
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
**p1.** The least prime factor of  $a$  is  $3$ , the least prime factor of  $b$  is  $7$ . Find the least prime factor of  $a + b$ .**p2.** In a Cartesian coordinate system, the two tangent lines from  $P = (39, 52)$  meet the circle defined by  $x^2 + y^2 = 625$  at points  $Q$  and  $R$ . Find the length  $QR$ .**p3.** For a positive integer  $n$ , there is a sequence  $(a_0, a_1, a_2,..., a_n)$  of real values such that  $a_0 = 11$  and  $(a_k + a_{k+1}) (a_k - a_{k+1}) = 5$  for every  $k$  with  $0 \le k \le n-1$ . Find the maximum possible value of  $n$ . (Be careful that your answer isn’t off by one!)**p4.** Persons  $A$  and  $B$  stand at point  $P$  on line  $\ell$ . Point  $Q$  lies at a distance of  $10$  from point  $P$  in the direction perpendicular to  $\ell$ . Both persons intially face towards  $Q$ . Person  $A$  walks forward and to the left at an angle of  $25^o$  with  $\ell$ , when he is again at a distance of  $10$  from point  $Q$ , he stops, turns  $90^o$  to the right, and continues walking. Person  $B$  walks forward and to the right at an angle of  $55^o$   with line  $\ell$ , when he is again at a distance of  $10$  from point  $Q$ , he stops, turns  $90^o$  to the left, and continues walking. Their paths cross at point  $R$ . Find the distance  $PR$ .**p5.** Compute  $$ \frac{lcm (1,2, 3,..., 200)}{lcm (102, 103, 104, ..., 200)}. $$  **p6.** There is a unique real value  $A$  such that for all  $x$  with  $1 < x < 3$  and  $x \ne 2$ ,   $$ \left| \frac{A}{x^2-x - 2} +\frac{1}{x^2 - 6x + 8} \right|< 1999. $$  Compute  $A$ .**p7.** Nine poles of height  $1, 2,..., 9$  are placed in a line in random order. A pole is called *dominant* if it is taller than the pole immediately to the left of it, or if it is the pole farthest to the left. Count the number of possible orderings in which there are exactly  $2$  dominant poles.**p8.**  $\tan (11x) = \tan (34^o)$  and  $\tan (19x) = \tan (21^o)$ . Compute  $\tan (5x)$ .


PS. You had better use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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