MathDB

3

Part of 2014 India IMO Training Camp

Problems(4)

Iterating functions

Source: IMOTC 2014 Practice Test 2 Problem 3

7/11/2014
For integers a,ba,b we define f((a,b))=(2a,b−a)f((a,b))=(2a,b-a) if a<ba<b and f((a,b))=(a−b,2b)f((a,b))=(a-b,2b) if a≥ba\geq b. Given a natural number n>1n>1 show that there exist natural numbers m,km,k with m<nm<n such that fk((n,m))=(m,n)f^{k}((n,m))=(m,n),where fk(x)=f(f(f(...f(x))))f^{k}(x)=f(f(f(...f(x)))),ff being composed with itself kk times.
functionnumber theory unsolvednumber theory
Orthocentres lying on the circumcircle

Source: IMOTC 2014 Practice Test 1 Problem 3

7/11/2014
In a triangle ABCABC, points XX and YY are on BCBC and CACA respectively such that CX=CYCX=CY,AXAX is not perpendicular to BCBC and BYBY is not perpendicular to CACA.Let Γ\Gamma be the circle with CC as centre and CXCX as its radius.Find the angles of triangle ABCABC given that the orthocentres of triangles AXBAXB and AYBAYB lie on Γ\Gamma.
geometrycircumcirclegeometric transformationtrigonometryfunctionangle bisectorgeometry unsolved
Limit of a sequence involving square roots

Source: Indian TST Day 1 Problem 3

7/11/2014
Starting with the triple (10072,20142,100714)(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14}), define a sequence of triples (xn,yn,zn)(x_{n},y_{n},z_{n}) by xn+1=xn(yn+zn−xn)x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})} yn+1=yn(zn+xn−yn)y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})} zn+1=zn(xn+yn−zn) z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})} for n≥0n\geq 0.Show that each of the sequences ⟨xn⟩n≥0,⟨yn⟩n≥0,⟨zn⟩n≥0\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0} converges to a limit and find these limits.
geometryinvariantalgebra unsolvedalgebra
Placing of rooks

Source: Indian TST Day 3 Problem 3

7/11/2014
In how many ways rooks can be placed on a 88 by 88 chess board such that every row and every column has at least one rook? (Any number of rooks are available,each square can have at most one rook and there is no relation of attacking between them)
combinatorics unsolvedcombinatorics