
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>**5.1.** A triangle has lengths such that one side is  $12$  less than the sum of the other two sides, the semi-perimeter of the triangle is  $21$ , and the largest and smallest sides have a difference of  $2$ . Find the area of this triangle.**5.2.** A rhombus has side length  $85$  and diagonals of integer lengths. What is the sum of all possible areas of the rhombus?**5.3.** A drink from YAKSHAY’S SHAKE SHOP is served in a container that consists of a cup, shaped like an upside-down truncated cone, and a semi-spherical lid. The ratio of the radius of the bottom of the cup to the radius of the lid is  $\frac23$  , the volume of the combined cup and lid is  $296\pi$ , and the height of the cup is half of the height of the entire drink container. What is the volume of the liquid in the cup if it is filled up to half of the height of the entire drink container?


<u>Round 6</u>

*Each answer in the next set of three problems is required to solve a different problem within the same set. There is one correct solution to all three problems; however, you will receive points for any correct answer regardless whether other answers are correct.***6.1.** Let the answer to problem  $2$  be  $b$ . There are b people in a room, each of which is either a truth-teller or a liar. Person  $1$  claims “Person  $2$  is a liar,” Person  $2$  claims “Person  $3$  is a liar,” and so on until Person  $b$  claims “Person  $1$  is a liar.” How many people are truth-tellers?**6.2.** Let the answer to problem  $3$  be  $c$ . What is twice the area of a triangle with coordinates  $(0, 0)$ ,  $(c, 3)$  and  $(7, c)$  ?**6.3.** Let the answer to problem  $ 1$  be  $a$ . Compute the smaller zero to the polynomial  $x^2 - ax + 189$  which has  $2$  integer roots.


<u>Round 7</u>**7.1.**Sir Isaac Neeton is sitting under a kiwi tree when a kiwi falls on his head. He then discovers Neeton’s First Law of Kiwi Motion, which states:

*Every minute, either  $\left\lfloor \frac{1000}{d} \right\rfloor$  or  $\left\lceil \frac{1000}{d} \right\rceil$  kiwis fall on Neeton’s head, where d is Neeton’s distance from the tree in centimeters.*

Over the next minute,  $n$  kiwis fall on Neeton’s head. Let  $S$  be the set of all possible values of Neeton’s distance from the tree. Let m and M be numbers such that  $m < x < M$  for all elements  $x$  in  $S$ . If the least possible value of  $M - m$  is  $\frac{2000}{16899}$  centimeters, what is the value of  $n$ ?

Note that  $\lfloor x \rfloor$  is the greatest integer less than or equal to  $x$ , and  $\lceil x \rceil$  is the least integer greater than or equal to  $x$ .**7.2.** Nithin is playing chess. If one queen is randomly placed on an  $ 8 \times 8$  chessboard, what is the expected number of squares that will be attacked including the square that the queen is placed on? (A square is under attack if the queen can legally move there in one move, and a queen can legally move any number of squares diagonally, horizontally or vertically.)**7.3.** Nithin is writing binary strings, where each character is either a  $0$  or a  $1$ . How many binary strings of length  $12$  can he write down such that  $0000$  and  $1111$  do not appear?


<u>Round 8</u>**8.** What is the period of the fraction  $1/2018$ ? (The period of a fraction is the length of the repeated portion of its decimal representation.) Your answer will be scored according to the following formula, where  $X$  is the correct answer and  $I$  is your input.  $$ max  \left\{ 0, \left\lceil min  \left\{13 - \frac{|I-X|}{0.1 |I|}, 13 - \frac{|I-X|}{0.1 |I-2X|} \right\} \right\rceil \right\} $$  PS. You should  use hide for answers. Rounds 1-4 have been posted [here](https://artofproblemsolving.com/community/c3h2765571p24215461). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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