Gaston and Jordon always misread recipe cooking times. For example, $1{:}32$ means 1 hour 32 minutes; Jordon reads it as 132 minutes, while Gaston reads it as 1.32 hours. For one recipe their misread times differ by exactly 90 minutes. What is the actual cooking time in minutes?
An integer has three digits in base 10. Interpreting its digits in base 6 and multiplying by 4 gives 2 more than three times the value of the same digits in base 7. Find the largest such integer in base 10.
Jenny has 1000 equal-sized marbles. At least one is red and the rest are green. If two are picked at random, the probability that both are red equals the probability that they have different colours. How many red marbles are there?
A beetle starts at $A$ in square $ABCD$. Each minute it traverses exactly one side to an adjacent corner. In two minutes its four possible walks are $ABA,ADA,ABC,ADC$. How many walks go from $A$ to $C$ in ten minutes?
A leaky boat initially contains some water and more enters at a constant rate. Each person operates a manual pump at the same constant rate. From a given starting time, five people would pump the boat dry in ten hours; twelve would do so in three hours. How many people are needed to pump it dry in two hours from that starting time?
A triangle $PQR$ is constructed so that the perpendicular bisector of $PQ$ meets side $QR$ at $N$, and $PN$ divides $\angle QPR$ into two angles in the ratio $1:22$ (not necessarily integer numbers of degrees). If $\angle QPR=p^\circ$ with integer $p$, find its maximum value.
A botanic garden has a line of 999 equally spaced positions, each containing at most one plant. In year 1 only the middle position has a plant. In year $n$, for every $n\ge2$, a position $X$ has a plant if and only if the total number of plants at $X$ and its two immediate neighbouring positions in the previous year is exactly one. Determine, with proof, the number of plants in year 128.
An even number of teams play a round robin: every pair meets exactly once, every team plays exactly once in each round, and there are no draws.
(a) Prove that after round 2 the number of teams with two wins equals the number with two losses.
(b) Find the least number of teams for which, after round 3, the number with three wins may differ from the number with three losses.
(c) Prove that after round 4 the number with four wins equals the number with four losses if and only if the number with exactly three wins equals the number with exactly three losses. (5 marks)
Investigation: prove that for every even $k\ge4$ there is a round robin of $k$ teams in which the number winning $k-1$ matches differs from the number losing $k-1$ matches. (3 bonus marks, separate from the base score.)