# Homework 1: Mathematical induction

Problem 1

Prove that $$\displaystyle 1^3+2^3+3^3+\cdots+ n^3=\frac{n^2(n+1)^2}4$$.

Problem 2

Determine the maximal possible number of regions in which the plane can be partitioned with $$n$$ lines.

Problem 3

Given $$3n$$ points $$A_1,A_2,\dots,A_{3n}$$ in the plane, assume that no three of them collinear. Prove that it is possible to construct $$n$$ disjoint triangles with vertices at the points $$A_i$$.

Problem 4

Find all positive integers $$n$$ such that $$\displaystyle 5^n> n!$$.

Problem 5

Consider the polynomial $$p(x)=a_0x^k+a_1x^{k-1}+\dots+ a_k$$ with integer coefficients. The polynomial $$p$$ is said to be divisible by an integer $$m$$ if $$p(x)$$ is divisible by $$m$$ for all integers $$x$$. Prove that if $$p(x)$$ is divisible by $$m$$, then $$k!a_0$$ is also divisible by $$m$$.

Problem 6

Prove that for every positive integer $$n$$ there exist positive integers $$a_{11}$$, $$a_{21}$$, $$a_{22}$$, $$a_{31}$$, $$a_{32}$$, $$a_{33}$$, $$\dots$$, $$a_{n1}$$, $$a_{n2}$$, $$\dots$$, $$a_{nn}$$ such that $a_{11}^2=a_{21}^2+a_{22}^2=a_{31}^2+a_{32}^2+a_{33}^2=\cdots=a_{n1}^2+\cdots+a_{nn}^2.$

Problem 7

Consider the set $$\mathcal F$$ of all injective functions $$f:\mathbb N\to\mathbb N$$ that satisfy $f(2x)+f(x)f(y)=f(x\cdot y)+2f(x),\;\;\mbox{for all }x,y\in\mathbb N.$ Determine $$\displaystyle\min_{f\in\mathcal F}\left\{f(2012)\right\}.$$

Problem 8

The sequence $$(a_n)_{n=1}^{\infty}$$ is defined as $$a_0=2$$, $$a_1=\frac52$$, and $$a_{n+1}=a_n(a_{n-1}^2-2)-\frac52$$, for $$n\geq 1$$. Prove that $3\log_2{[a_n]}=2^n-(-1)^n,$ where $$[x]$$ is the integral part of $$x$$.

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