x0,x1,x2,…x_0, x_1, x_2, \ldots defined by x0=1x_0 = 1 and xn+1=xnp(xn)q(xn) x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} for n≥0n \geq 0. Find all nn such that xn=1995x_n = 1995.
inductionalgebrapolynomialIterationSequenceIMO ShortlistHi