[/quote]
,
be a positive integer. Determine all sequences 
of positive integers such that
for all positive integers 
.
and 
be two circles with respective centres 
and 
intersecting in two distinct points 
and 
such that 
is an obtuse angle. Let the circumcircle of 
intersect 
and 
.
intersect 
; let the line 
intersect 
.
are concyclic.
be an acute scalene triangle with circumcenter 
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
.
be the circle with diameter 
.
be the intersection of 
and 
.
be the orthocenter of 
be the intersection of 
and 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
.
be the intersection of 
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
,
and 
are both divisible by 
be an odd prime. Define 
over the positive integers as follows:
where 
such that 
.
such that 
and 
is a perfect square.
denote the set of all positive integers less than 
that are relatively prime to 
.
reducible over the rational numbers, given that it has integer coefficients?
students, each pair of students have watched a movie together. Let the set of movies watched by a student be his movie collection. If every student has watched every movie at most once, at least how many different movie collections can these students have?
be non-negative real numbers such that 
Prove that
and 
.
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 
.
,
such that 
.
.
has 
,
,
.
.
Y by Awesome_Twin1
I am an eighth grader in North Carolina and I am sort of pressured to make nats this year, I only got 26 on last year's state test (orz to @mathprodigy2011) and I have mocked 39, 40, and 37 on the 2014, 2015, and 2022 tests respectively. I am aiming for a 40 this year because that seems to be the cutoff. How should I sufficiently improve? My test is on March 14th, 2025.
To be more clear, last year I got 16/10 and on the tests I have mocked, I got 27/12, 26/14, and 25/12.
To be more clear, last year I got 16/10 and on the tests I have mocked, I got 27/12, 26/14, and 25/12.
This post has been edited 1 time. Last edited by AMathCountsguy10, Feb 21, 2025, 2:13 PM
