AoPS Academy
be real numbers satisfying 
Prove that 
and determine all cases of equality.
be the real polonomial such that all roots are real and distinct. Prove that there is a rational number 
that all roots of 
are real numbers
be an acute angled triangle with orthocenter 
and 
. Let 
be any other point on the arc 
of the circumcircle of 
and let line 
intersect line 
at 
and let line 
intersect line 
at 
. Let the circumcircles of 
and intersect again at 
(
). Let the lines 
intersect the circumcircle of 
again at 
(
). Prove that the lines 
concur.
, 
be normed spaces with a Banach space. Suppose 
is a family of continuous linear maps. Prove that the set![\[
X = \left\{ x \in E \mid \sup_{n\geq 1}|| A_n(x)||< +\infty \right\}
\]](http://latex.artofproblemsolving.com/e/c/d/ecd2cd83e90604ce128e8efb5e5b0748ad354ee3.png)
is either of first category in or is equal to the whole space .
be a positive integer. At most how many distinct unit vectors can be selected in 
such that from any three of them, at least two are orthogonal?
![$f \in C^2([0,1],[0,1])$](http://latex.artofproblemsolving.com/e/5/3/e536883b91a3d5e0c1703712edc76fc5ef28e274.png)
diffeomorphisme bijectif.
?
be real square matrices of the same order, and suppose 
is invertible. Prove that![\[ (A-B)C=BA^{-1}\implies C(A-B)=A^{-1}B \]](http://latex.artofproblemsolving.com/0/6/0/06007271017c2c48faa708366f1adc67db91f245.png)
![$f,g \in C^1([0,1])$](http://latex.artofproblemsolving.com/6/1/7/617c5be50035205b21074842d7266911aa1dc4d8.png)
.
?
a list of 
real numbers 
is said to be area definite for if the inequality ![\[\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0\]](http://latex.artofproblemsolving.com/e/7/7/e778445aa49547e4a227d16cc8685370a5d1a7d9.png)
holds for every choice of 
points 
in 
For example, the list of four numbers 
is area definite for 
Prove that if a list of numbers is area definite for 
then it is area definite for 
be a ring with unity such that for every 
there exist 
such that 
. Prove that
then 
such that 
then the result from a) may no longer hold.
be the orthocenter of an obtuse triangle and 
arbitrary points on the sides 
respectively.Prove that the tangents drawn from to the circles with diametrs 
are equal.
the intersection of the altitudes with the respective side.
we will notice that 
is on the circle with diameter 
.
is the tangent from to this circle we need to prove that 
is equal to the others.
which is obvious using the fact that 
are altitudes and they form inscribed quadrilaterals (
)
wrt to the circles are all equal.
are arbitrary points on the sides, by fixing one point say 
we can prove that for all points 
the power of wrt to the circle with diameter 
is a constant and then for a specific point 
we prove that 
. It goes something like this.
, where 
is the midpoint of 
. By law of cosines 
where 
is the foot of the altitude from to 
. Then we have 
. Because 
is cyclic we get that 
so we have 
.
there exists positive integers 
such that
is the radical center of the circles (with the cevians as diameters) is found in Coxeter's Geometry Revisited, page 37:
be the orthic triangle of 
and let 
be the circles with diameters
,
and 
,
is cyclic and 
and 
.
.![\[HA\cdot HD = HB \cdot HE = HC \cdot HF = k\]](http://latex.artofproblemsolving.com/f/a/c/faca4146f10505d009a6547f9e55f6ba20b30d4b.png)
Now, observe that 
,
and 
.![\[k = \text{Pow}_{\omega_A} (H) = \text{Pow}_{\omega_B} (H) = \text{Pow}_{\omega_C} (H)\]](http://latex.artofproblemsolving.com/4/d/0/4d0aafa3bec8c3837da8dde74dfec5a3323e7599.png)
and 
.