[/quote]
,
![\[
f_k(x) = 1 + 2x + 3x^2 + \dots + (k+1)x^k,
\]](http://latex.artofproblemsolving.com/f/3/5/f35ea75e6415a63a40fb2c3c95fa5801e6fb891f.png)
![\[
\begin{vmatrix}
f_0(1) & f_1(1) & f_2(1) & \dots & f_{2023}(1) \\
f_0(2) & f_1(2) & f_2(2) & \dots & f_{2023}(2) \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
f_0(2024) & f_1(2024) & f_2(2024) & \dots & f_{2023}(2024)
\end{vmatrix}
= \prod_{k=1}^{2024} k!.
\]](http://latex.artofproblemsolving.com/4/7/f/47ff41069b9444fe794076787b8890352929047a.png)
that has 8097 distinct points with area of a triangle that has 3 points belong to 
.
that has the area 
contains at least 2025 points that belong to 
crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least 
be the incentre of acute triangle 
with 
.
of 
,
at 
and 
,
perpendicular to 
meets 
.
meets 
.
and 
meet again at 
.
and 
meet on the line through 
perpendicular to 
.
and 
Prove that
,
.
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
and 
Prove that
integers are such that 
is prime and 
.
such that 
.
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
satisfying 
.
and 
be a continuous function for which there exists an antiderivative 
,
,
,
.
for all 
be a positive integer, ![$g \in \mathbb{R}[X]$](http://latex.artofproblemsolving.com/0/1/e/01e7ebcebcb9f16935716ee859f7acfaa66c29d4.png)
,
be a polynomial with all of its roots being real, and 
for any 
for all 
I can prove that the sequence 
is convergent by monotone bounded convergence theorem.
Y by
Let 
be an odd prime number, and 
be an odd number not divisible by . Consider a field 
be a field with 
elements, and 
be the set of elements of 
, whose order is not in the multiplicative group 
. Prove that the polynomial 
has at least coefficients equal to 
.
be an odd prime number, and 
be an odd number not divisible by . Consider a field 
be a field with 
elements, and 
be the set of elements of 
, whose order is not in the multiplicative group 
. Prove that the polynomial 
has at least coefficients equal to 
.
