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 an odd prime and 
is a bijection and 
is an integer such that 
Show that 
is a multiple of .
. Let 
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of 
.
such that 
divides 
.
satisfies
for some 
and for all 
, then there exists a unique solution 
of the functional equation 
for all such that
for all 
.
be distinct polynomials of degree 
with non-zero coefficients. Suppose that they have 
common real roots counting multiplicity and 
common coefficients. Determine the maximum possible value of 
.
such that:
is strictly increasing,
such that 
for all 
,
, there exists 
such that![\[
f(y) = \frac{f(x) + f(x + 2024)}{2}.
\]](http://latex.artofproblemsolving.com/b/e/2/be26213154bb74bd5a35b8d160011351871bfa9b.png)
Proposed by Pavle Martinović
, how many digits does 
have?
not 
has the most factors?
has the most factors?
Prove that
Where 
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 
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