[/quote]
,
and 
.
be the incircle and circumcircle of the acute triangle 
,
so that all of its sides are tangent to 
,
are both on 
.
and 
,
and 
,
and 
intersects on 
and 
Prove that
Let 
and 
Prove that
be natural numbers such that![\[
\frac{a^n - 1}{(a - 1)^n + 1}
\]](http://latex.artofproblemsolving.com/6/2/7/627f1d5f55e7594f2e6dcf5d541e796b7a5bfae2.png)
is a natural number.
is odd.
be a prime divisor of ![\[
a^{(q - 1)/2} \equiv 1 \pmod{q}.
\]](http://latex.artofproblemsolving.com/b/2/1/b2135d8c5526e577722718e6d02bcc52559b0555.png)
3. Prove that if a is prime and 
,![\[
2^{(a - 1)/2} \equiv 1 \pmod{a}.
\]](http://latex.artofproblemsolving.com/2/d/c/2dc6cc1fe27cc72e72d63b13cf07939e19051245.png)
.
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
such that 
and![\[
f(af(x) - f(y) + y) \leq x + f(y) - y, \quad \forall x, y \in \mathbb{R}^+.
\]](http://latex.artofproblemsolving.com/e/2/9/e296b71ad8b8bd977e2bb2a3630a5bc020b8e03a.png)
such that
is not a polynomial .
for all 
, 
.
and 
Prove that
be a regular heptagon with its center 
is the orthocenter of triangle 
,
is the incenter of triangle 
.
be the midpoint of 
and 
and 
.
.
concur at a single point.
as the escenter (inside the angle 
)![\[3|AI_A|\geq 2(m_B+m_C),\]](http://latex.artofproblemsolving.com/1/a/f/1affae2d9cd5be0adf8134faa739cbfbaaad5bab.png)
where 
is the length of median, and so is 
.
of positive integers such that 
and 
.
and orthocenter 
and 
be the feet of the altitudes from 
and 
onto sides 
and 
,
be the midpoint of 
.
meet 
and 
.
meets 
and 
.
and 
.
be a positive integer. We say that 
such that:
for which![\[
a \mid s^b - l.
\]](http://latex.artofproblemsolving.com/5/d/9/5d9db7678c54d1c6a909b4aca39a329cadd71e2c.png)
2. For every proper divisor 
of 
and 
)![\[
a' \mid s^b - l.
\]](http://latex.artofproblemsolving.com/4/b/2/4b26b9bd3da8958287703c3f575bf5d5da7fd477.png)
Determine all bao-good positive integers 
Y by Adventure10, Mango247
Let 
be real number with 
, and let 
be a convex cyclic quadrilateral whose circumcentre 
lies in its interior. The quadrilateral's circumcircle 
has radius 
, and the longest and shortest sides of the quadrilateral have length and 
, respectively. Lines 
are tangent to at 
, respectively.
Let lines
and 
, and 
, and 
, and intersect at 
respectively. Determine the minimum value of 
.
be real number with 
, and let 
be a convex cyclic quadrilateral whose circumcentre 
lies in its interior. The quadrilateral's circumcircle 
has radius 
, and the longest and shortest sides of the quadrilateral have length and 
, respectively. Lines 
are tangent to at 
, respectively.Let lines
and 
, and 
, and 
, and intersect at 
respectively. Determine the minimum value of 
.
be 
.
.
.
and 
.
and 
.
or 
.
and 
.
(I am considering only acute 
)
where 
so this function is concave.So 
and 
and 
.
would have become the diameter.However there was no guarantee that in this case the ratio would be at its minimum.On the other hand the above is the general case which can be applied in any situation. So this is the required minimal value.
