[/quote]
,
index this yr (118.5 on amc10+ 9 on AIME) i’m in 7th rn btw first year comp math also. I will grind so hard until like 30 hrs/week. I’m ok at proofs. made mc nats
be a set of integers with the following properties:
.
and 
,
.
,
is composite, then all positive divisors of 
L
be an arbitrary point on 
such that 
be an arbitrary point on 
again at 
and 
cuts 
again at 
and 
is 
and 
be the intersection of 
with 
and 
respectively such that 
,
,
are coaxial circles
,
,
,
be 
integers. If 
is an integer, prove that the only solution to
is is 
.
be a cyclic quadrilateral an let 
The diagonals 
at 
The line through 
mmets the extension of the side 
beyond 
at 
The line through 
meets the extension of the side 
Prove that the circumcircles of the triangles 
and 
are tangent .
such that 
,
such that 
such that 
is a perfect square but none of the integers 
are perfect squares.
and ![$P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/c/6/4/c64176e52d94032e571b4579e35f9e6adfde675a.png)
such that 
.
such that every equation: 
has no real roots
,
and 
e
are tangent to 
at 
and 
intersects line 
.
intersects 
intersects 
.
intersects 
.
.
with four distinct elements is special if there exists a rearrangement 
of 
such that 
.
cannot be partitioned into 
(first player) and 
,
chips from the remaining pile of chips, where 
. No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?
Y by centslordm
A school has 
students and 
teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are 
and . Let 
be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let 
be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is 
?
students and 
teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are 
and . Let 
be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let 
be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is 
?
This post has been edited 1 time. Last edited by fidgetboss_4000, Nov 11, 2021, 5:09 PM
My aime qualification got oofed
,
,
and 
Then, 
![\[t=\frac{50+20+20+5+5}{5}=\frac{100}{5}=20\]](http://latex.artofproblemsolving.com/9/9/0/9908abfb058aa2fbb6109c605c4c2722520d3e6b.png)
.![\[s=\frac{50\cdot50+20\cdot20+20\cdot20+5\cdot5+5\cdot5}{100}=\frac{50\cdot50+20\cdot20+20\cdot20+10\cdot5}{100}=25+4+4+0.5=33.5\]](http://latex.artofproblemsolving.com/a/8/0/a8055c054285f508a66414108145a0b2f21cc3f6.png)
.
.
.
.
.
.
