
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.** David runs at  $3$  times the speed of Alice. If Alice runs  $2$  miles in  $30$  minutes, determine how many minutes it takes for David to run a mile.**p2.** Al has  $2019$  red jelly beans. Bob has  $2018$  green jelly beans. Carl has  $x$  blue jelly beans. The minimum number of jelly beans that must be drawn in order to guarantee  $2$  jelly beans of each color is  $4041$ . Compute  $x$ .**p3.** Find the  $7$ -digit palindrome which is divisible by  $7$  and whose first three digits are all  $2$ .**p4.** Determine the number of ways to put  $5$  indistinguishable balls in  $6$  distinguishable boxes.**p5.** A certain reduced fraction  $\frac{a}{b}$  (with  $a,b > 1$ ) has the property that when  $2$  is subtracted from the numerator and added to the denominator, the resulting fraction has  $\frac16$  of its original value. Find this fraction.**p6.** Find the smallest positive integer  $n$  such that  $|\tau(n +1)-\tau(n)| = 7$ . Here,  $\tau(n)$  denotes the number of divisors of  $n$ .**p7.** Let  $\vartriangle ABC$  be the triangle such that  $AB = 3$ ,  $AC = 6$  and  $\angle BAC = 120^o$ . Let  $D$  be the point on  $BC$  such that  $AD$  bisect  $\angle BAC$ . Compute the length of  $AD$ .**p8.**  $26$  points are evenly spaced around a circle and are labeled  $A$  through  $Z$  in alphabetical order. Triangle  $\vartriangle LMT$  is drawn. Three more points, each distinct from  $L, M$ , and  $T$  , are chosen to form a second triangle. Compute the probability that the two triangles do not overlap.**p9.** Given the three equations $a +b +c = 0$  $a^2 +b^2 +c^2 = 2$  $a^3 +b^3 +c^3 = 19$ find  $abc$ .**p10.** Circle  $\omega$  is inscribed in convex quadrilateral  $ABCD$  and tangent to  $AB$  and  $CD$  at  $P$  and  $Q$ , respectively. Given that  $AP = 175$ ,  $BP = 147$ , $CQ = 75$ , and  $AB \parallel CD$ , find the length of  $DQ$ .**p11.**Let  $p$  be a prime and m be a positive integer such that  $157p = m^4 +2m^3 +m^2 +3$ . Find the ordered pair  $(p,m)$ .**p12.** Find the number of possible functions  $f : \{-2,-1, 0, 1, 2\} \to \{-2,-1, 0, 1, 2\}$  that satisfy the following conditions.
(1)  $f (x) \ne f (y)$  when  $x \ne y$  
(2) There exists some  $x$  such that  $f (x)^2 = x^2$ **p13.** Let  $p$  be a prime number such that there exists positive integer  $n$  such that  $41pn -42p^2 = n^3$ . Find the sum of all possible values of  $p$ .**p14.** An equilateral triangle with side length  $ 1$  is rotated  $60$  degrees around its center. Compute the area of the region swept out by the interior of the triangle.**p15.** Let  $\sigma (n)$  denote the number of positive integer divisors of  $n$ . Find the sum of all  $n$  that satisfy the equation  $\sigma (n) =\frac{n}{3}$ .**p16**. Let  $C$  be the set of points  $\{a,b,c\} \in Z$  for  $0  \le a,b,c \le 10$ . Alice starts at  $(0,0,0)$ . Every second she randomly moves to one of the other points in  $C$  that is on one of the lines parallel to the  $x, y$ , and  $z$  axes through the point she is currently at, each point with equal probability. Determine the expected number of seconds it will take her to reach  $(10,10,10)$ .**p17.** Find the maximum possible value of   $$ abc \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^3 $$   where  $a,b,c$  are real such that  $a +b +c = 0$ .**p18.** Circle  $\omega$  with radius  $6$  is inscribed within quadrilateral  $ABCD$ .  $\omega$  is tangent to  $AB$ ,  $BC$ ,  $CD$ , and  $DA$  at  $E, F, G$ , and  $H$  respectively. If  $AE = 3$ ,  $BF = 4$  and  $CG = 5$ , find the length of  $DH$ .**p19.** Find the maximum integer  $p$  less than  $1000$  for which there exists a positive integer  $q$  such that the cubic equation   $$ x^3 - px^2 + q x -(p^2 -4q +4) = 0 $$   has three roots which are all positive integers.**p20.** Let  $\vartriangle ABC$  be the triangle such that  $\angle ABC = 60^o$ , $\angle ACB = 20^o$ . Let  $P$  be the point such that  $CP$  bisects  $\angle ACB$  and  $\angle PAC = 30^o$ . Find  $\angle PBC$ .


PS. You had better use hide for answers.

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