Prove that
Is there an elegant way to prove this?
through 
.
.
,
,
with probability 
and to pad 
with probability 
.
,
,
be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of 
-
as many lattice points as does 
,
of the lattice points contained in 
is 27 times the fraction of lattice points in 
.
Product of all even divisors of 
If 
dice are rolled, Then number of ways of getting sum at least 
is
,
,
,
such that ![\[\left(a_1+\frac1{a_1}\right)\left(a_2+\frac1{a_2}\right)\cdots\left(a_n+\frac1{a_n}\right)=2020.\]](http://latex.artofproblemsolving.com/3/6/2/362fccfa6353ffcbc4d29f24ccbfce73269080bf.png)
Compute 
.
be an isosceles triangle such that 
.
be a dot on the 
side.
at point 
at 
.
A
have angles 
and 
such that 
and 
. Moreover, suppose that the product of the side lengths of the triangle is equal to its area. Let 
denote the circumcircle of 
intersect 
at 
intersect 
at 
,
intersect 
at 
.
can be written as 
for relatively prime integers 
and 
and squarefree 
,
is 
)
divisors.
does the number 
leave a remainder of 
when divided by 
[/quote]
be a trapezoid with 
parallel to 
,
,
,
.
and 
on segment 
,
[/quote]
of positive integers less than or equal to 20 are there such that 
,
,
are all integers?
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Hello AoPS!
STEAM For All is excited to announce its third virtual
The virtual Orange Mathematics Competition (OMC) 10/12, is a Mock American Mathematics Competition (AMC) 10/12 with questions written by volunteers with years of experience in competition math! Students may only take one exam. Like the AMCs, students in and below 10th grade can take the OMC 10 while students in and below 12th grade can take the OMC 12. Students will have a great opportunity to challenge themselves with new questions, listen to experienced competitors explain the most difficult questions, and be recognized in an awards ceremony!
Register here! Registration ends Friday, October 28, 2022, at 11:59 PDT.
The competition will be held on Saturday, October 29th, 2022 from 2 - 4:30 PM PDT. The general schedule will be as follows (note that all times below are in PDT):
2:00 - 2:15 PM: Introduction
2:15 - 3:30 PM: OMC 10/12 Competition
3:30 - 4:15 PM: Problem Review Session
4:15 - 4:30 PM: Awards and Closing Ceremony
For sample problems, here are a few problems from past OMC 10/12s! Feel free to discuss any of them in this thread (or any other problems from past SFA contests).
More Questions? Contact us at ocmathcircle@gmail.com or ask in this thread!
We hope that you join us for the upcoming competition!
STEAM For All is excited to announce its third virtual
OMC 10/12!
The virtual Orange Mathematics Competition (OMC) 10/12, is a Mock American Mathematics Competition (AMC) 10/12 with questions written by volunteers with years of experience in competition math! Students may only take one exam. Like the AMCs, students in and below 10th grade can take the OMC 10 while students in and below 12th grade can take the OMC 12. Students will have a great opportunity to challenge themselves with new questions, listen to experienced competitors explain the most difficult questions, and be recognized in an awards ceremony!
Register here! Registration ends Friday, October 28, 2022, at 11:59 PDT.
The competition will be held on Saturday, October 29th, 2022 from 2 - 4:30 PM PDT. The general schedule will be as follows (note that all times below are in PDT):
2:00 - 2:15 PM: Introduction
2:15 - 3:30 PM: OMC 10/12 Competition
3:30 - 4:15 PM: Problem Review Session
4:15 - 4:30 PM: Awards and Closing Ceremony
For sample problems, here are a few problems from past OMC 10/12s! Feel free to discuss any of them in this thread (or any other problems from past SFA contests).
More Questions? Contact us at ocmathcircle@gmail.com or ask in this thread!
We hope that you join us for the upcoming competition!
works. But then you have the sneakier cases to work with, and that's where I think things can get messy and easy to miss stuff (reminiscent of 2018 AIME I #9).
,
,
,
,
,
Did anyone actually get their scores back?
