MathDB

Problem 2

Part of 1999 VJIMC

Problems(2)

11|a^n+b^n -> 11|a,11|b for all a,b, find n

Source: VJIMC 1999 1.2

8/1/2021
Find all natural numbers n≥1n\ge1 such that the implication (11∣an+bn)  ⟹  (11∣a∧11∣b)(11\mid a^n+b^n)\implies(11\mid a\wedge11\mid b)holds for any two natural numbers aa and bb.
number theory
x_(n+1)=avg(x_n,f(x_n)) converges to fixed point of f

Source: VJIMC 1999 2.2

7/29/2021
Let a,b∈Ra,b\in\mathbb R, a≤ba\le b. Assume that f:[a,b]→[a,b]f:[a,b]\to[a,b] satisfies f(x)−f(y)≤∣x−y∣f(x)-f(y)\le|x-y| for every x,y∈[a,b]x,y\in[a,b]. Choose an x1∈[a,b]x_1\in[a,b] and define xn+1=xn+f(xn)2,n=1,2,3,….x_{n+1}=\frac{x_n+f(x_n)}2,\qquad n=1,2,3,\ldots.Show that {xn}n=1∞\{x_n\}^\infty_{n=1} converges to some fixed point of ff.
limitsreal analysisfunction