
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
<u>Round 1</u>**1.1.** A classroom has  $29$  students. A teacher needs to split up the students into groups of at most  $4$ . What is the minimum number of groups needed?**1.2.** On his history map quiz, Eric recalls that Sweden, Norway and Finland are adjacent countries, but he has
forgotten which is which, so he labels them in random order. The probability that he labels all three countries
correctly can be written as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .**1.3.** In a class of  $40$  sixth graders, the class average for their final test comes out to be  $90$  (out of a  $100$ ). However, a student brings up an issue with problem  $5$ , and  $10$  students receive credit for this question, bringing the class average to a  $90.75$ . How many points was problem  $5$  worth?

<u>Round 2</u>**2.1.** Compute  $1 - 2 + 3 - 4 + ... - 2022 + 2023$ .**2.2.** In triangle  $ABC$ ,  $\angle ABC = 75^o$ . Point  $D$  lies on side  $AC$  such that  $BD = CD$  and  $\angle BDC$  is a right angle. Compute the measure of  $\angle A$ .**2.3.** Joe is rolling three four-sided dice each labeled with positive integers from  $1$  to  $4$ . The probability the sum of the numbers on the top faces of the dice is  $6$  can be written as  $\frac{p}{q}$  where  $p$  and  $q$  are relatively prime integers. Find  $p + q$ .

<u>Round 3</u>**3.1.** For positive integers  $a, b, c, d$  that satisfy  $a + b + c + d = 23$ , what is the maximum value of  $abcd$ ?**3.2.** A buckball league has twenty teams. Each of the twenty teams plays exactly five games with each of the other teams. If each game takes 1 hour and thirty minutes, then how many total hours are spent playing games?**3.3.** For a triangle  $\vartriangle ABC$ , let  $M, N, O$  be the midpoints of  $AB$ ,  $BC$ ,  $AC$ , respectively. Let  $P, Q, R$  be points on  $AB$ ,  $BC$ ,  $AC$  such that  $AP =\frac13 AB$ ,  $BQ =\frac13 BC$ , and  $CR =\frac13 AC$ . The ratio of the areas of  $\vartriangle MNO$  and  $\vartriangle P QR$  can be expressed as  $\frac{m}{n}$  , where  $ m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .

<u>Round 4</u>**4.1.**  $2023$  has the special property that leaves a remainder of  $1$  when divided by  $2$ ,  $21$  when divided by  $22$ , and  $22$  when divided by  $23$ . Let  $n$  equal the lowest integer greater than  $2023$  with the above properties. What is  $n$ ?**4.2.** Ants  $A, B$  are on points  $(0, 0)$  and  $(3, 3)$  respectively, and ant A is trying to get to  $(3, 3)$  while ant  $B$  is trying to get to  $(0, 0)$ . Every second, ant  $A$  will either move up or right one with equal probability, and ant  $B$  will move down or left one with equal probability. The probability that the ants will meet each other be  $\frac{a}{b}$ , where  $a$  and  $b$  are relatively prime positive integers. Find  $a + b$ .**4.3.** Find the number of trailing zeros of  $100!$  in base  $ 49$ .


PS. You should  use hide for answers. Rounds 5-9 have been posted [here](https://artofproblemsolving.com/community/c3h3129723p28347714). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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