# The Problem of the week

## Overview

Each Friday the Fall and spring semesters, we will post a problem to encourage undergraduate students to enjoy mathematics outside of the classroom. If you have a solution and want to be a part of it, e-mail your solution to Dr. Wesolek (pwesolek+mathclub@binghamton.edu ) by the due date. We will post solutions (from us) as well as novel solutions from students and record the names of students who've got the most number of novel solutions throughout each semester.

When you submit your answer, please provide some detailed solution rather than just an answer. Also, please include your major and the year you are in so that we can record.

### Fall 2017

### Fall 2017 Solutions

### Spring 2017

Champion: Alex Thornton

**Problem 1.** Circle A rolls one time around the circle B whose radius is three times that of

circle A. A letter A is drawn inside the circle A. How many times will the letter A rotate?

(Hint: It’s not 3)

**Problem 2. **Let f(x) : R → R be a function which has a continuous second derivative and satisfies the following condition:

2f(x + 1) = f(x) + f(2x), x ∈ R (1)

Suppose further that f(0) = 1. What is f(x)?

**Problem 3.** An ant is sitting at the vertex of a cube of length 1. It needs to get to the opposite vertex by crawling along the surface of the cube.

(1) What is the shortest distance that the ant needs to crawl?

(2) How many different shortest-distance routes can it take?

**Problem 4.** Let a0 = 0.91, a1 = 0.9901, a2 = 0.99990001, and in general ak = 0.999 · · · 0000 · · · 001 (there are 2k 9’s, 2k − 1 0’s and then 1).

Compute the following numbers: a0 · a1 · a2 · · · = Product(ai,0,infinity)

**Problem 5.** Let l be a line on a plane P. Then l divides P into two regions. Let l1 and l2 be two lines on a plane P. Then when l1 = l2, two lines {l1, l2} divide P into two regions. When l1 6= l2, two lines divide P into four regions. Suppose that we have n lines {l1, ..., ln} on a plan. Let R(n) be the maximum number of regions we can get from n lines on a plane P. For instance, R(1) = 2, R(2) = 4, R(3) = 7.

What is R(4) and R(5)? What is the general formula for R(n)?

**Problem 6.** A rectangle can be divided into n equal squares. It can also be divided into n + 76 equal squares. What is n?

**Problem 7.** This is an additional question to Problem 6. Describe all k which satisfies the following condition: if a rectangle can be divided into n equal squares and n+k equal squares, then n is unique

**Problem 8.** Is there an even number k > 2017 such that p 2 + k is composite for every odd prime number p?

**Problem 9.** Consider the numbers 19^m −5 ^n for positive integers m, n. What is the smallest positive number among them?