
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
<span style="font-size:200%"><u>Test 1</u></span>

1. Let  ${a_n}$  be the sequence of rational numbers defined by  $a_0 = 2021,$  and  $a_{n+1}=a_{n}+\frac{2}{a_{n}},$  for all  $n \geq 0.$  Can any  $a_{n}, n \geq 1,$  be a square of a rational number?

2. Let  $a, b, c, d$  be the roots of the equation  $x^4+x+1=0.$  Let  $a^5+2a+1, b^5+2b+1, c^5+2c+1, d^5+2d+1$  be roots of the equation  $x^4+px^3+qx^2+rx+s=0.$  Find the value of  $p+2q+4r+8s.$ 3. Four non-overlapping families dine at a restaurant together. Each family consists of one father, one mother and three children. There are five tables, which are red, blue, yellow, green and brown respectively. Every table can accomodate four people, and every person must sit at some table. How many ways are there to assign the 20 people to tables so that each child sits at the same table as at least one of his/her parents? (We do not distinguish between the four seats at the same table.)

4. Let  $ABCD$  be a cyclic quadrilateral with circumcenter  $O.$  Diagonals  $AC$  and  $BD$  meet at  $E.$   $F$  and  $G$  are points on segments  $AB$  and  $CD$  respectively. Suppose  $AF=15,$   $FB=13,$   $BE=30,$   $ED=13,$   $DG=7.5,$  and  $GC=6.5.$  Let  $P$  be a point such that  $PF\perp AB$  and  $PG\perp CD.$  Find  $\frac{PE}{PO}.$ 5. Let  $f(n)=\left|\binom{n}{0}+\binom{n}{3}+\dots +\binom{n}{3[\frac{n}{3}]}-\frac{2^n}{3}\right|,$  where  $[x]$  is the greatest integer not exceeding  $x.$  Find  $f(1)+f(2)+\dots+f(2021).$  

6. There is a set of  $n$   $01$ -sequences of length  $200.$  Every pair of  $01$ -sequences differ at least at  $101$  positions. (For example, the two  $01$ -sequences of length  $6,$   $111100$  and  $010001$  differ at four positions,  $1$ st,  $3$ rd,  $4$ th and  $6$ th positions, counting from the left.) Is it possible that  $n \geq 101?$ 

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