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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