
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
*20 problems for 20 minutes.***p1.** Determine how many digits the number  $10^{10}$  has.**p2.** Let  $ABC$  be a triangle with  $\angle ABC = 60^o$  and  $\angle BCA = 70^o$ . Compute  $\angle CAB$  in degrees.**p3.** Given that  $x : y = 2012 : 2$  and  $y : z = 1 : 2013$ , compute  $x : z$ . Express your answer as a common fraction.**p4.** Determine the smallest perfect square greater than  $2400$ .**p5.** At  $12:34$  and  $12:43$ , the time contains four consecutive digits. Find the next time after 12:43 that the time contains four consecutive digits on a 24-hour digital clock.**p6.** Given that  $ \sqrt{3^a \cdot 9^a  \cdot 3^a} = 81^2$ , compute  $a$ .**p7.** Find the number of positive integers less than  $8888$  that have a tens digit of  $4$  and a units digit of  $2$ .**p8.** Find the sum of the distinct prime divisors of  $1 + 2012 + 2013 + 2011 \cdot 2013$ .**p9.** Albert wants to make  $2\times 3$  wallet sized prints for his grandmother. Find the maximum possible number of prints Albert can make using one  $4 \times 7$  sheet of paper.**p10.** Let  $ABC$  be an equilateral triangle, and let  $D$  be a point inside  $ABC$ . Let  $E$  be a point such that  $ADE$  is an equilateral triangle and suppose that segments  $DE$  and  $AB$  intersect at point  $F$ . Given that  $\angle CAD = 15^o$ , compute  $\angle DFB$  in degrees.**p11.** A palindrome is a number that reads the same forwards and backwards; for example,  $1221$  is a palindrome. An almost-palindrome is a number that is not a palindrome but whose first and last digits are equal; for example,  $1231$  and  $1311$  are an almost-palindromes, but  $1221$  is not. Compute the number of  $4$ -digit almost-palindromes.**p12.** Determine the smallest positive integer  $n$  such that the sum of the digits of  $11^n$  is not  $2^n$ .**p13.** Determine the minimum number of breaks needed to divide an  $8\times 4$  bar of chocolate into  $1\times 1 $ pieces. (When a bar is broken into pieces, it is permitted to rotate some of the pieces, stack some of the pieces, and break any set of pieces along a vertical plane simultaneously.)**p14.** A particle starts moving on the number line at a time  $t = 0$ . Its position on the number line, as a function of time, is   $$ x = (t-2012)^2 -2012(t-2012)-2013. $$   Find the number of positive integer values of  $t$  at which time the particle lies in the negative half of the number line (strictly to the left of  $0$ ).**p15.** Let  $A$  be a vertex of a unit cube and let  $B$ , $C$ , and  $D$  be the vertices adjacent to A. The tetrahedron  $ABCD$  is cut off the cube. Determine the surface area of the remaining solid.**p16.** In equilateral triangle  $ABC$ , points  $P$  and  $R$  lie on segment  $AB$ , points  $I$  and  $M$  lie on segment  $BC$ , and points  $E$  and  $S$  lie on segment  $CA$  such that  $PRIMES$  is a equiangular hexagon. Given that  $AB = 11$ ,  $PS = 2$ ,  $RI = 3$ , and  $ME = 5$ , compute the area of hexagon  $PRIMES$ .**p17.** Find the smallest odd positive integer with an odd number of positive integer factors, an odd number of distinct prime factors, and an odd number of perfect square factors.**p18.** Fresh Mann thinks that the expressions  $2\sqrt{x^2 -4} $ and  $2(\sqrt{x^2} -\sqrt4)$  are equivalent to each other, but the two expressions are not equal to each other for most real numbers  $x$ . Find all real numbers  $x$  such that  $2\sqrt{x^2 -4} = 2(\sqrt{x^2} -\sqrt4)$ .**p19.** Let  $m$  be the positive integer such that a  $3 \times 3$  chessboard can be tiled by at most  $m$  pairwise incongruent rectangles with integer side lengths. If rotations and reflections of tilings are considered distinct, suppose that there are  $n$  ways to tile the chessboard with  $m$  pairwise incongruent rectangles with integer side lengths. Find the product  $mn$ .**p20.** Let  $ABC$  be a triangle with  $AB = 4$ ,  $BC = 5$ , and  $CA = 6$ . A triangle  $XY Z$  is said to be friendly if it intersects triangle  $ABC$  and it is a translation of triangle  $ABC$ . Let  $S$  be the set of points in the plane that are inside some friendly triangle. Compute the ratio of the area of  $S$  to the area of triangle  $ABC$ .


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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