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.
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 circumscribed about circle 
, and let be the orthocenter of 
. The circle touches line at . The tangent to the circle 
at meets at 
. Let be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
be two nonzero real numbers such that 
Consider the functions 
and 
satisfying the condition ![$$g(f(x+by))=a[f(x)+2yg(x)]+2xg(y)+bg(y)[y+g(y)],\quad\forall x,y\in\mathbb{R}.$$](http://latex.artofproblemsolving.com/3/c/9/3c974cb76aae7f529dcd85202b0c7d441a07d002.png)
(a) Prove that there exists a real number such that 
(b) Find all functions 
and 
that satisfy the given condition.
will be denoted by ![$ [\mathcal{F}] $](http://latex.artofproblemsolving.com/a/8/3/a8310f7269efa4a158b09927762b1d9b1e11be63.png)
and 
respectively.![$ r=\frac{2[\triangle AEF]}{p(\triangle AEF)}=\frac{2[\triangle BDF]}{p(\triangle BDF)}=\frac{2[\triangle CDE]}{p(\triangle CDE)}=\frac{2[\triangle ABC]-2[\triangle DEF]}{p(\triangle ABC)+p(\triangle DEF)}= $](http://latex.artofproblemsolving.com/0/3/b/03babbd8af4ddebf2c91fa307ed67fd58906370c.png)
.
touches 
at 
, then we have 
and thus 
. But since 
is the homothetic center of incircles of and 
, we get 
. Thus, we have 
. Similarly, we get 
and 
. Adding, we have 
.
. 
and 
Prove that
, such that 
Prove that :
equality cases : ?
lie on sides 
respectively. Moreover, the radii of incircles of 
are equal to 
.
and 
the radii of incircles of 
and 
.