Here are three problems to enjoy over the break!

## Problem 1

Three friends make the following statements:

• Ben says, “Exactly one of Dan and Cam is telling the truth.”
• Dan says, “Exactly one of Ben and Cam is telling the truth.”
• Cam says, “Neither Ben nor Dan is telling the truth.”

Which of the three friends is lying?

Source: Senior Maths Challenge 2016

## Problem 2

Compute the remainder when $10002000400080016003200640128025605121024204840968192$ is divided by $100020004000800160032$.

Source: HMMT Guts Round, Feb 2022

## Problem 3

Find the positive integer $n$ such that $32$ is the product of the real number solutions of $x^{\log_2{(x^3)}−n} = 13$.

Source: Purple Comet Math Meet, 2019

Happy problem solving!

Feel free to email your solutions to [email protected] or check the solutions page in a few weeks’ time!