
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.** Vincent the Bug is at the vertex  $A$  of square  $ABCD$ . Each second, he moves to an adjacent vertex with equal probability. The probability that Vincent is again on vertex  $A$  after  $4$  seconds is  $\frac{p}{q}$  , where  $p$  and  $q$  are relatively prime positive integers. Compute  $p + q$ .**p14.** Let  $ABC$  be a triangle with  $AB = 2$ ,  $AC = 3$ , and  $\angle BAC = 60^o$ . Let  $P$  be a point inside the triangle such that  $BP = 1$  and  $CP =\sqrt3$ , let  $x$  equal the area of  $APC$ . Compute  $16x^2$ .**p15.** Let  $n$  be the number of multiples of $ 3$  between  $2^{2020}$  and  $2^{2021}$ . When  $n$  is written in base two, how many digits in this representation are  $1$ ?


<u>Round 6</u>**p16.** Let  $f(n)$  be the least positive integer with exactly n positive integer divisors. Find  $\frac{f(200)}{f(50)}$  .**p17.** The five points  $A, B, C, D$ , and  $E$  lie in a plane. Vincent the Bug starts at point  $A$  and, each minute, chooses a different point uniformly at random and crawls to it. Then the probability that Vincent is back at  $A$  after  $5$  minutes can be expressed as  $\frac{p}{q}$  , where  $p$  and  $q$  are relatively prime positive integers. Compute  $p + q$ .**p18.** A circle is divided in the following way. First, four evenly spaced points  $A, B, C, D$  are marked on its perimeter. Point  $P$  is chosen inside the circle and the circle is cut along the rays  $PA$ ,  $PB$ ,  $PC$ ,  $PD$  into four pieces. The piece bounded by  $PA$ ,  $PB$ , and minor arc  $AB$  of the circle has area equal to one fifth of the area of the circle, and the piece bounded by  $PB$ ,  $PC$ , and minor arc  $BC$  has area equal to one third of the area of the circle. Suppose that the ratio between the area of the second largest piece and the area of the circle is  $\frac{p}{q}$ , where  $p$  and  $q$  are relatively prime positive integers. Compute  $p + q$ .


<u>Round 7</u>**p19.** There exists an integer  $n$  such that  $|2^n - 5^{50}|$  is minimized. Compute  $n$ .**p20.** For nonnegative integers  $a = \overline{a_na_{n-1} ... a_2a_1}$ ,  $b = \overline{b_mb_{m-1} ... b_2b_1}$ , define their distance to be   $$ d(a, b) = \overline{|a_{\max\,\,(m,n)} - b_{\max\,\,(m,n)}||a_{\max\,\,(m,n)-1} - b_{\max\,\,(m,n)-1}|...|a_1 - b_1|} $$   where  $a_k = 0$  if  $k > n$ ,  $b_k = 0$  if  $k > m$ . For example,  $d(12321, 5067) = 13346$ . For how many nonnegative integers  $n$  is  $d(2021, n) + d(12345, n)$  minimized?**p21.** Let  $ABCDE$  be a regular pentagon and let  $P$  be a point outside the pentagon such that  $\angle PEA = 6^o$  and  $\angle PDC = 78^o$ . Find the degree-measure of  $\angle PBD$ .


<u>Round 8</u>**p22.** What is the least positive integer  $n$  such that  $\sqrt{n + 3} -\sqrt{n} < 0.02$  ?**p23.** What is the greatest prime divisor of  $20^4 + 21 \cdot 23 - 6$ ?**p24.** Let  $ABCD$  be a parallelogram and let  $M$  be the midpoint of  $AC$ . Suppose the circumcircle of triangle  $ABM$  intersects  $BC$  again at  $E$ . Given that  $AB = 5\sqrt2$ ,  $AM = 5$ ,  $\angle BAC$  is acute, and the area of  $ABCD$  is  $70$ , what is the length of  $DE$ ?


PS. You should use hide for answers. Rounds 1-4 have been posted [here](https://artofproblemsolving.com/community/c3h2949414p26408213). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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