[/quote]
,
be an odd prime and 
be integers so that the integers 
are divisible by 
such that:
for any positive integer 
,
are all distinct, and
,
?
lies on a line segment 
between 
and 
and circles are drawn having 
and 
as diameters. A common tangent to both circles touches the circle with 
and the circle with 
.
and the common tangent to both circles at 
and 
denote three distinct integers, and let 
denote a polynomial having integer coefficients. Show that it is impossible that 
and 
.
is a minuan pair if it satisfies the following two conditions:
is even.
are all the positive divisors of 
,
is precisely
mod 3 can be written as the sum of one square and 2 cubes.
,
be a point on the segment 
and let 
be the intersection point of 
with the diagonal 
.
at point 
.
be a point on the segment 
such that 
.
be the intersection point of the diagonal 
.
and 
,
and 
be the respective intersections with BC of the internal angle bisector, external angle bisector, and the median from A. The circumcircle of 
intersects 
a second time at point 
different from A. Define 
and 
analogously. Prove that the circumcenter of 
lies on the Euler line of ABC.
be a polynomial of degree 
with integer coefficients and let 
be a positive integer. Consider the polynomial 
,
such that 
.
,
and 
meet at the orthocenter 
.
from 
.
meets 
at 
.
is the midpoint of 
is perpendicular to the angle bisector of 
be one to one maps. Show that the function 
defined by 
,
,
be a surjective function. Show that there exist surjective functions 
such that 
,
prove that for each 
there exists 
such that either 
or 
is multiple of 
.
and 
at 
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This is just for fun. I bet it would probably stop after a while when the questions become too difficult, so please make it at most IMO Q1 standard (preferably lower). Anyway, the rules are the same as other marathons, and here is a sample:
Problem 0
Does there exist real numbers
, both not equal to 0, such that 
and 
?
Solution 0
Yes, take
. Then 
and .
Please don't just write the answer yes, or no. If yes, give an example or show it exists, and if no, prove it.
Ok, now for the real one.
Problem 1
Does there exist positive integers
such that 
and 
are both squares?
Problem 0
Does there exist real numbers
, both not equal to 0, such that 
and 
?Solution 0
Yes, take
. Then 
and .Please don't just write the answer yes, or no. If yes, give an example or show it exists, and if no, prove it.
Ok, now for the real one.
Problem 1
Does there exist positive integers
such that 
and 
are both squares?
and 
.
and 
are positive, 
or 
or 
which means 
, similarly 
which, obviously isn't possible. Hence, there don't exist any such positive integers.
for any nonnegative integer 
points of the same color are distance 
black if 
is odd and white if 
be any triangle. Does there always exists a point 
?
on the sides 
respectively. Let the second intersection of their circumcircles be 
.
.
.
which is divisible by every no. lesser than its square root?
then we have 
divides 
fo
.
all divide 
but it means ![$lcm[1, 2, 3, \cdots, k]|n$](http://latex.artofproblemsolving.com/3/1/f/31fe3e1d8ba8366aae3b17b4584be056dee40c98.png)
.
and now, ![$n\Longrightarrow [1, 2, \cdots, k]$](http://latex.artofproblemsolving.com/f/0/a/f0a0e4b835738d879984c5c4a24bb530e84034ca.png)
divide 
be the 
prime where 
is natural. And also, take natural 
.![$[1, 2, \cdots, k]>\prod_{i=1}^m p_i>8p_{m}p_{m-1}$](http://latex.artofproblemsolving.com/1/0/3/1039f9099d43969c174df51a460398b77a45dbad.png)
since 
and multiplying the above two inequalities gives 
and we have a clear contradiction.
such that 
divides 
for all natural 
which is impossible if 
b
so that 
so that 
c
,
.
.
.
wich implies that there exists 
positive integers such that 
By Zsigmondy on the third equation, we get 
wich does not yield a solution for the first equation.
is true and 
is not 
works.
or 
cannot be partitioned into disjoint 
s
:
be the 
cyclotomic polynomial. Let 
denote the polynomial formed by homogenising the 
such that 
are all perfect squares?
