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that exist function 
such 
continuous on 
and discontinuous on 
be a continuous function defined on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
with 
and 
. We define the point 
to be the projection of point on the x-axis. Prove that there exist points 
such that 
is a square.
satisfy 
for all 
. Prove that
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
, there exists some constant 
, for any 
, among the sequence 
and 
could be chosen to satisfy 
and 
.
be an acute, scalene triangle, and let 
, 
, and 
be the midpoints of 
, 
, and 
, respectively. Let the perpendicular bisectors of and 
intersect ray 
in points 
and 
respectively, and let lines 
and 
intersect in point 
, inside of triangle . Prove that points 
, , , and all lie on one circle.
be a parallelogram with 
. A circle tangent to sides 
, 
, and 
intersects diagonal 
at points 
and 
with 
, as shown. Suppose that 
, 
, and 
. Then the area of can be expressed in the form 
, where 
and 
are positive integers, and is not divisible by the square of any prime. Find 
.
and 
intersect at points and 
. Line 
is tangent to and at 
and 
, respectively, with line 
closer to point than to . Circle 
passes through and intersecting again at 
and intersecting again at 
. The three points , , 
are collinear, 
, 
, and 
. Find 
.
, we know that lines 
and 
intersect at 
, lines and 
intersect at 
, and lines 
and 
intersect at 
. Suppose that the circumcircle of 
intersects line 
at and , and the circumcircle of 
intersects line at and , where 
and 
are collinear in that order. Prove that if lines 
and 
intersect at 
, then 
.
and 
are written on a blackboard, where 
and 
are relatively prime positive integers. At any point, Evan may pick two of the numbers 
and 
written on the board and write either their arithmetic mean 
or their harmonic mean 
on the board as well. Find all pairs 
such that Evan can write 1 on the board in finitely many steps.![\[ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots \]](http://latex.artofproblemsolving.com/a/c/c/acc097f524dc9fe9bd85e8eb659ea4a75aec475d.png)
in the plane with the following property: For any three distinct integers 
and 
, points 
, 
, and 
are collinear if and only if 
.
![$f:[0,1]->R$](http://latex.artofproblemsolving.com/4/4/6/4462365b0a7312aba97a0ac040c492240c3884f8.png)
one function diferențiale whirt 
integable and 
.
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