[/quote]
,
![\[ \int_{-1}^{1} e^x \sin \sqrt{1-x^2} \, \mathrm dx .\]](http://latex.artofproblemsolving.com/0/f/0/0f01e4820da5328db8b1b55947de2148cbdd3e83.png)
be a positive semi-definite matrix. Prove that the matrix 
,
.
be an acute triangle with incenter 
and circumcenter 
.
and 
at 
and 
respectively, and 
is the reflection of 
and 
meet at 
,
and 
,
is the midpoint of 
.
and 
,
.
with right angles at 
has an inscribed circle with center 
.
and 
be 
and 
,
and 
be 
.
be a point on the Neuberg cubic. Show that as 
increasing with
?
?
such that 
for all real numbers 
is the greatest integer less than or equal to 
)
be the circumcircle of a non-isosceles triangle 
be a tangent line to 
)
be midpoint of 
and 
.
and 
intersect at points 
.
and 
intersects 
,
intersects 
.
,
.
and 
are chosen on lines 
and 
respectively such that 
is a parallelogram. Points 
and 
and 
are the angle bisectors of angles 
and 
respectively. Let 
past 
past 
.
at 
.
on 
and 
on 
is cyclic, then 
is tangent to the circumcircle of 
.
Y by
Let 
and 
be integers and 
a square matrix of order 
with integer entries. Prove that for any permutation 
there is a function 
such that replacing the entries 
of 
respectively by 
the determinant of the matrix 
thus obtained is not divisible by 
and 
be integers and 
a square matrix of order 
with integer entries. Prove that for any permutation 
there is a function 
such that replacing the entries 
of 
respectively by 
the determinant of the matrix 
thus obtained is not divisible by 
This post has been edited 3 times. Last edited by Filipjack, Apr 6, 2025, 10:58 AM
be a square matrix with integer entries modulo 
,
.
chosen uniformly at random. The determinant of the modified matrix is a degree-1 polynomial in each 
,
it is not identically zero (since it includes a monomial corresponding to the sign of 
)
such that the determinant is nonzero modulo 
sends 
to 
is trivial (determinant can be either 
or 
)
with 
,
rows and columns; by the induction hypothesis, there exists a function 
that the determinant of this function is not divisible by 
;
,
except that 
sends 
to 
sends 
