
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.** Evaluate  $1+3+5+··· +2019$ .**p2.** Evaluate  $1^2 -2^2 +3^2 -4^2 +...· +99^2 -100^2$ .**p3.**Find the sum of all solutions to  $|2018+|x -2018|| = 2018$ .**p4.** The angles in a triangle form a geometric series with common ratio  $\frac12$  . Find the smallest angle in the triangle.**p5.** Compute the number of ordered pairs  $(a,b,c,d)$  of positive integers  $1 \le a,b,c,d \le  6$  such that  $ab +cd$  is a multiple of seven.**p6.** How many ways are there to arrange three birch trees, four maple, and five oak trees in a row if trees of the same species are considered indistinguishable.**p7.** How many ways are there for Mr. Paul to climb a flight of 9 stairs, taking steps of either two or three at a time?**p8.** Find the largest natural number  $x$  for which  $x^x$  divides  $17!$ **p9.** How many positive integers less than or equal to  $2018$  have an odd number of factors?**p10.** Square  $MAIL$  and equilateral triangle  $LIT$  share side  $IL$  and point  $T$  is on the interior of the square. What is the measure of angle  $LMT$ ?**p11.** The product of all divisors of  $2018^3$  can be written in the form  $2^a \cdot 2018^b$  for positive integers  $a$  and  $b$ . Find  $a +b$ .**p12.** Find the sum all four digit palindromes. (A number is said to be palindromic if its digits read the same forwards and backwards.**p13.** How ways are there for an ant to travel from point  $(0,0)$  to  $(5,5)$  in the coordinate plane if it may only move one unit in the positive x or y directions each step, and may not pass through the point  $(1, 1)$  or  $(4, 4)$ ?**p14.** A certain square has area  $6$ . A triangle is constructed such that each vertex is a point on the perimeter of the square. What is the maximum possible area of the triangle?**p15.** Find the value of ab if positive integers  $a,b$  satisfy  $9a^2 -12ab +2b^2 +36b = 162$ .**p16.**  $\vartriangle ABC$  is an equilateral triangle with side length  $3$ . Point  $D$  lies on the segment  $BC$  such that  $BD = 1$  and  $E$  lies on  $AC$  such that  $AE = AD$ . Compute the area of  $\vartriangle ADE$ .**p17**. Let  $A_1, A_2,..., A_{10}$  be  $10$  points evenly spaced out on a line, in that order. Points  $B_1$  and  $B_2$  lie on opposite sides of the perpendicular bisector of  $A_1A_{10}$  and are equidistant to  $l$ . Lines  $B_1A_1,...,B_1A_{10}$  and  $B_2A_1,...· ,B_2A_{10}$  are drawn. How many triangles of any size are present?**p18.** Let  $T_n = 1+2+3··· +n$  be the  $n$ th triangular number. Determine the value of the infinite sum  $\sum_{k\ge 1} \frac{T_k}{2^k}$ .**p19.** An infinitely large bag of coins is such that for every  $0.5 < p \le 1$ , there is exactly one coin in the bag with probability  $p$  of landing on heads and probability  $1- p$  of landing on tails. There are no other coins besides these in the bag. A coin is pulled out of the bag at random and when flipped lands on heads. Find the probability that the coin lands on heads when flipped again.**p20.** The sequence  $\{x_n\}_{n\ge 1}$  satisfies  $x1 = 1$  and  $(4+ x_1 + x_2 +··· + x_n)(x_1 + x_2 +··· + x_{n+1}) = 1$  for all  $n \ge 1$ . Compute  $\left \lfloor \frac{x_{2018}}{x_{2019}} \right \rfloor$ .


PS. You had better use hide for answers.

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