Prove that
Prove that
and 
.![$$ a^2-ab+b^2 \geq \sqrt[3]{2} $$](http://latex.artofproblemsolving.com/7/9/8/798484c6b985ebbace2a3dde816f11a46635feba.png)
Let 
.![$$ 3\geq a^2+ab+b^2 \geq \sqrt[3]{4} $$](http://latex.artofproblemsolving.com/d/7/7/d77fbdc50a88395259e58d31413b9ac453e5dfd0.png)
of odd degree with real coefficients such that 
.
where 
,
the tangents at the points of intersection are concurrent.
be an odd integer greater than 
Let 
be an 
symmetric matrix such that each row and column consists of some permutation of the integers 
Show that each of the integers 
must appear in the main diagonal of 
convergent?
and radius 
and the base of a hemisphere of radius 
This body is placed with the hemispherical end on a horizontal table, with the axis of the cylinder in a vertical position, and then slightly oscillated. It is intuitively evident that if 
is large as compared to 
which enables the body to rest in neutral equilibrium in any position?
![\[ \begin{vmatrix} 0 & a_1 - x & a_2 - x \\ -a_1 - x & 0 & a_3 - x \\ -a_2 - x & -a_3 - x & 0\end{vmatrix} = 0 \qquad (a_i \neq 0) \]](http://latex.artofproblemsolving.com/7/2/1/721620c769b5622dc4172dd38b7a924b5d95e694.png)
shall have a multiple root.
by 
South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude 
S.In the very close vicinity of this parallel how much are East and West distances exaggerated as compared to North and South distances?
and 
be relatively prime positive integers, 
,
be chosen so that
is a minimum. Prove that
be a permutation of the integers 
Call 
a big integer if 
for all 
Find the mean number of big integers over all permutations on the first 
Y by GeoKing, Rounak_iitr
Let 
and 
be the circumcenter and orthocenter of a scalene triangle 
, respectively. Let 
be the intersection point of the lines 
and 
. Suppose the line 
meets the side at 
. Let 
and 
be the second intersection points of the circumcircles of 
and 
with the circumcircle of 
, respectively. Show that the four points 
and lie on a circle.
and 
be the circumcenter and orthocenter of a scalene triangle 
, respectively. Let 
be the intersection point of the lines 
and 
. Suppose the line 
meets the side at 
. Let 
and 
be the second intersection points of the circumcircles of 
and 
with the circumcircle of 
, respectively. Show that the four points 
and lie on a circle.This post has been edited 1 time. Last edited by a_507_bc, May 3, 2024, 10:18 AM
and 
Also, let 
be the midpoint of 
and let 
be the midpoint of 
Note that 
is the nine-point circle of 
which we will denote with 
are collinear.
be the antipode of 
.
are collinear, so if 
intersects 
for a second time at point 
,
However, we know that 
so 
and we are done.
are collinear.
centered at 
Note that under this inversion, we have that 
and 
,
Hence, 
and 
.
Since 
,
Note that 
and in particular 
Therefore, 
is cyclic iff 
is cyclic. However, points 
all lie on the circle with diameter 
and we may conclude.
be the orthic triangle, let 
where 
are midpoints of 
,
inversion the queue points go to 
are collinear but this follows as 
are antipodes of 
and 
.
be circle with diameter 
other then 
we have 
cyclic.
inversion about 
, 
and 
.
.
are cyclic 
cyclic
Let 
be the antipodes of 
on 
be the foot of the altitude from 
to 
.
is cyclic.![\[\frac{r+a+b+c}{2}=a\iff r=a-b-c\]](http://latex.artofproblemsolving.com/c/0/5/c051a406274e309a65d0de4a89bb599563e6f8df.png)
![\[s=\frac{\overline{(a+b+c)}(a-b-c)+(a+b+c)\overline{(a-b-c)}}{2\overline{(a+b+c)}}=\frac{(ab+bc+ca)(a-b-c)+(a+b+c)(bc-ab-ac)}{2(ab+bc+ca)}\]](http://latex.artofproblemsolving.com/3/8/3/383e4c5c5a7ca65986086aba5a692fb8272bc82f.png)
![\[s=\frac{-ab^2-abc-ac^2}{(ab+bc+ca)}\]](http://latex.artofproblemsolving.com/a/f/6/af66ef05323a993327c979578b110720881e2403.png)
![\[\frac{k-r}{k-s}.\frac{l-s}{l-r}=\frac{c-a}{-b+\frac{ab^2+abc+ac^2}{(ab+bc+ca)}}.\frac{-c+\frac{ab^2+abc+ac^2}{(ab+bc+ca)}}{b-a}\]](http://latex.artofproblemsolving.com/c/1/4/c146933f20d5e61696444ac8f8da2c3de745e0c3.png)
![\[=\frac{c-a}{b-a}.\frac{ab^2-bc^2}{ac^2-b^2c}=\frac{b(c-a)(ab-c^2)}{c(b-a)(ac-b^2)}\]](http://latex.artofproblemsolving.com/4/c/b/4cb2084c042395f4ce7ba514fef4f67c30a41312.png)
![\[\frac{b(c-a)(ab-c^2)}{c(b-a)(ac-b^2)}\overset{?}{=}\overline{(\frac{b(c-a)(ab-c^2)}{c(b-a)(ac-b^2)})}=\frac{(a-c)(c^2-ab)b}{(a-b)(b^2-ac)c}\]](http://latex.artofproblemsolving.com/a/c/d/acd7c2dba9886660414589d983fadc86dd17e313.png)
Which is true as desired
be the orthic triangle of 
.
,
and 
.
and 
are the midpoints of 
and 
is 
,
lies on line 
.
,
:
;
.
is cyclic which means 
.
is circle with diameter 
means 
at 
and 
be the point 
.
be a point on the line 
so that 
.
.
