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 a right-angled triangle where ∠
=90°.Let 
be an altitude of that triangle and points 
and 
be the midpoints of and 
, respectively.If 
is the circumcenter of the triangle 
, prove that 
and are paralel.
, let 
be a point (
). Let 
be the anticomplement of , and let 
be the 
-anticomplementary conjugate of . Prove that the line 
is parallel to the real asymptote of the circular pivotal isocubic with pivot .
and 
. Prove that there always exists a positive integer 
such that 
.
is a power of a prime number, but for arbitrary then well.....
be a positive integer. Define a sequence by setting 
and, for each 
, letting 
be the unique integer in the range 
for which 
is divisible by 
. For instance, when 
the obtained sequence is 
. Prove that for any the sequence 
eventually becomes constant.
, points 
lie on sides 
, respectively, such that 
. Altitudes 
meet at orthocenter 
. Points and lie on segment 
such that 
and 
. Lines 
and 
intersect at point 
. Compute 
. be a right angled triangle with 
. Point is the midpoint of side And be an arbitrary point on 
. The reflection of 
over 
intersects lines 
and at 
and , respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
satisfy the equality, 
prove that the triangle must have a right angle.
such that for any real numbers 
and 
, the following inequality holds:![\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq (f(x+y))^2
\]](http://latex.artofproblemsolving.com/8/4/3/8437449cd3f8f0da24d17a60b55dcb563e6b5655.png)
defined by 
and![\[
u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/9/9/4/994aa754cc1288ce4f28a95a0276e64282fb5f66.png)
a) Prove that there exist infinitely many positive integers 
such that 
.![\[
\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.
\]](http://latex.artofproblemsolving.com/f/f/1/ff174e7431cbfbc17c650d109651241286756a1a.png)
is a cyclic quadrilateral. Let 
and 
. Denote by 
and 
the orthocenters of triangles 
and 
, respectively. Prove that the points 
, , are collinear.
be a triangle. A circle passing through 
and 
intersects the sides 
and 
again at 
and 
respectively. Prove that 
, 
and 
are concurrent, where 
and 
are the orthocentres of triangles and 
respectively.
.The Incircle touches sides and in 
and ,respectively. Lines CI and EF intersect at . The point ≠ is on the line AI so that 
=
.If 
is the foot of the altitude from 
in triangle ,prove that points , and are colinear.
and 
are same then obviously your answer is wrong. Actually the answer would be 
I guess. Trying to prove.....
case is trivial which gives us 
....and the rest as follows
doesn't necessarily divide that part....
be a prime such that 
for 
. Using![\[
a^3+b^3+1 - 3ab = (a+b+1)(a^2+b^2+1-ab-a-b)
\]](http://latex.artofproblemsolving.com/e/9/7/e978baa1bbf867ec90243096ff0c1edf3725f438.png)
we get 
. Suppose first that 
(the case 
is symmetric). Then, 
forces 
which together with 
forces 
. So, 
, where 
are arbitrary are both solution families.
. Then 
and 
. Since 
, we must have 
, giving the family 
where 
are arbitrary.
such that 
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
.
,
,
,
,
.
divides 
and 
,
,
for all 
for some non-negative integers 
.
.
is odd then one of this numbers has to be even and one odd and then this pair works in this problem 
)
.
.
and