[/quote]
,
Prove that
and 
Find the value of 
be an infinite sequence of real numbers, for which there exists a real number 
with 
for all 
,![\[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \]](http://latex.artofproblemsolving.com/a/4/d/a4d87553590a450365b6525ca62a5d0416fe41cc.png)
Prove that 
.
.
are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers 
and 
are on the board, then 
can be erased as 
,
as 
,
as 
.
be non-empty subsets of the positive integers, and let 
denote their union. Prove that![\[\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).\]](http://latex.artofproblemsolving.com/1/c/d/1cdf561d3af92f524628e0724e0c5ee24d8d6a09.png)
Proposed by Dávid Matolcsi, Berkeley
Prove that
satisfying
Prove
.![\[
\sum_{\text{cyc}} \frac{x^3}{y^2 + z^2} \geq \frac{x + y + z}{2}
\]](http://latex.artofproblemsolving.com/9/0/3/903220c471bdd3028dda2fcb13874a4bcf752872.png)
without using the Rearrangement Inequality or Chebyshev's Inequality.
which satisfy the following set of equations
inscribed in 
with 
being the midpoint of 
.
of 
.
be the projection of 
onto 
.
that is not on 
intersects 
at 
.
and 
intersect at 
.
is an isosceles triangle.
,
be positive real numbers, such that 
.![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
such that 
divides 
for any positive integers 
and 
.
of real numbers such that 
for all positive integers 
denotes the greatest integer less than or equal to 
.
Y by kiyoras_2001
Problem. Let 
be the nonnegative integers in 
where 
. Define 
. Prove that 
.
Bonus problem (Open). Can we prove a tighter result than the one above? That is, is there a function
such that 
where 
, and the statement is still true when 
?
Or, is there a function such that 
where 
, and the statement is still true when ?.
be the nonnegative integers in 
where 
. Define 
. Prove that 
.Bonus problem (Open). Can we prove a tighter result than the one above? That is, is there a function
such that 
where 
, and the statement is still true when 
?Or, is there a function
such that 
where 
, and the statement is still true when ?.This post has been edited 10 times. Last edited by CBMaster, Apr 27, 2025, 5:43 AM
Reason: .
Reason: .
