MathDB

2

Part of 2010 Saudi Arabia IMO TST

Problems(4)

AM x AN / AB x AC + BM x BN/ BA x BC + CM x CN / CA x CB = 1

Source: 2010 Saudi Arabia IMO TST III p2

12/27/2021
Points MM and NN are considered in the interior of triangle ABCABC such that ∠MAB=∠NAC\angle MAB = \angle NAC and ∠MBA=∠NBC\angle MBA = \angle NBC. Prove that AM⋅ANAB⋅AC+BM⋅BNBA⋅BC+CM⋅CNCA⋅CB=1\frac{AM \cdot AN}{AB \cdot AC}+ \frac{BM\cdot BN}{BA \cdot BC}+ \frac{CM \cdot CN }{CA \cdot CB}=1
ratiogeometryequal angles
f(m ) - f(n) = (m - n)(g(m) + g(n))

Source: 2010 Saudi Arabia IMO TST I p2

12/27/2021
Find all functions f,g:N→Nf,g : N \to N such that for all m,n∈Nm ,n \in N the following relation holds: f(m)−f(n)=(m−n)(g(m)+g(n))f(m ) - f(n) = (m - n)(g(m) + g(n)). Note: N={0,1,2,...}N = \{0,1,2,...\}
functional equationfunctionalalgebra
_|_ diagonals if AC^2 BD^2 = 2 AB x BC x CD x DA, <ABC = <ADC =135^o

Source: 2010 Saudi Arabia IMO TST III p2

12/27/2021
Let ABCDABCD be a convex quadrilateral such that ∠ABC=∠ADC=135o\angle ABC = \angle ADC =135^o and AC2BD2=2AB⋅BC⋅CD⋅DA.AC^2 BD^2=2AB\cdot BC \cdot CD\cdot DA. Prove that the diagonals of ABCDABCD are perpendicular.
perpendiculargeometry
(1 + \sqrt5)^n =\sqrt{a_n} + \sqrt{a_n+4^n}

Source: 2010 Saudi Arabia IMO TST V p2

12/28/2021
a) Prove that for each positive integer nn there is a unique positive integer ana_n such that (1+5)n=an+an+4n.(1 + \sqrt5)^n =\sqrt{a_n} + \sqrt{a_n+4^n} . b) Prove that a2010a_{2010} is divisible by 5⋅420095\cdot 4^{2009} and find the quotient
number theoryalgebraSequence