Maximum number of nice subsets
by FireBreathers, Apr 23, 2025, 10:27 PM
Given a set 
of natural numbers with 
elements with odd number. A nonempty subset 
of is called 
if the product of the elements of divisible by the sum of the elements of , but not by its square. It is known that the set itself is good. Determine the maximum number of subsets (including itself).
of natural numbers with 
elements with odd number. A nonempty subset 
of is called 
if the product of the elements of divisible by the sum of the elements of , but not by its square. It is known that the set itself is good. Determine the maximum number of subsets (including itself).
be an integer greater than zero. Then, the value of the sum of the reciprocals of the cubes of the roots of the equation![\[
mx^4 + 8x^3 - 139x^2 - 18x + 9 = 0
\]](http://latex.artofproblemsolving.com/a/3/a/a3a9bca412e927632e4921c51f2a6593409031d9.png)
is equal to:
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
be triangle, inscribed in parabola. Tangents in points 
forms triangle 
.
.
)
for which there exists an even natural number 
such that the number![\[
(a - 1)(a^2 - 1)\cdots(a^n - 1)
\]](http://latex.artofproblemsolving.com/9/c/a/9caf4eeb82ff46b5ba55ab4b6bc28f0cace586ec.png)
is a perfect square.
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.
and 
where 
What is the sum of the digits of the least possible value of 
is a good choice, because once you stumble upon 
it's pretty easy to convince yourself this is optimal. But this problem doesn't really have the same "impact" as some of my previous proposals.
be the kite formed by joining two right triangles with legs 
and 
along a common hypotenuse. Eight copies of 
?
,
,
was intentional: I wanted to make sure most (if not all) of the kites were part of the triangle.
cells wide and 
cells tall consisting of 
squares. Carl places ![[asy] size(6cm); for (int i=0; i<9; ++i) { draw((i,0)--(i,3),dotted); } for (int i=0; i<4; ++i){ draw((0,i)--(8,i),dotted); } for (int i=0; i<8; ++i) { for (int j=0; j<3; ++j) { if (j==1) { label("1",(i+0.5,1.5)); }}} [/asy]](http://latex.artofproblemsolving.com/b/1/c/b1cfd38f58057d8f0ed435b054143237ba4fac8c.png)
s
is rotated about its center by angle 
,
,
.
is 
.
?![[asy]
defaultpen(fontsize(13)); size(170);
pair O=(0,0),A=dir(225),B=dir(-15),C=dir(105),D=rotate(38.21,O)*A,E=rotate(38.21,O)*B,F=rotate(38.21,O)*C;
draw(A--B--C--A,gray+0.4);draw(D--E--F--D,gray+0.4); draw(A--D--B--E--C--F--A,black+0.9); dot(O); dot("$A$",A,dir(A)); dot("$B$",B,dir(B)); dot("$C$",C,dir(C)); dot("$D$",D,dir(D)); dot("$E$",E,dir(E)); dot("$F$",F,dir(F));
[/asy]](http://latex.artofproblemsolving.com/a/7/2/a72db31d1d76e376654dd13a3f9a166d0d6056ce.png)
is the square of an integer
.
is 
.
-
satisfy the following properties?
.
.
satisfying 
.
,
and 
?
.
is rotated 
clockwise about its center to obtain square 
,
?![[asy] size(170); defaultpen(linewidth(0.6)); real r = 25; draw(dir(135)--dir(45)--dir(315)--dir(225)--cycle); draw(dir(135-r)--dir(45-r)--dir(315-r)--dir(225-r)--cycle); label("$A$",dir(135),NW); label("$B$",dir(45),NE); label("$C$",dir(315),SE); label("$D$",dir(225),SW); label("$E$",dir(135-r),N); label("$F$",dir(45-r),E); label("$G$",dir(315-r),S); label("$H$",dir(225-r),W); [/asy]](http://latex.artofproblemsolving.com/4/0/4/404883b79d8ef1c910971e24f1655976ce3afbae.png)
and 
,
by ![\[w\otimes z=ac+bd\,i.\]](http://latex.artofproblemsolving.com/c/8/e/c8ecf215ad3e92354fba57bc8ffdd4526021f183.png)
Suppose 
.
?
![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
and 
be on segment 
of an acute triangle 
and 
.
and 
,
and 
.
and 
is on the circumference of triangle 
April 2025
November 2024