AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
, in square units, is![\[ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 46 \qquad
\textbf{(D)}\ 66 \qquad
\textbf{(E)}\ 74
\]](http://latex.artofproblemsolving.com/8/2/f/82ffc15072ca99f6cd1abc312781eb4827817356.png)
+1 w[hide=P#]a1b2c3[/hide] [hide=S#]a1b2c3[/hide]
and 
are numbers that work, but 
and 
don't work.)
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of are either all odd or all even.
is interesting if 2018 divides 
(the number of positive divisors of ). Determine all positive integers 
such that there exists an infinite arithmetic progression with common difference whose terms are all interesting.
with its center 
. There are two fixed points 
and one moving point 
on . The midpoint of the line segment 
is 
. 
is a fixed point on . Line 
intersects
at 
, and line 
intersects 
at 
. such that is always tangent to 
when varies.
such that:![\[
a^2 + b^2 \mid a^3 + b^3
\]](http://latex.artofproblemsolving.com/a/1/d/a1ddbd20c854e291da5670de449229d1664fa56c.png)
be an isosceles triangle with 
. The circle 
, passing through 
and 
, intersects segment 
at 
. The circle 
is tangent to at and passes through 
. Let and 
be the midpoints of segments and 
, respectively. The line 
intersects and at points 
and 
, respectively, where and are the intersections closer to . Prove that 
.
such that for any 
:
, then either travels [list] units vertically (up or down) and 
units horizontally (left or right); or units horizontally (left or right) and units vertically (up or down)., the ant can choose to go to one of eight possible points.
and 
, the ant can travel from 
to 
using at most 
moves.
, the sides and 
are parallel and the angles 
and 
are acute. Show that it is possible to divide the triangle into 4 disjoint triangle 
and the triangle 
into 4 disjoint triangles 
such that the triangles 
and 
are congruent for all 
.
is given in the plane, let 
be the set of points which lie strictly inside it. Also let 
be the set of circles in the plane. For a point 
, let 
be the reflection of in sides 
respectively. Define a function 
such that 
is the circumcircle of 
. Is injective? is called injective if for any 
, 
)
20Test
231.pdf]these[/url] (change the last number in http://agmath.com/media//DIR_57727/2003-200420Test231.pdf to get more) around half an hour before the test started. I didn't score it or anything, nor did I actually care about the results, but I found it was a good way to "wake up" before a competition.
20Test231.pdf]these[/url] (change the last number in http://agmath.com/media//DIR_57727/2003-200420Test231.pdf to get more) around half an hour before the test started. I didn't score it or anything, nor did I actually care about the results, but I found it was a good way to "wake up" before a competition.
:o
so 
and since 
is just 
therefore 
so 
