AoPS Academy
+4 w
, let 
be the result.
, where 
is the Euler's totient theorem, which calculates the number of integers less than relatively prime to . again because why not, let the result be 
.
for 
. Let 
be the remainder 
leaves when you divide it by 
.
be your predicted USA(J)MO score.![$\lfloor \frac{x + \sqrt[r]{r^2} + r \ln(r)}{r} \rfloor$](http://latex.artofproblemsolving.com/c/d/3/cd3f7d917b0ec944ec3017e1f737c538e1edd10e.png)
.
, and 
, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance 
away. What is the least number of moves that Carina can make in order for triangle 
to have area 2021?
in the coordinate plane where and 
are both integers, not necessarily positive.)
be a positive integer, and 
be two complex numbers such that 
and 
, for any 
. The matrices 
satisfy the relation 
. Prove that 
and 
are invertible.
is solvable over 
for every 
. Here 
, 
.
be collinear points (in order) and 
a point in plane. Consider the disc 
of center and radius 
, for some 
. Prove that ![$\mathcal{D}\cap [AC]$](http://latex.artofproblemsolving.com/0/5/6/0569d4130943000380a87b72be211965b5ff02ad.png)
is either the empty set or a segment of length at most 
. be a positive integer and ![$P(X)\in\mathbb{C}[X]$](http://latex.artofproblemsolving.com/9/9/e/99e8cde28b0b9bc9414425ebc30fa2dc0386fb37.png)
be a polynomial of degree . Prove that ![\[\sup_{x\in[0,1]}|P(x)|\le(2n+1)^{n+1}\int\limits_{0}^{1}|P(x)|\mathrm{d}x.\]](http://latex.artofproblemsolving.com/9/1/6/91647cd75888c5e9d291280c8ae3f199cc6d1bfd.png)
![$f \colon [0,1] \to \mathbb{R} $](http://latex.artofproblemsolving.com/2/1/8/21859d37ed035abd07d0be112f49d52598fa477e.png)
be a differentiable function such that its derivative is an integrable function on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
, and 
. Prove that ![\[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]](http://latex.artofproblemsolving.com/2/3/e/23e33eed39fc1e20486bbb86261d3566e8fd2017.png)
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 .
and 
be a continuous function for which there exists an antiderivative 
, such that 
, for any 
, and
holds for any 
. Prove that 
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 a polynomial function such that 
for any . Prove that 
for all .
be two matrices such that 
. Prove that: is odd, then 
;
, then .
matrices which has only 
and for their entries?
, with their second derivative being continuous, such that the following holds for all 
: ![\[(f(x)-g(y))(f'(x)-g'(y))(f''(x)-g''(y))=0\]](http://latex.artofproblemsolving.com/0/2/9/029a935e28ca0a9b36e65702689cca91eeb8f614.png)
unless of your age can be written as 
.
