AoPS Academy
and 
, then we can deduce 
\le 2 $ ?
that simultaneously satisfy the conditions 
and![\[
\frac{x^4}{a} - \frac{y^4}{b} = \frac{1}{a - b}.
\]](http://latex.artofproblemsolving.com/4/5/e/45e29c956890cb0bd869c67e7ad94c92d868d2e2.png)
, we have![\[
\left( \frac{x^2}{a} \right)^n + \left( \frac{y^2}{b} \right)^n = \frac{2}{(a - b)^n}.
\]](http://latex.artofproblemsolving.com/4/0/5/405491f7aae19de4c7ad1f90da3d66e4038e3299.png)
for all 
.
be an odd positive integer satisfying 
with exactly 2 distinct prime factors. Prove that is a prime number or the square of a prime number.
that satisfy the following condition: For every integer k such that 
there always exists a positive integer 
that is divisible by n and 
. 
: sum of elements of
is drawn externally on side 
of 
. Point 
is taken on 
. Find 
if 
.
be a quadrilateral with 
, 
, 
and 
. The measure of 
(in degrees) is:
so 
angle sum property
,
,
and 
.
Equality holds when 
or 
or 
has 
,
,
.
be the angle bisector of angle 
such that 
lies on side 
.
at 
and 
.
?
of the numbers 
satisfy 
and 
?
be 
and 
degrees
,
and then 
![[asy]
import markers;
size(8cm);
pair A = (0,0);
pair D = (2,2);
pair C = (4,0);
pair B = (2, -3);
draw(A--B--C--D--cycle);
draw(A--C--D);
draw(A--C--B);
markangle(Label("$110^\circ$"), B, A, D, radius=15); // ∠BAD
markangle(Label("$70^\circ$"), A, D, C, radius=15); // ∠ADC
markangle(Label("$70^\circ$"), D, C, A, radius=15); // ∠ACD
markangle(Label("$10^\circ$"), A, C, B, radius=15); // ∠ACB
// Label the points
label("$A$", A, SW);
label("$B$", B, S);
label("$C$", C, SE);
label("$D$", D, N);
draw(anglemark(B, A, D, 15));
draw(anglemark(A, D, C, 15));
draw(anglemark(D, C, A, 15));
draw(anglemark(A, C, B, 15));
[/asy]](http://latex.artofproblemsolving.com/2/4/b/24b33efb089d2f48afad11040b7f68943430a9f6.png)