
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
Since guts has 36 questions, they will be combined into posts.

1.**[5]** Farmer Yang has a  $2015$  ×  $2015$  square grid of corn plants. One day, the plant in the very center
of the grid becomes diseased. Every day, every plant adjacent to a diseased plant becomes diseased.
After how many days will all of Yang's corn plants be diseased?

2. **[5]** The three sides of a right triangle form a geometric sequence. Determine the ratio of the length of
the hypotenuse to the length of the shorter leg.

3. **[5]** A parallelogram has  $2$  sides of length  $20$  and  $15$ . Given that its area is a positive integer, ﬁnd the
minimum possible area of the parallelogram.

4. **[6]** Eric is taking a biology class. His problem sets are worth  $100$  points in total, his three midterms are
worth  $100$  points each, and his ﬁnal is worth  $300$  points. If he gets a perfect score on his problem sets
and scores  $60\%$ , $70\%$ , and  $80\%$  on his midterms respectively, what is the minimum possible percentage
he can get on his ﬁnal to ensure a passing grade? (Eric passes if and only if his overall percentage is
at least  $70\%$ ).

5. **[6]** James writes down three integers. Alex picks some two of those integers, takes the average of them,
and adds the result to the third integer. If the possible ﬁnal results Alex could get are  $42$ ,  $13$ , and  $37$ ,
what are the three integers James originally chose?

6. **[6]** Let  $AB$  be a segment of length  $2$  with midpoint  $M$ . Consider the circle with center  $O$  and radius $r$  that is externally tangent to the circles with diameters  $AM$  and  $BM$  and internally tangent to the
circle with diameter  $AB$ . Determine the value of  $r$ .

7. **[7]** Let n be the smallest positive integer with exactly  $2015$  positive factors. What is the sum of
the (not necessarily distinct) prime factors of n? For example, the sum of the prime factors of  $72$  is $2 + 2 + 2 + 3 + 3 = 14$ .

8. **[7]** For how many pairs of nonzero integers  $(c, d)$  with  $-2015 \le c,d \le 2015$  do the equations  $cx = d$ and  $dx = c$  both have an integer solution?

9. **[7]** Find the smallest positive integer n such that there exists a complex number z, with positive real
and imaginary part, satisfying  $z^n = (\overline{z})^n$ .

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