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.
, 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.
be a set of 
elements and 
are subsets of (
), such that every one of them has at least 
elements.
and 
, with 
, such that the number of common elements of 
and 
is greater or equal to: 
, 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 .
[hide=P#]a1b2c3[/hide] [hide=S#]a1b2c3[/hide]
has side lengths , , and 
. Select a point 
inside , and construct the incenters of 
, 
, and 
and denote them as 
, 
, 
. What is the maximum area of the triangle 
?
be non-negative real numbers such that 
Prove that
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.
). We have 
distribute the 
factors of 
among the values. Continuing for all primes gives the product 
. For the example, the answer is 
. The number of unordered pairs would be messy because you have to do casework with pairs containing repeated values.
, how many pairs of numbers multiply to 
:o
so 
and since 
is just 
therefore 
so 
,![\[ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 46 \qquad
\textbf{(D)}\ 66 \qquad
\textbf{(E)}\ 74
\]](http://latex.artofproblemsolving.com/8/2/f/82ffc15072ca99f6cd1abc312781eb4827817356.png)
such that
with prime factorization 
,
numbers that multiply to 
,
,
then their product far exceed 
)
I think?