[/quote]
,
points in the plane such that all pairwise distances between them are distinct, prove that there exist two points 
,
has at least two points on each of its sides (i.e., each side contains at least one point other than 
is a sequence of real numbers with 
such that for all 
Prove that
be natural numbers. As the New Year arrives, it is a cherished tradition among Iranian families to call their relatives and exchange Nowruz greetings. Just moments after the New Year begins, a family with 
members calls a related family with 
members to celebrate the occasion. Since the two families are close, every member of one family wishes to speak with every member of the other family to exchange greetings. Thus, during this single call, all 
distinct one-on-one conversations must take place exactly once.
.
meets 
at points 
,
is 
.
intersects line 
,
intersects line 
,
intersect at 
.
is tangent to 
.
?
students want to share 
pencils. If every student gets at least one pencil, how many ways are there to distribute the pencils?
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 
.
.
has 
,
,
.
.
such that 
Prove that 
Where 
then find minimal value of 
be a positive integer and let 
be an irrational number in 
.
,
.
is a family of non-empty subsets of 
is separated if each subset in the partition does not contain consecutive integers. Prove that, for every positive integer 
,
is equal to the number of separated partitions of the set 
.
is a separated partition of the set 
.
is a partition of the same set, but it is not separated since 
contains consecutive integers.
Prove that
Where 
Y by PikaPika999, Sedro
Hi people!
I have been and aops user for only a few months and not an active one either, just a user checking other posts and competitions. This is actually my first post. I am in 7th grade and only this year I have actually started locking in on competition math, before that, I didn’t really care about Mathcounts and AMCs. So, I hone my problem-solving skills by lightly doing some math problems. I have signed up for a lot of math competitions this month and am struggling to prepare properly with the resources I have. Do you guys have any resources you use to prepare for math competitions and that are actually helpful to use to prepare for competitions in a short amount of time. I am looking for resources that first teach you stuff and give you problems, like guided notes and handouts. I have few aops books like “Competition Math for Middle School” and “Intermediate Algebra”, but they’re not enough. I have learnt Algebra 1 and started doing high school geometry and a bit of Algebra 2 at home. Do let me know if you have any suggestions!
I have been and aops user for only a few months and not an active one either, just a user checking other posts and competitions. This is actually my first post. I am in 7th grade and only this year I have actually started locking in on competition math, before that, I didn’t really care about Mathcounts and AMCs. So, I hone my problem-solving skills by lightly doing some math problems. I have signed up for a lot of math competitions this month and am struggling to prepare properly with the resources I have. Do you guys have any resources you use to prepare for math competitions and that are actually helpful to use to prepare for competitions in a short amount of time. I am looking for resources that first teach you stuff and give you problems, like guided notes and handouts. I have few aops books like “Competition Math for Middle School” and “Intermediate Algebra”, but they’re not enough. I have learnt Algebra 1 and started doing high school geometry and a bit of Algebra 2 at home. Do let me know if you have any suggestions!
