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.
satisfy that: 
and 
. Find the maximum: 
opposite the vertex 
be tangent to the side 
at the point 
. Define the points 
on 
and 
on 
analogously, using the excircles opposite 
and 
, respectively. Suppose that the circumcentre of triangle 
lies on the circumcircle of triangle . Prove that triangle is right-angled.
satisfying the condition![\[
2f(n)^2 = n^2 + g(n)^2 \quad \text{for all } n \in \mathbb{N}^*.
\]](http://latex.artofproblemsolving.com/6/a/1/6a112ea3fe44273fdd4dc445e9bd7387d365925f.png)
Prove that if 
for all 
, then 
has infinitely many fixed points.
be a positive integer. Given a sequence 
, 
, 
with 
or 
for each 
, , 
, the sequences 
, , 
and 
, , 
are constructed by the following rules: ![\[a_0 = b_0 = 1, \quad a_1 = b_1 = 7,\]](http://latex.artofproblemsolving.com/d/f/d/dfd49833078b6b676cd261cc358e8114b2c4b7f4.png)
![\[\begin{array}{lll}
a_{i+1} =
\begin{cases}
2a_{i-1} + 3a_i, \\
3a_{i-1} + a_i,
\end{cases} &
\begin{array}{l}
\text{if } \varepsilon_i = 0, \\
\text{if } \varepsilon_i = 1, \end{array}
& \text{for each } i = 1, \dots, n - 1, \\[15pt]
b_{i+1}=
\begin{cases}
2b_{i-1} + 3b_i, \\
3b_{i-1} + b_i,
\end{cases} &
\begin{array}{l}
\text{if } \varepsilon_{n-i} = 0, \\
\text{if } \varepsilon_{n-i} = 1, \end{array}
& \text{for each } i = 1, \dots, n - 1.
\end{array}\]](http://latex.artofproblemsolving.com/8/9/0/8908021c60992e39acdb4fcf6b587f96b88539e0.png)
Prove that 
.
such that for all 
is great if the following holds: for any point 
on the side , if 
and 
are the feet of the perpendiculars from to the lines and 
, respectively, then the reflection of in the line 
lies on the circumcircle of the triangle . Prove that triangle is great if and only if 
and 
.
board, each square is colored either white or black, with all the squares on the border of the board being black. Additionally, no 
square within the board has all four squares of the same color. Prove that the board contains a square colored like a chessboard.
. Let be the foot of the altitude of triangle issuing from the vertex , and let 
be the circumcenter of triangle . Assume that 
. Prove that 
.
Prove that 
When does equality hold?
be a parallelogram such that 
Let 
and 
be the midpoints of and 
respectively. Assuming that 
is a cyclic quadrilateral, find 
be a polynomial with integer coefficients. Define 
and 
for 
.
such that 
, then prove that either 
or 
.
have 
distinct real roots, which sum up to 
. If 
, find the values of 
and their corresponding roots.
be a positive integer greater than . Find all functions 
satisfying:
for all integers 
on the circle, then make points and on such circle such that 
so 
. Then, an entire semicircle (and an extra segment) is not available for a point to lie in. Now pick a point on the circle outside the segment. It will encapsulate a semicircle of the original circle and an extra segment also. To maximize the available remaining area, make sure these two overlap as much as possible, so for example is arbitrarily close to or on the circle. This would still mean more than three quarters of the original circle is not available to be placed a point anymore, so we can only populate the last sector and we are out of space.
in such a way that the distance between every two points is greater than 
.