[/quote]
,
be real numbers drawn uniformly at random from the interval ![$[0,10]$](http://latex.artofproblemsolving.com/c/e/e/cee1594683f8cde0e7ba367764ac2d336b54a6f5.png)
.
be the unique monic quadratic whose roots are 
.
.
is not an integer.
be a triangle with circumcircle 
.
and 
intersect at 
.
intersects the line 
at 
and 
is the midpoint of 
.
and 
intersect on 
,
,
,
be distinct circles such that 
,
,
,![\[
\frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}.
\]](http://latex.artofproblemsolving.com/0/4/e/04ead9928d0c767a2209f610885e30f6b6e0a4e9.png)
,
.
and 
of a convex quadrilateral are perpendicular to each other. Draw a line that passes through point 
,
,
;
,
,
.
the equation for positive real numbers 
with 
.
and 
is a prime number and 
.
is also a prime number. For example: 
and 
are absolute primes. Prove that no absolute prime contains all of the digits 
and 
in base 
is good because 
and 
.
?
,
and let 
be the intersection point of 
with the diagonal 
.
.
be a point on the segment 
such that 
.
be the intersection point of the diagonal 
.
and 
Y by GA34-261
P1. Given that 
is a monic polynomial of smallest degree satisfying 
If coefficients of are rational then which of the following option(s) is/are true
A)
B)
C)
D)
P2. Which of the following option(s) is/are correct?
A) There exists a
degree polynomial with rational coefficients satisfying 
B) There exists a
degree polynomial with rational coefficients satisfying 
C) If
D) If
P3. If denotes a cyclic permutation of 
. Use the idea of quadratic equation as an identity to prove that
(i)
(ii)
P4. Let denote a cyclic permutation of 
. Then find the value of 
is a monic polynomial of smallest degree satisfying 
If coefficients of
are rational then which of the following option(s) is/are trueA)
B)
C)
D)
P2. Which of the following option(s) is/are correct?
A) There exists a
degree polynomial with rational coefficients satisfying 
B) There exists a
degree polynomial with rational coefficients satisfying 
C) If
D) If
P3. If
denotes a cyclic permutation of 
. Use the idea of quadratic equation as an identity to prove that(i)
(ii)
P4. Let
denote a cyclic permutation of 
. Then find the value of 
This post has been edited 2 times. Last edited by P162008, Jun 7, 2025, 2:07 AM
Reason: Typo
Reason: Typo
for 
follows from 
hence it equals 
.
we find 
hence 
.
.
has degree 4 over 
.
where ![$p\in Q[x]$](http://latex.artofproblemsolving.com/e/a/e/eae22acf9c23ccb39b12eadaebc8d5b25f492f70.png)
and 
.
.
then 
and 
.
has roots 
.
we find 
.
then we find 
.
.
Hence if 
then 
.
hence 
.
and 
then 
.
