1 A point 
in the first quadrant lies on a line with intercepts 
and 
, with ![]()
0 a,b > 0″ />. Rectangle 
has vertices 
, 
, , and 
, while rectangle 
has vertices , 
, 
, and 
. What is the ratio of the area of to that of ?
2 Two sequences 
and 
are defined as follows: 
and 
. If both sequences are defined with 
ranging across the natural numbers, how many numbers belong to both sequences?
3 Billy and Bobby are located at points 
and 
, respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point ; the second time they meet, they are located 10 units from point . Find all possible values for the distance between and .
4 In the following alpha-numeric puzzle, each letter represents a different non-zero digit. What are all possible values for 
?
5 We have eight light bulbs, placed on the eight lattice points (points with integer coordinates) in space that are 
units away from the origin. Each light bulb can either be turned on or off. These lightbulbs are unstable, however. If two light bulbs that are at most 2 units apart are both on simultaneously, they both explode. Given that no explosions take place, how many possible configurations of on/off light bulbs exist?
6 Circle 
with diameter 
has chord 
drawn such that is perpendicular to at 
. Another circle 
is drawn, sharing chord . A point 
on minor arc of is chosen so that 
. Line 
is tangent to through and a point 
on is chosen such that 
. If 
and 
, find 
.
7 The number 
has 
trailing zeroes when expressed in base .
a) Find the minimum possible .
b) Find the maximum possible .
c) Find the total number of possible .
8 Define 
to be the nearest integer to 
, with the greater integer chosen if two integers are tied for being the nearest. For example, 
, 
, and 
. Define ![[A]](http://data.artofproblemsolving.com/images/latex/b/d/6/bd686ccc367c41afe174cdfd587b04224e0aeacc.gif "[A]")
to be the area of region . Define region 
, for each positive integer 
, to be the region on the Cartesian plane which satisfies the inequality <img title=”f(|x|) + f(|y|) < n” src=”http://data.artofproblemsolving.com/images/latex/d/c/3/dc30d44d8951d826bd03abacc3ea16d5410bdb23.gif” alt=”f(|x|) + f(|y|) . We pick an arbitrary point 
on the perimeter of , and mark every two units around the perimeter with another point. Region 
is defined by connecting these points in order.
a) Prove that the perimeter of is always congruent to 
.
b) Prove that ![[S_{nO}]](http://data.artofproblemsolving.com/images/latex/d/3/0/d30c9a07454b9292490497ebdecbcd08efa8fa80.gif "[S_{nO}]")
is constant for any .
c) Prove that ![[R_n] + [S_{nO}] = (2n-1)^2](http://data.artofproblemsolving.com/images/latex/0/8/4/084db52827725c71509c677d95bcffe80747f290.gif "[R_n] + [S_{nO}] = (2n-1)^2")
.
soal kompetisi matematika AMC 10
