
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
<u>Round 5</u>**p13.** Sally is at the special glasses shop, where there are many different optical lenses that distort what she sees and cause her to see things strangely. Whenever she looks at a shape through lens  $A$ , she sees a shape with  $2$  more sides than the original (so a square would look like a hexagon). When she looks through lens  $B$ , she sees the shape with  $3$  fewer sides (so a hexagon would look like a triangle). How many sides are in the shape that has  $200$  more diagonals when looked at from lense  $A$  than from lense  $B$ ?**p14.** How many ways can you choose  $2$  cells of a  $5$  by  $5$  grid such that they aren't in the same row or column?**p15.** If  $a + \frac{1}{b} = (2015)^{-1}$  and  $b +  \frac{1}{a} = (2016)^2$  then what are all the possible values of  $b$ ?


<u>Round 6</u>**p16.** In Canadian football, linebackers must wear jersey numbers from  $30 -35$  while defensive linemen must wear numbers from  $33 -38$  (both intervals are inclusive). If a team has  $5$  linebackers and  $4$  defensive linemen, how many ways can it assign jersey numbers to the  $9$  players such that no two people have the same jersey number?**p17.** What is the maximum possible area of a right triangle with hypotenuse  $8$ ?**p18.**  $9$  people are to play touch football. One will be designated the quarterback, while the other eight will be divided into two (indistinct) teams of  $4$ . How many ways are there for this to be done?


<u>Round 7</u>**p19.** Express the decimal  $0.3$  in base  $7$ .**p20.**  $2015$  people throw their hats in a pile. One at a time, they each take one hat out of the pile so that each has a random hat. What is the expected number of people who get their own hat?**p21.** What is the area of the largest possible trapezoid that can be inscribed in a semicircle of radius  $4$ ?


<u>Round 8</u>**p22.** What is the base  $7$  expression of  $1211_3  \cdot 1110_2  \cdot 292_{11} \cdot 20_3$  ?**p23.** Let  $f(x)$  equal the ratio of the surface area of a sphere of radius  $x$  to the volume of that same sphere. Let  $g(x)$  be a quadratic polynomial in the form  $x^2 + bx + c$  with  $g(6) = 0$  and the minimum value of  $g(x)$  equal to  $c$ . Express  $g(x)$  as a function of  $f(x)$  (e.g. in terms of  $f(x)$ ).**p24.** In the country of Tahksess, the income tax code is very complicated. Citizens are taxed  $40\%$  on their first  $\$ 20, 000 $ and $ 45\% $ on their next $ \ $40, 000$  and  $50\%$  on their next  $\$ 60, 000 $ and so on, with each $ 5\% $ increase in tax rate aecting $ \ $20, 000$  more than the previous tax rate. The maximum tax rate, however, is  $90\%$ . What is the overall tax rate (percentage of money owed) on  $1$  million dollars in income?


PS. You should use hide for answers. Rounds 1-4 have been posted [here ](https://artofproblemsolving.com/community/c3h3157009p28696627) and 9-12 [here](https://artofproblemsolving.com/community/c3h3158564p28715928). .Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Present the answer in LaTex format: \boxed{Your answer}
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