[/quote]
,
such that the following inequality holds for all 
:![\[
ab + bc + ca \leq \frac{(a + b + c)^2}{3} + k \cdot \max \{ (a - b)^2, (b - c)^2, (c - a)^2 \} \leq a^2 + b^2 + c^2
\]](http://latex.artofproblemsolving.com/1/c/e/1ce66163068b499383adb4a4ecec86c5f4edaec3.png)
be positive real numbers such that : 
. Prove that : 
where 
are lengths of parallel sides and 
are other two sides.
,
,
,
,
in some order.
board that is entirely covered by dominoes, with no overlap. He jumps along the vertical lines of the board, always going from the midpoint of the vertical segment of a 
square to another midpoint of the vertical segment, according to the rules:
When the domino is horizontal, the cricket jumps to the opposite vertical segment (such as from 
to 
)
When the domino is vertical downwards in relation to its position, the cricket jumps diagonally downwards (such as from 
to 
When the domino is vertically upwards relative to its position, the cricket jumps diagonally upwards (such as from 
)
board.
Prove that
candies. A starts first. On a move, if in the box there are 
candies, the player chooses positive integer 
so that 
and 
,
.
such that 
and 
.
be a trapezoid with parallel sides 
.
and 
lie on the line segments 
and 
,
.
and 
on the line segment 
satisfying ![\[\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.\]](http://latex.artofproblemsolving.com/b/1/a/b1ae208955104081bafe221832509d9dd20eeb83.png)
Prove that the points 
and 
are concyclic.
,
with center 
and radius 
the midpoint of side 
intersects the circle 
.
be a point on the circle 
and 
.
.
.
and 
as the reflections of 
be the circumcircle of 
.
the second intersection of line 
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
![\[\sqrt{a+h_B}+\sqrt{b+h_C}+\sqrt{c+h_A}=\sqrt{a+h_C}+\sqrt{b+h_A}+\sqrt{c+h_B}\]](http://latex.artofproblemsolving.com/5/d/6/5d6881be29327bcdc3e7bd76c221968578c7f50b.png)
is isosceles.
be all the divisors of 
are all divisors for some 
(
)
Y by Adventure10, Rounak_iitr
In a right-angled triangle 
(
), the perpendicular bisector of 
intersects the line 
in 
and the perpendicular bisector of 
intersects the line 
in 
. If the line 
be the internal bisector of angle 
, find all possible values for angles 
and .
by Mahdi Etesami Fard
(
), the perpendicular bisector of 
intersects the line 
in 
and the perpendicular bisector of 
intersects the line 
in 
. If the line 
be the internal bisector of angle 
, find all possible values for angles 
and .by Mahdi Etesami Fard
This post has been edited 1 time. Last edited by parmenides51, Jul 22, 2018, 12:15 PM
Reason: typo
Reason: typo
is a solution, but this is not correct. In this case angle bisector of 
and perpendicular bisector of 
