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.
, such that 
and 
.
such that 
is prime, 
with 
and ?
be a positive integer. Egor has 
cards with the number “
” written on them, and cards with the number “
” written on them. Egor wants to paint each card red or blue so that no subset of cards of the same color has the sum of the numbers equal to . Find the greatest such that Egor will not be able to paint the cards in such a way.
that has 8097 distinct points with area of a triangle that has 3 points belong to all 
. Prove that there exists a triangle 
that has the area 
contains at least 2025 points that belong to ( each of that 2025 points can be inside the triangle or lie on the edge of triangle )X
is even it's contradiction. I mean 
.
and 
,we can see the only solution is
and it is contradiction.
and 
. So 
.İt means 
.
but 
.İt is contradiction.
we have 2 option. Firstly 
but 
is contradiction. Secondly 
but is contradiction. I mean 
. So 
. If 
but is contradiction. I mean 
. The only solution is 
.
of positive integers, a partition of into two disjoint nonempty subsets 
and 
is 
if the least common multiple of the elements in is equal to the greatest common divisor of the elements in . Determine the minimum value of 
such that there exists a set of positive integers with exactly 
good partitions.
are lengths of sides of some triangle. Prove the inequality
where 
. Point 
is the circumcenter of triangle , and 
is the projection of point 
onto line 
. The midpoints of , 
, and 
are 
, 
, and 
, respectively. The line 
intersects 
and 
at points 
and 
, respectively. Prove that is the incenter of triangle 
.
-tuple 
, 
, of symbols from , we define a sequence 
according to the following rule: If 
, then 
, where 
if 
(taking 
) and 
is the symbol other than 
if 
. Find all positive integers 
for which there exists some integer 
such that 
.
be an acute triangle with orthocenter . Let , 
and 
be the feet of the altitudes of triangle opposite to vertices , 
, and 
respectively. Let 
and 
be the midpoints of 
and 
, respectively. Let 
be the intersection of lines 
and 
. Prove that is the circumcenter of triangle if and only if is the midpoint of 
.
and 
be two prime numbers, with being a divisor of 
. Prove that there exist integers 
such that the polynomial 
is divisible by the polynomial 
with 
is a multiple of and 
.
be the circumcircle of acute triangle . Points 
and 
are on segments 
and 
respectively such that 
. The perpendicular bisectors of 
and 
intersect minor arcs and of at points 
and respectively. Prove that lines 
and 
are either parallel or they are the same line. be a scalene acute triangle with incenter 
and circumcircle 
. 
is the midpoint of small arc 
on and 
is the projection of onto the line passing through the midpoints of and . A circle 
with center 
is internally tangent to at , and touches segment . If the circle with diameter 
meets again at 
, prove that 
bisects 
.
be a right-angled triangle with 
and let be the altitude from on to . Draw 
and 
. Let 
and 
be respectively the incentres of triangle 
and 
. Suppose 
are collinear. Prove that 
lie on a circle.
?
.
centered at taking triangle 
to 
.
and thus the corresponding incenters will also be mapped to each other that is
maps line 
to 
.
.
are collinear we get 
are concyclic
and 
Prove that
so it suffices to show that 
.
and 
.
,
as desired.