Find the maximum positive integer 
such that for every subset 
with n elements, there always exist two elements a, b in T such that:
of integers satisfying![\[x_0 = 0, |x_n| = |x_{n-1} + 1|\]](http://latex.artofproblemsolving.com/4/8/e/48e7b6f98f2062879377a9a2e507a63eac9269a8.png)
.
.
,
and 
and so forth.
.
be a tetrahedron such that 
A sphere of radius 
is circumscribed about tetrahedron 
Given that 
,
,
,![$$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge \frac{9\sqrt[3]{4}}{4}$$](http://latex.artofproblemsolving.com/6/c/6/6c665b4611f5fc8ebeba03ac649bfee4e2634950.png)
with equality at ![$a=b=c=\sqrt[3]{2}.$](http://latex.artofproblemsolving.com/b/b/9/bb9c9671b23cb45e5f85ae01bf25d566c2e43e91.png)
and 
be points on a plane such that 
,
is a positive integer. Let 
be the set of all points 
such that 
,
is a real number. The path that 
be reals such that 
. Prove that
Let 
. Prove that
Y by
Jerry likes to play with numbers. One day, he wrote all the integers from 
to 
on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them with either their sum or their product. (For example, Jerry's first step might have been to erase 
, and 
, and then write either 
, their sum, or 
, their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the board were odd. What is the maximum possible number of integers on the board at that time?
to 
on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them with either their sum or their product. (For example, Jerry's first step might have been to erase 
, and 
, and then write either 
, their sum, or 
, their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the board were odd. What is the maximum possible number of integers on the board at that time?
This post has been edited 1 time. Last edited by ostriches88, Nov 13, 2024, 6:08 PM
it was bc that you can only lose or no gain, so I thought 1012 was the answer, but it wasn't achievable
.
evens, and there are 
evens, meaning we must make at least 
operations and therefore lose at least 
numbers.
evens into 
and operate on each one. 
evens into 
evens in 
takes 
operations. Now operate the final even with three random odds and we are done.
.
(some odd numbers will be removed on the last operation).
mod 3 or 1010.
,
