MathDB

Problem 3

Part of 1996 VJIMC

Problems(2)

finite number of solutions to a diophantine equation

Source: VJIMC 1996 1.3

10/13/2021
Prove that the equation x1+x2+y1+y2+z1+z2=11996\frac x{1+x^2}+\frac y{1+y^2}+\frac z{1+z^2}=\frac1{1996}has finitely many solutions in positive integers.
Diophantine equationnumber theory
sum of digits in base ten, limit of sequence

Source: VJIMC 1996 2.3

10/19/2021
Let cif⁡(x)\operatorname{cif}(x) denote the sum of the digits of the number xx in the decimal system. Put a1=199719961997a_1=1997^{1996^{1997}}, and an+1=cif⁡(an)a_{n+1}=\operatorname{cif}(a_n) for every n>0n>0. Find lim⁡n→∞an\lim_{n\to\infty}a_n.
real analysislimitsnumber theory