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.
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
be the set of real numbers. Find all functions 
such that, for all real numbers 
and 
, the following equation holds:
be a positive integer. If 
is a power of a prime number, prove that is also prime.
positive real numbers 
. For each 
we let 
(here we define 
to be 
and 
to be 
). Assume that for all 
and 
in the range 
to 
, we have 
if and only if 
.
.
such that for any 
, 
such that 
is a perfect square.
, how many digits does 
have?
not 
has the most factors?
has the most factors?
a perfect square?
Prove that
have the digit 
and are divisible by 
?
now, we have covered all the numbers in the 7 ones place. (27, if I counted right) now, lets find numbers with tens digit 7 that do NOT have 7 in its one place, since we already counted the _77 in the first case.
this totals up to 25 numbers, so the total is
so 
?
without a calculator.
for 
:
:
.
and 
where the rest has been omitted because of the floor function. Now 
and multiplying by 
yields 
grid is made. 
unit squares are created by separating the side with length 
in 
and 
,
rectangles can be created.
grid, each unit square is created by dividing the sides of lengths 
and 
.
horizontal lines and 
vertical lines:
that can be formed, we multiply the number of ways to choose the horizontal lines and the ways to choose the vertical lines:
satisfy 
,
and 
are positive integers, what is the probability that 
is divisible by 
?
?
.
,
represents a distinct digit from 
to 
.
is our answer. (By python, 
does not work and surprisingly, 
is the only permutation number that is divisible by 
8907