
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.** Chris goes to Matt's Hamburger Shop to buy a hamburger. Each hamburger must contain exactly one bread, one lettuce, one cheese, one protein, and at least one condiment. There are two kinds of bread, two kinds of lettuce, three kinds of cheese, three kinds of protein, and six different condiments: ketchup, mayo, mustard, dill pickles, jalape~nos, and Matt's Magical Sunshine Sauce. How many different hamburgers can Chris make?**p2.** The degree measures of the interior angles in convex pentagon  $NICKY$  are all integers and form an increasing arithmetic sequence in some order. What is the smallest possible degree measure of the pentagon's smallest angle?**p3.** Daniel thinks of a two-digit positive integer  $x$ . He swaps its two digits and gets a number  $y$  that is less than  $x$ . If  $5$  divides  $x-y$  and  $7$  divides  $x+y$ , find all possible two-digit numbers Daniel could have in mind.**p4.** At the Lio Orympics, a target in archery consists of ten concentric circles. The radii of the circles are  $1$ ,  $2$ ,  $3$ ,  $...$ ,  $9$ , and  $10$  respectively. Hitting the innermost circle scores the archer  $10$  points, the next ring is worth  $9$  points, the next ring is worth 8 points, all the way to the outermost ring, which is worth  $1$  point. If a beginner archer has an equal probability of hitting any point on the target and never misses the target, what is the probability that his total score after making two shots is even?**p5.** Let  $F(x) = x^2 + 2x - 35$  and  $G(x) = x^2 + 10x + 14$ . Find all distinct real roots of  $F(G(x)) = 0$ .**p6.** One day while driving, Ivan noticed a curious property on his car's digital clock. The sum of the digits of the current hour equaled the sum of the digits of the current minute. (Ivan's car clock shows  $24$ -hour time; that is, the hour ranges from  $0$  to  $23$ , and the minute ranges from  $0$  to  $59$ .) For how many possible times of the day could Ivan have observed this property?**p7.** Qi Qi has a set  $Q$  of all lattice points in the coordinate plane whose  $x$ - and  $y$ -coordinates are between  $1$  and  $7$  inclusive. She wishes to color  $7$  points of the set blue and the rest white so that each row or column contains exactly  $1$  blue point and no blue point lies on or below the line  $x + y = 5$ . In how many ways can she color the points?**p8.** A piece of paper is in the shape of an equilateral triangle  $ABC$  with side length  $12$ . Points  $A_B$  and  $B_A$  lie on segment  $AB$ , such that  $AA_B = 3$ , and  $BB_A = 3$ . Define points  $B_C$  and  $C_B$  on segment  $BC$  and points  $C_A$  and  $A_C$  on segment  $CA$  similarly. Point  $A_1$  is the intersection of  $A_CB_C$  and  $A_BC_B$ . Define  $B_1$  and  $C_1$  similarly. The three rhombi -  $AA_BA_1A_C$ , $BB_CB_1B_A$ ,  $CC_AC_1C_B$  - are cut from triangle  $ABC$ , and the paper is folded along segments  $A_1B_1$ ,  $B_1C_1$ ,  $C_1A_1$ , to form a tray without a top. What is the volume of this tray?**p9.** Define  $\{x\}$  as the fractional part of  $x$ . Let  $S$  be the set of points  $(x, y)$  in the Cartesian coordinate plane such that  $x + \{x\} \le y$ ,  $x \ge 0$ , and  $y \le 100$ . Find the area of  $S$ .**p10.** Nicky likes dolls. He has  $10$  toy chairs in a row, and he wants to put some indistinguishable dolls on some of these chairs. (A chair can hold only one doll.) He doesn't want his dolls to get lonely, so he wants each doll sitting on a chair to be adjacent to at least one other doll. How many ways are there for him to put any number (possibly none) of dolls on the chairs? Two ways are considered distinct if and only if there is a chair that has a doll in one way but does not have one in the other.


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

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