[/quote]
,
is strictly increasing : 
such that for all 
. 
. 
q
answer choices. A participant who answers all questions correctly will receive a gift. To ensure that at least one member of my family answers all questions correctly, how many family members need to take the quiz?
,
.
Let 
.
be positive real numbers such that 
.
.
satisfying the following conditions:
divides 
,
divides 
,
.
and 
. Prove that
be positive reals such that 
.![\[\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.\]](http://latex.artofproblemsolving.com/3/b/d/3bd44cede90261ab949beabd32f8c08411ddc051.png)
such that for all 
,![\[
f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y).
\]](http://latex.artofproblemsolving.com/d/1/0/d10c01d7d416243614b24f04218d157e8b025ff2.png)
such that for all 
.
such that 
there exist 
such that 
and 
. Note : 
and 
for all 
. 
.
determine the minimum and maximum value of![\[\frac{\sum_{i,j=1}^n|a_i+a_j|}{\sum_{i=1}^n|a_i|}.\]](http://latex.artofproblemsolving.com/6/f/2/6f2427edf9a5e8e8f2e12276d0a841765e8322f1.png)
Y by
The sequence 
is a permutation of 
. Each time, Alice swaps two neighbouring numbers in the sequence. Her goal is to make any three consecutive numbers in the sequence non-monotone. Find the smallest positive integer 
such that Alice can always achieve her goal in at most swaps.
Remark. A monotonic sequence is either an increasing or decreasing sequence.
proposed by chengbilly
is a permutation of 
. Each time, Alice swaps two neighbouring numbers in the sequence. Her goal is to make any three consecutive numbers in the sequence non-monotone. Find the smallest positive integer 
such that Alice can always achieve her goal in at most swaps.Remark. A monotonic sequence is either an increasing or decreasing sequence.
proposed by chengbilly
This post has been edited 2 times. Last edited by chengbilly, Apr 26, 2025, 1:51 PM
