be a quadrilateral inscribed in a circle 
.
,
,
,
,
,
,
,
.
lies over 
.
table filled with natural numbers, we say it is a divisor table if:
-
,
-
,
for every 
.
is given. Determine the smallest natural number 
,
be integers so that the integers 
are divisible by 
be a fixed integer. There are 
beads on a circular necklace. You wish to paint the beads using 
consecutive beads every color appears at least once. Find the largest value of 
for which this task is 
possible.
are in arithmetic progression. Find the value of 
.
and 
such that![\[
\frac{11}{\log_2 x} + \frac{1}{2 \log_{25} x} - \frac{3}{\log_8 x} = \frac{1}{\log_b x} \quad \text{for all } x > 1.
\]](http://latex.artofproblemsolving.com/2/d/7/2d76319b3d4fcd1f193ea0f9f37cb4bf513376f3.png)
![\[ x^{\log{x}} = 10^{2 - 3\log{x} + 2(\log{x})^2}\]](http://latex.artofproblemsolving.com/b/f/b/bfb8185992867a8926b5c62198a799c935771d53.png)
where 
is the logarithm of 
to the base 
.
Prove that
that satisfies![\[
a^3+b^3=1957
\]](http://latex.artofproblemsolving.com/5/d/8/5d8d2a40734ec6b0a9149f9a4ae3a5828d5055f3.png)
![\[
(a+b)(a+1)(b+1)=2014
\]](http://latex.artofproblemsolving.com/3/0/0/300f2d16e669a1d8d9c54b357a6eaf7a8eb5d826.png)
Hence, find the value of 
Y by Adventure10, Mango247
Let 
be a triangle and 
an arbitrary point in the plane. Let 
be interior angles of the triangle and its area is denoted by 
Prove:
When does equality occur?
be a triangle and 
an arbitrary point in the plane. Let 
be interior angles of the triangle and its area is denoted by 
Prove:\[ \ov{AP}^2 \cdot \sin 2\alpha + \ov{BP}^2 \cdot \sin 2\beta + \ov{CP}^2 \cdot \sin 2\gamma \geq 2F
\]When does equality occur?
at points 
respectively, then
- 
When does equality occur?
