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.
, 
is the midpoint of 
. 
is an arbitrary point on 
. Let 
be the intersection of 
and 
. The line 
intersects with the circumcircle of 
, for the second time at 
. 
is the circumcenter of triangle 
. Prove that 
.
be an acute triangle, let 
and 
be its orthocentre and circumcentre, respectively,
and 
be the feet of the altitudes from 
to 
and from 
to 
, respectively.
be the midpoint of the segment 
, and let 
be the midpoint of the segment 
. The line, parallel to 
, crosses the sides and at 
and 
, respectively. The line 
again at 
, and the line 
meets the circle 
again at 
. Prove that
and 
are similar.
be the midpoint of on triangle . Point 
lies on segment such that 
and 
on line 
such that 
. Prove that 
.
be an odd prime number. Prove that for any integer 
, the equation 
has no solution in positive integers 
such that for all 
,
is a prime number, 
are natural numbers (
). It is known that 
. Prove that 
.
, how many digits does 
have?
not 
has the most factors?
has the most factors?
Prove that
to 
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 
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 
,
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