MathDB

1

Part of 2006 India IMO Training Camp

Problems(4)

S(j)+S^(-1)(j)=n+1

Source: India tst 2006 p1

6/26/2012
Let nn be a positive integer divisible by 44. Find the number of permutations σ\sigma of (1,2,3,⋯ ,n)(1,2,3,\cdots,n) which satisfy the condition σ(j)+σ−1(j)=n+1\sigma(j)+\sigma^{-1}(j)=n+1 for all j∈{1,2,3,⋯ ,n}j \in \{1,2,3,\cdots,n\}.
linear algebramatrixcombinatorics proposedcombinatorics
P is a point in the plane of ABC

Source: India tst 2006 p4

6/27/2012
Let ABCABC be a triangle and let PP be a point in the plane of ABCABC that is inside the region of the angle BACBAC but outside triangle ABCABC.
(a) Prove that any two of the following statements imply the third.
(i) the circumcentre of triangle PBCPBC lies on the ray PA→\stackrel{\to}{PA}.
(ii) the circumcentre of triangle CPACPA lies on the ray PB→\stackrel{\to}{PB}.
(iii) the circumcentre of triangle APBAPB lies on the ray PC→\stackrel{\to}{PC}.
(b) Prove that if the conditions in (a) hold, then the circumcentres of triangles BPC,CPABPC,CPA and APBAPB lie on the circumcircle of triangle ABCABC.
geometrycircumcirclegeometric transformationreflectionhomothetyratiogeometry unsolved
R \geq 2r, A little generalised

Source: India tst 2006 p7

6/26/2012
Let ABCABC be a triangle with inradius rr, circumradius RR, and with sides a=BC,b=CA,c=ABa=BC,b=CA,c=AB. Prove that R2r≥(64a2b2c2(4a2−(b−c)2)(4b2−(c−a)2)(4c2−(a−b)2))2.\frac{R}{2r} \ge \left(\frac{64a^2b^2c^2}{(4a^2-(b-c)^2)(4b^2-(c-a)^2)(4c^2-(a-b)^2)}\right)^2.
geometryinradiuscircumcircleinequalitiesinequalities proposed
a^2+b^2=c^2, gcd(a,b,c)=1, Number of triples=?

Source: India tst 2006 p10

6/27/2012
Find all triples (a,b,c)(a,b,c) such that a,b,ca,b,c are integers in the set {2000,2001,…,3000}\{2000,2001,\ldots,3000\} satisfying a2+b2=c2a^2+b^2=c^2 and gcd(a,b,c)=1\text{gcd}(a,b,c)=1.
number theorygreatest common divisormodular arithmeticsymmetrynumber theory proposed