
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
**p13.** Let  $\vartriangle ABC$  be an equilateral triangle with side length  $1$ . Let the unit circles centered at  $A$ ,  $B$ , and  $C$  be  $\Omega_A$ ,  $\Omega_B$ , and  $\Omega_C$ , respectively. Then, let  $\Omega_A$  and  $\Omega_C$  intersect again at point  $D$ , and  $\Omega_B$  and  $\Omega_C$  intersect again at point  $E$ . Line  $BD$  intersects  $\Omega_B$  at point  $F$  where  $F$  lies between  $B$  and  $D$ , and line  $AE$  intersects  $\Omega_A$  at  $G$  where  $G$  lies between  $A$  and  $E$ .  $BD$  and  $AE$  intersect at  $H$ . Finally, let  $CH$  and  $FG$  intersect at  $I$ . Compute  $IH$ .**p14.** Suppose Bob randomly fills in a  $45 \times 45$  grid with the numbers from  $1$  to  $2025$ , using each number exactly once. For each of the  $45$  rows, he writes down the largest number in the row. Of these  $45$  numbers, he writes down the second largest number. The probability that this final number is equal to  $2023$  can be expressed as  $\frac{p}{q}$  where  $p$  and  $q$  are relatively prime positive integers. Compute the value of  $p$ .**p15.**  $f$  is a bijective function from the set  $\{0, 1, 2, ..., 11\}$  to  $\{0, 1, 2, ... , 11\}$ , with the property that whenever  $a$  divides  $b$ ,  $f(a)$  divides  $f(b)$ . How many such  $f$  are there? 
*A bijective function maps each element in its domain to a distinct element in its range.*

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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