Dear Readers, welcome to this year-ending issue of Parabola!
It offers excellent problems as well as four articles that each look at numbers: primes, integers and reals. Enjoy!
Using only simple combinatorial arguments and binomial coefficients, this paper proves upper and lower bounds for the number of primes up to $x$.
Let us play Factorial Countdown, a game in which you are given numbers $1,2,\ldots,n$, a set of arithmetic operations and a target number $N$. The aim of the game if to use all of the numbers, and some of the operations, to reach your target $N$. Can you win this game?
This article presents a simple inequality that relates arithmetic means to root mean square. It is simple to prove, has nice visual proofs in the two-variable case, and has equally nice applications.
Highly composite numbers, denoted by $H_n$, are positive integers with more factors than any smaller positive integer. Based on an investigation of numeric data, this paper shows that there seem to be unusually many primes among numbers of the form $H_n\pm1$.
A new pun comic by the author of $2\mathbb{Z}$ Or Not $2\mathbb{Z}$.
Q1571 Find positive integers $a,b,c$ such that $a \leq b \leq c$ and
\[
\frac{1}{a}
+ \frac{1}{ab}
+ \frac{1}{abc}
= \frac{5}{26}\,.
\]
Q1561 Let $a,b,c$ be positive numbers for which
$\frac{a+b}c = 2018$ and $\frac{b+c}a = 2019$.
Evaluate $\frac{a+c}b$.