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.
+1 w
(use in a problem for junior) to practice. Can anyone share to me?
be a square with 
side length and with center 
. A rhombus 
is drawn where point 
lies on line 
. What is the area of ?
be an odd integer, 
. Consider 
integers 
such that for any 
, 
and 
. Prove that the number of 
for which satisfy 
for some 
, 
is greater than 
.
such that![\[f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy\]](http://latex.artofproblemsolving.com/e/b/c/ebccae7e765d9fc3c8605e58a7399a70127b39b1.png)
, such that 
Prove that :
equality cases : ?
distinct positive integers such that adding the product of any two of them to 
yields a perfect square?
be a sequence of positive integers with 
, 
and![\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]](http://latex.artofproblemsolving.com/1/b/f/1bf1a88979e7d2b33530de979d66a4e81032953d.png)
for all 
. Show that if 
is a prime less than 
for some positive integer 
, then there exists 
such that 
.
for which
be an acute angled scalene triangle with circumcircle 
. Let 
and 
be the circumcenter and orthocenter of 
respectively. Let 
and 
be points on segments 
and , respectively, such that
Prove that 
and 
intersect on the circumcircle of 
.
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let 
be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
is called feirense if it is possible to write on a sheet of paper some integers such that every positive divisor of less than is the difference between two numbers on the sheet, and no other positive number is.
bisector of angle 
intersects side 
in point 
. Bisector of line segment 
intersects circumcircle of triangle in points and 
. Show that circumcircle of triangle 
is tangent to line .
-appolonius circle of 
and notice that 
must lie on the perpendicular bisector of 
, since the intersection of the tangent to at and 
is the center of the -appolonius circle and is the center of the -appolonius circle.
in 
.since 
it is enough to prove 
which is equivalent to prove 
.
are bisectors in 
so also bisect 
therefore, as desired .
, 
, 
and 
. Now observe, 
is the reflection of over 
, hence, is the orthocenter of 
and since, 
is the circum-orthic triangle WRT , hence, is the incenter WRT 
are isogonal WRT 
hence, 
, hence, is tangent to 
bisector of angle intersects side in point . Bisector of line segment intersects circumcircle of triangle in points and . Show that circumcircle of triangle is tangent to line .
= 
and 
=
=1/2. + 
and 
=1/2. + 
now if we look at last 2 equations we notice that = - = and now proof is completed
to circumcircle meet at 
. 
so 
is isosceles so lies on so 
so 
is tangent to .
meet 
by pop we have 
so done.
.
.Since 
,
.
by 
,
gives 
tangent to 
.![[asy]
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