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1. Greatest integer in a set of five ●○○○○ 10 ptsFive positive integers have an average of 15. What is the greatest possible value of one of these integers?
2. Smallest average of distinct even integers ●○○○○ 10 ptsWhat is the smallest possible average of four distinct positive even integers?
3. Most economical car speed ●○○○○ 10 ptsA car's fuel use is given by $f(v)=\frac{3}{80}v^{2}-6v+245$, where $v$ is the travelling speed. Find the speed $v$ that minimises the fuel use.
4. Largest sum for an involutory matrix ●●○○○ 20 ptsLet $a,$ $b,$ $c$ be integers such that $\mathbf{A} = \frac{1}{5} \begin{pmatrix} -3 & a \\ b & c \end{pmatrix}$ and $\mathbf{A}^2 = \mathbf{I}.$ Find the largest possible value of $a + b + c.$
5. Least difference between geometric and arithmetic terms ●●○○○ 20 ptsSequence $A$ is a geometric sequence and sequence $B$ is an arithmetic sequence. Each sequence stops as soon as one of its terms is greater than $300$. Sequence $A$: $2, 4, 8, 16, 32, \ldots$. Sequence $B$: $20, 40, 60, 80, 100, \ldots$. Find the least positive difference between a number selected from sequence $A$ and a number selected from sequence $B$.
6. Greatest third term of an arithmetic sequence ●●○○○ 20 ptsThe sum of four positive integers that form an arithmetic sequence is 46. Of all such possible sequences, what is the greatest possible third term?
7. Maximum of a piecewise function ●●○○○ 20 ptsDefine a new operation $\oplus$ as follows: when $a \geq b$, $a \oplus b = a$; when $a < b$, $a \oplus b = b^2$. Let $f(x) = (1 \oplus x)\cdot x - (2 \oplus x)$ for $x \in [-2, 2]$. Find the maximum value of $f(x)$.
8. Minimum area with given hypotenuse altitude ●●○○○ 20 ptsIn right triangle $ABC$, the altitude to the hypotenuse has length 4. Find the minimum value of the area of triangle $ABC$.
9. Maximum of a trigonometric quadratic in cosine ●●○○○ 20 ptsFind the maximum value of the function $f(x)=\sin^{2}x+\sqrt{3}\cos x-\frac{3}{4}$ for $x\in\left[0,\frac{\pi}{2}\right]$.
10. Corner squares for maximum box volume ●●○○○ 20 ptsAn open-top box is made from a square sheet of metal with sides of 20 cm by cutting equal squares from the corners and folding up the sides. Find the side length (in cm) of the square cut from each corner so that the volume of the box is maximised. Round to two decimal places.

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