
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.** We define  $a \oplus b = \frac{ab}{a+b}$ . Compute  $(3 \oplus 5) \oplus (5 \oplus 4)$ .**p2.** Let  $ABCD$  be a quadrilateral with  $\angle A = 45^o$   and  $\angle B = 45^o$ . If  $BC = 5\sqrt2$ ,  $AD = 6\sqrt2$ , and  $AB = 18$ , find the length of side  $CD$ .**p3.** A positive real number  $x$  satisfies the equation  $x^2 + x + 1 + \frac{1}{x }+\frac{1}{x^2} = 10$ . Find the sum of all possible values of  $x + 1 + \frac{1}{x}$ .**p4.** David writes  $6$  positive integers on the board (not necessarily distinct) from least to greatest. The mean of the first three numbers is  $3$ , the median of the first four numbers is  $4$ , the unique mode of the first five numbers is  $5$ , and the range of all 6 numbers is  $6$ . Find the maximum possible value of the product of David’s  $6$  integers.**p5.** Let  $ABCD$  be a convex quadrilateral such that  $\angle A = \angle B = 120^o$  and  $\angle C = \angle D = 60^o$ . There exists a circle with center  $I$  which is tangent to all four sides of  $ABCD$ . If  $IA \cdot IB \cdot  IC \cdot ID = 240$ , find the area of quadrilateral  $ABCD$ .**p6.** The letters  $EXETERMATH$  are placed into cells on an annulus as shown below. How many ways are there to color each cell of the annulus with red, blue, green, or yellow such that each letter is always colored the same color and adjacent cells are always colored differently?
![Image](https://cdn.artofproblemsolving.com/attachments/3/5/b470a771a5279a7746c06996f2bb5487c33ecc.png)**p7.** Let  $ABCD$  be a square, and let  $\omega$  be a quarter circle centered at  $A$  passing through points  $B$  and  $D$ . Points  $E$  and  $F$  lie on sides  $BC$  and  $CD$  respectively. Line  $EF$  intersects  $\omega$  at two points,  $G$  and  $H$ . Given that  $EG = 2$ ,  $GH = 16$  and  $HF = 9$ , find the length of side  $AB$ .**p8.** Let x be equal to  $\frac{2022! + 2021!}{2020! + 2019! + 2018!}$  . Find the closest integer to  $2\sqrt{x}$ .**p9.** For how many ordered pairs of positive integers  $(m, n)$  is the absolute difference between  $lcm(m, n)$  and  $gcd(m, n)$  equal to  $2023$ ?**p10.** There are  $2023$  distinguishable frogs sitting on a number line with one frog sitting on  $i$  for all integers  $i$  between  $-1011$  and  $1011$ , inclusive. Each minute, every frog randomly jumps either one unit left or one unit right with equal probability. After  $1011$  minutes, over all possible arrangements of the frogs, what is the average number of frogs sitting on the number  $0$ ?**p11.** Albert has a calculator initially displaying  $0$  with two buttons: the first button increases the number on the display by one, and the second button returns the square root of the number on the display. Each second, he presses one of the two buttons at random with equal probability. What is the probability that Albert’s calculator will display the number  $6$  at some point?**p12.** For a positive integer  $k \ge 2$ , let  $f(k)$  be the number of positive integers  $n$  such that n divides  $(n-1)!+k$ . Find   $$ f(2) + f(3) + f(4) + f(5) + ... + f(100). $$  **p13.** Mr. Atf has nine towers shaped like rectangular prisms. Each tower has a  $1$  by  $1$  base. The first tower as height  $1$ , the next has height  $2$ , up until the ninth tower, which has height  $9$ . Mr. Atf randomly arranges these  $9$  towers on his table so that their square bases form a  $3$  by  $3$  square on the surface of his table. Over all possible solids Mr. Atf could make, what is the average surface area of the solid?**p14.** Let  $ABCD$  be a cyclic quadrilateral whose diagonals are perpendicular. Let  $E$  be the intersection of  $AC$  and  $BD$ , and let the feet of the altitudes from  $E$  to the sides  $AB$ ,  $BC$ ,  $CD$ ,  $DA$  be  $W, X, Y , Z$  respectively. Given that  $EW = 2EY$  and  $EW \cdot   EX \cdot EY \cdot EZ = 36$ , find the minimum possible value of  $\frac{1}{[EAB]} +\frac{1}{[EBC]}+\frac{1}{[ECD]} +\frac{1}{[EDA]}$ . The notation  $[XY Z]$  denotes the area of triangle  $XY Z$ .**p15.** Given that  $x^2 - xy + y^2 = (x + y)^3$ ,  $y^2 - yz + z^2 = (y + z)^3$ , and  $z^2 - zx + x^2 = (z + x)^3$  for complex numbers  $x, y, z$ , find the product of all distinct possible nonzero values of  $x + y + z$ .


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