[i]20 problems for 20 minutes.[/i]


[b]p1.[/b] Determine how many digits the number $10^{10}$ has.


[b]p2.[/b] Let $ABC$ be a triangle with $\angle ABC = 60^o$ and $\angle BCA = 70^o$. Compute $\angle CAB$ in degrees.


[b]p3.[/b] Given that $x : y = 2012 : 2$ and $y : z = 1 : 2013$, compute $x : z$. Express your answer as a common fraction.


[b]p4.[/b] Determine the smallest perfect square greater than $2400$.


[b]p5.[/b] 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.


[b]p6.[/b] Given that $ \sqrt{3^a \cdot 9^a  \cdot 3^a} = 81^2$, compute $a$.


[b]p7.[/b] Find the number of positive integers less than $8888$ that have a tens digit of $4$ and a units digit of $2$.


[b]p8.[/b] Find the sum of the distinct prime divisors of $1 + 2012 + 2013 + 2011 \cdot 2013$.


[b]p9.[/b] 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.


[b]p10.[/b] 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.


[b]p11.[/b] 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.


[b]p12.[/b] Determine the smallest positive integer $n$ such that the sum of the digits of $11^n$ is not $2^n$.


[b]p13.[/b] 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.)


[b]p14.[/b] 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 $0$).


[b]p15.[/b] 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.


[b]p16.[/b] 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 $PRIMES$.


[b]p17.[/b] 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.


[b]p18.[/b] 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 $2\sqrt{x^2 -4} = 2(\sqrt{x^2} -\sqrt4)$.


[b]p19.[/b] 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 $mn$.


[b]p20.[/b] 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 [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
### Problem 2
1. Given that $\angle ABC = 60^\circ$ and $\angle BCA = 70^\circ$ in triangle $ABC$.
2. The sum of the angles in a triangle is $180^\circ$.
3. Therefore, $\angle CAB = 180^\circ - (\angle ABC + \angle BCA)$.
4. Substituting the given angles, we get:
   \[
   \angle CAB = 180^\circ - (60^\circ + 70^\circ) = 180^\circ - 130^\circ = 50^\circ.
   \]

The final answer is $\boxed{81}$.