Featured Problems

Crux Corner

February 2025

Crux Corner

February 2025 (Vol. 57, No. 1)

Get ready for a challenge! “Crux Corner” is a new feature in CMS Notes, bringing you problems from Crux Mathematicorum (CRUX), the CMS’s world-class problem-solving journal. CRUX is a fantastic resource for secondary and undergraduate students, packed with challenging problems and elegant solutions. Plus, it’s free online! See a selection of problems below.

Click here to access Crux problems, solutions, problem solving articles and more: https://cms.math.ca/publications/crux/

December issue, Crux 50 (10):

S16.

Prove that a positive integer

n

is prime if and only if there is a unique pair of positive integers

j

and

k

such that

\frac{1}{j}=\frac{1}{n}+\frac{1}{k}

5000. Proposed by Bill Sands. Dedicated in memoriam to Andy Liu.

There is a straight row of vertical cylinders stretching in both directions, each of radius 1 metre, and equally spaced at a distance of $s>2$ metres apart (centre to centre). You are standing at a point $d>1$ metres from the line through the centres of the cylinders. From your position, you can see a number of the cylinders completely, but eventually the cylinders in both directions become partly covered by cylinders closer to you.

a) Show that you can see at most $2\left\lceil\frac{d+1}{2}\right\rceil$ complete cylinders, that is, not partly obscured by other cylinders.

b) Suppose that $d$ is an integer and that $s>\frac{2d}{\sqrt{2d-1}}$ . Show that you can see at least $2\left\lceil\frac{d-1}{2}\right\rceil$ complete cylinders.

c) Suppose in addition that the perpendicular drawn from you to the line of centres hits that line at a point exactly halfway between two neighbouring cylinders. How many cylinders can you see completely?

January issue, Crux 51 (1):

MA304.

At a picnic, there are

c

children,

m

mothers, and

f

fathers, with

2 \leq f < m < c

. Every person shakes hand with every other person. The sum of the number of handshakes amongst the children, amongst the mothers, and amongst the fathers is 80. How many persons attended the picnic?

OC715.

A mathematician has 19 different weights, the masses of which in kilograms are equal to $\ln 2$ , $\ln 3$ , $\ln 4$ , $\ldots$ , $\ln 20$ , and an absolutely precise two-pan scale. He puts several weights on the scale so that equilibrium is established. What is the greatest number of weights that could be on the scale?