
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. Bahri lives quite close to the clock gadang in the city of Bukit Tinggi West Sumatra. Bahri has an antique clock. On Monday  $4$ th March  $2013$  at  $10.00$  am, Bahri antique clock is two minutes late in comparison with Clock Tower. A day later, the antique clock was four minutes late compared to the Clock Tower. March  $6$ ,  $2013$  the clock is late six minutes compared to Jam Gadang. The following days Bahri observed that his antique clock exhibited the same pattern of delay. On what day and what date in  $2014$  the antique Bahri clock (hand short and long hands) point to the same number as the Clock Tower?


p2. In one season, the Indonesian Football League is participated by  $20$  teams football. Each team competes with every other team twice. The result of each match is  $3$  if you win,  $ 1$  if you draw, and  $0$  if you lose. Every week there are  $10$  matches involving all teams. The winner of the competition is the team that gets the highest total score. At the end what week is the fastest possible, the winner of the competition on is the season certain?


p3. Look at the following picture. The quadrilateral  $ABCD$  is a cyclic. Given that  $CF$  is perpendicular to  $AF$ ,  $CE$  is perpendicular to  $BD$ , and  $CG$  is perpendicular to  $AB$ . Is the following statements true?
Write down your reasons.   $$ \frac{BD}{CE}=\frac{AB}{CG}+ \frac{AD}{CF} $$  ![Image](https://cdn.artofproblemsolving.com/attachments/b/0/dbd97b4c72bc4ebd45ed6fa213610d62f29459.png)


p4. Suppose  $M=2014^{2014}$ . If the sum of all the numbers (digits) that make up the number  $M$  equals  $A$  and the sum of all the digits that make up the number  $A$  equals  $B$ , then find the sum of all the numbers that make up  $B$ .


p5. Find all positive integers  $n < 200$  so that  $n^2 + (n + 1)^2$  is square of an integer.

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