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Curvy Equation

This problem asks you to use your curve sketching knowledge to find all the solutions to an equation.

Euler's Totient Function

How many numbers are there less than $n$ which have no common factors with $n$?

Frosty the Snowman

Frosty the Snowman is melting. Can you use your knowledge of differential equations to find out how his volume changes as he shrinks?

Digital Equation

Age 16 to 18
Challenge Level
  • If $k$ is an integer between 0 and 9, what possible values can $10k-k^2$ take? What are the minimum and maximum values?
  • What do you know about the factors $k$, $k-1$ and $k+1$? What does this mean about the product of these three factors?
  • What is $S$ in terms of $a$, $b$ and $c$? Make sure you read the question carefully!
  • Can you use $S=N$ to write an equation involving $a$, $b$ and $c$?
  • What values can $a$, $b$, $c$ take? What does the fact that $N$ is a 3-digit number tell us about $a$?
  • Can you link the first parts of the question to the last part?
  • Can you put bounds on the values of the parts of the equation involving $a$, $b$, and $c$?
  • Can you rule out any values of $a$, $b$ or $c$?