[/quote]
,
such that the equation ![\[x^{2}-(a^{2}+b^{2}+c^{2}+d^{2}+1)x+ab+bc+cd+da=0 \]](http://latex.artofproblemsolving.com/9/0/4/904d1b23d5c744118a7d68df1e2e94bddac8c1cd.png)
has an integer solution. Prove that the other solution is integer too and both solutions are perfect squares.
be a triangle such that![\[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \]](http://latex.artofproblemsolving.com/6/9/b/69b1964be5f92eb44a36b0b8604bf473fe27e210.png)
and 
denote its semiperimeter and its inradius, respectively. Prove that triangle 
whose side lengths are all positive integers with no common divisors and determine these integers.
.
and 
intersect 
and 
.
be the intersection point of these angle bisectors. Let 
and 
be the respective reflections of 
and 
.
.
,
of 
cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions:
cards: if the cards are laid down in increasing order face down and we are told which card belongs to which player, but not what numbers are written on them, we can still decide the winner.
and 
,
for all 
,
from the deck, 
then 
and 
for 
.
is increasing.
,
,
of 
such that ![\[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\]](http://latex.artofproblemsolving.com/9/8/0/98097fb721722b9103ef65d72322956877e262cb.png)
for every 
.
.
,
has at most 
be the sum of the digits of 
such that there are infinite positive integers 
.
be three diameters of the circumcircle of an acute triangle 
be an arbitrary point in the interior of 
,
be the orthogonal projection of 
,
be the point such that 
is the midpoint of 
,
be the point such that 
is the midpoint of 
,
be the point such that 
is the midpoint of 
.
is similar to triangle 
such that if 
and 
are relatively prime divisors of 
,
divides 
is an infinite sequence of integers satisfying the following two conditions:
divides 
for 
such that 
for all 
such that 
for all 
with complex coefficients for which all complex roots of the polynomials 
have absolute value 1.
Show that there exists a constant 
such that for all positive integer 
![\[\sum_{k\le n^{\varepsilon}}(n\text{ mod } k)>cn^{2\varepsilon}.\]](http://latex.artofproblemsolving.com/f/e/9/fe933e886e28c2ecf1fa880d0e650ac7a2998aaa.png)
Proposed by Cheng Jiang
such that 
with integer coefficients in the cells of an infinite grid on the plane, such that each such polynomial is placed in exactly one cell, and for every finite rectangular subgrid with an area greater than one, the sum of the polynomials within that rectangle has all roots real and distinct?
Y by fe.
Is it possible to place all monic quadratic polynomials with natural coefficients in the cells of an infinite grid on the plane, such that each polynomial is placed in exactly one cell, and for every finite rectangular subgrid with an area greater than one, the sum of the polynomials within that rectangle has a real root?
Proposed by Mojtaba Zare
Proposed by Mojtaba Zare
This post has been edited 1 time. Last edited by mojyla222, Apr 30, 2025, 7:59 AM
Reason: comment
Reason: comment
grid with two polynomials whose sum has a rational root. Then we iteratively expand the rectangle by one on each side. First, write the lowest indexed polynomial not already present in one cell. Then for the remaining cells, write the polynomial 
where 
)
