
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.** What is the units digit of the product of the first seven primes?**p2.**In triangle  $ABC$ ,  $\angle BAC$  is a right angle and  $\angle ACB$  measures  $34$  degrees. Let  $D$  be a point on segment  $ BC$  for which  $AC = CD$ , and let the angle bisector of  $\angle CBA$  intersect line  $AD$  at  $E$ . What is the measure of  $\angle BED$ ?**p3.** Chad numbers five paper cards on one side with each of the numbers from  $ 1$  through  $5$ . The cards are then turned over and placed in a box. Jordan takes the five cards out in random order and again numbers them from  $ 1$  through  $5$  on the other side. When Chad returns to look at the cards, he deduces with great difficulty that the probability that exactly two of the cards have the same number on both sides is  $p$ . What is  $p$ ?**p4.** Only one real value of  $x$  satisfies the equation  $kx^2 + (k + 5)x + 5 = 0$ . What is the product of all possible values of  $k$ ?**p5.** On the Exeter Space Station, where there is no effective gravity, Chad has a geometric model consisting of  $125$  wood cubes measuring  $ 1$  centimeter on each edge arranged in a  $5$  by  $5$  by  $5$  cube. An aspiring carpenter, he practices his trade by drawing the projection of the model from three views: front, top, and side. Then, he removes some of the original  $125$  cubes and redraws the three projections of the model. He observes that his three drawings after removing some cubes are identical to the initial three. What is the maximum number of cubes that he could have removed? (Keep in mind that the cubes could be suspended without support.)**p6.** Eric, Meena, and Cameron are studying the famous equation  $E = mc^2$ . To memorize this formula, they decide to play a game. Eric and Meena each randomly think of an integer between  $1$  and  $50$ , inclusively, and substitute their numbers for  $E$  and  $m$  in the equation. Then, Cameron solves for the absolute value of  $c$ . What is the probability that Cameron’s result is a rational number?**p7.** Let  $CDE$  be a triangle with side lengths  $EC = 3$ ,  $CD = 4$ , and  $DE = 5$ . Suppose that points  $ A$  and  $B$  are on the perimeter of the triangle such that line  $AB$  divides the triangle into two polygons of equal area and perimeter. What are all the possible values of the length of segment  $AB$ ?**p8.** Chad and Jordan are raising bacteria as pets. They start out with one bacterium in a Petri dish. Every minute, each existing bacterium turns into  $0, 1, 2$  or  $3$  bacteria, with equal probability for each of the four outcomes. What is the probability that the colony of bacteria will eventually die out?**p9.** Let  $a = w + x$ ,  $b = w + y$ ,  $c = x + y$ ,  $d = w + z$ ,  $e = x + z$ , and  $f = y + z$ . Given that  $af = be = cd$  and   $$ (x - y)(x - z)(x - w) + (y - x)(y - z)(y - w) + (z - x)(z - y)(z - w) + (w - x)(w - y)(w - z) = 1, $$   what is   $$ 2(a^2 + b^2 + c^2 + d^2 + e^2 + f^2) - ab - ac - ad - ae - bc - bd - bf - ce - cf - de - df - ef ? $$  **p10.** If  $a$  and  $b$  are integers at least  $2$  for which  $a^b - 1$  strictly divides  $b^a - 1$ , what is the minimum possible value of  $ab$ ?
Note: If  $x$  and  $y$  are integers, we say that  $x$  strictly divides  $y$  if  $x$  divides  $y$  and  $|x| \ne |y|$ .


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

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