Always Two

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

Surds

Find the exact values of x, y and a satisfying the following system of equations: 1/(a+1) = a - 1 x + y = 2a x = ay

Tennis Training

Age 14 to 16 ShortChallenge Level

Answer: Andy collected $16$ balls.

In terms of $a$
Suppose Andy collected $a$ balls. Then Roger collected $\frac 12 a$ and Maria collected $a-5$.

In total they collected $35$, so: \begin{align}&a + \tfrac 12 a + a - 5 = 35\\ \Rightarrow&\tfrac 52 a - 5 = 35\\ \Rightarrow&\tfrac 52 a = 40\\ \Rightarrow&a = 16\end{align}

In terms of $R$
Suppose Andy collected $A$ balls, Roger collected $R$ balls and Maria collected $M$ balls. Then $A=2R=M+5$.

In total they collected $35$ balls, so: $$A+R+M=35$$
Putting this in terms of $R$ gives: \begin{align}&2R+R+(2R-5)=35\\ \Rightarrow&5R=40\\ \Rightarrow&R=8\end{align}
Since Andy collected twice as many balls as Roger, he collected $2R=2\times8=16$ balls.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.