The history of renewable energy suggests there is a steep learning curve, meaning that, as more is produced, costs fall rapidly because of economies of scale and learning by doing. The firms' green innovation is path-dependent: the more a firm does, the more it is likely to do in the future. The strongest evidence for this is the collapse in the price of solar energy, which became about $90 \%$ cheaper during the 2010s, repeatedly beating forecasts. Moving early and gradually gives economies more time to adjust, allowing them to reap the benefits of pathdependent green investment without much disruption. A late, more chaotic transition is costlier.

A natural number $N$ is such that it can be expressed as $\mathbf{N}=\mathbf{p}+\mathbf{q}+\mathbf{r}$, where $\mathbf{p}, \mathbf{q}$ and $\mathbf{r}$ are distinct factors of $\mathbf{N}$. How many numbers below 50 have this property?

The history of renewable energy suggests there is a steep learning curve, meaning that, as more is produced, costs fall rapidly because of economies of scale and learning by doing. The firms' green innovation is path-dependent: the more a firm does, the more it is likely to do in the future. The strongest evidence for this is the collapse in the price of solar energy, which became about $90 \%$ cheaper during the 2010s, repeatedly beating forecasts. Moving early and gradually gives economies more time to adjust, allowing them to reap the benefits of pathdependent green investment without much disruption. A late, more chaotic transition is costlier.

Three prime numbers $p, q$ and $r$, each less than 20 , are such that $p-q=q-r$. How many distinct possible values can we get for $(p+q+r)$ ?

The history of renewable energy suggests there is a steep learning curve, meaning that, as more is produced, costs fall rapidly because of economies of scale and learning by doing. The firms' green innovation is path-dependent: the more a firm does, the more it is likely to do in the future. The strongest evidence for this is the collapse in the price of solar energy, which became about $90 \%$ cheaper during the 2010s, repeatedly beating forecasts. Moving early and gradually gives economies more time to adjust, allowing them to reap the benefits of pathdependent green investment without much disruption. A late, more chaotic transition is costlier.

How many possible values of $(p+q+r)$ are there satisfying $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1$, where $p, q$ and $r$ are natural numbers (not necessarily distinct)?

The history of renewable energy suggests there is a steep learning curve, meaning that, as more is produced, costs fall rapidly because of economies of scale and learning by doing. The firms' green innovation is path-dependent: the more a firm does, the more it is likely to do in the future. The strongest evidence for this is the collapse in the price of solar energy, which became about $90 \%$ cheaper during the 2010s, repeatedly beating forecasts. Moving early and gradually gives economies more time to adjust, allowing them to reap the benefits of pathdependent green investment without much disruption. A late, more chaotic transition is costlier.

Team $X$ scored a total of $N$ runs in 20 overs. Team $Y$ tied the score in $10 \%$ less overs. Had team Y's average run rate (runs per over) been $50 \%$ higher, the scores would have been tied in 12 overs. How many runs were scored by team X?