[/quote]
,
be a prime number. Prove that there exists a natural number 
such that![\[
p \mid 2024^n - n.
\]](http://latex.artofproblemsolving.com/1/4/a/14aea55ed2bcea62d2a0d8d0770513715d8c4fb8.png)
be positive real numbers . Prove that
and 
intersect at the point 
.
?
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
?
families, there are 
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
such that 
.
,
lies on diagonal 
such that 
.
.
has 
,
,
.
.
such that for all real 
:
Prove that: 
is the sum of 
Let 
and 
is the sum of 
be positive reals with 
.![\[\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}\]](http://latex.artofproblemsolving.com/f/6/5/f65f6fce4465125dfacf002f0659c898df7ae150.png)
Y by audio-on, avyaank, PikaPika999, GodGodGodGodGoose
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The 2025 MA4G competition is now open!
Apply Here: https://xmathandai4girls.submittable.com/submit
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