[/quote]
,
and 
Prove that
, 
is a point on 
such that 
is the internal bisector of 
. Now Suppose 
=
. Prove that 
.
![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
,
is expressed by the sum of 
consecutive odd numbers. Express the minimum term in terms of 
.
Prove that :
Where 
denote the set of positive integers. Find all functions 
such that for positive integers 
and 
![\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]](http://latex.artofproblemsolving.com/c/1/f/c1f8ffe04cfcc45b498f3931a4796d5c56dc04d0.png)
that satisfy![\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]](http://latex.artofproblemsolving.com/f/b/b/fbb693770cd90d8a144026608074122e54da5802.png)
for all 
.
are there such that 
,
.
satisfying the following properties
for all 
.
for all 
and 
.
.
and 
be the midpoints of 
and 
,
is chosen on the arc 
of the circumcircle of 
is cyclic. Given 
,
and 
,
may be expressed as 
for positive integers 
and 
,
.
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Hello AoPS!
After a successful mock contest with the OMC 8, STEAM For All is excited to announce its first ever
The virtual Orange Mathematics Competition (OMC) 10/12, is a Mock American Mathematics Competition (AMC) 10/12 with questions written by STEAM for All volunteers! Our problem writers and testsolvers consist of a two-time MOPper, a two-time USAJMO qualifier, and plenty of AIME qualifiers.
The mock contest offers two entirely free online math contests, the OMC 10 and the OMC 12. Students may only take one exam, and students in grades 10 and below are eligible for the OMC 10 while students in grades 12 and below are eligible for the OMC 12. It is a wonderful opportunity for students to engage in a practice test with fun, original questions, listen to experienced competitors explain solutions to the most difficult questions on the exam, and finish off with an awards ceremony to recognize some of the top scoring students!
Register at https://www.steamforall.org/omc-10-12. Registration ends on Thursday, January 21st, 2021 at 12 PM PST.
The competition will be held on Saturday, January 23rd, 2021 from 4 - 6:30PM PST. The general schedule will be as follows (note that all times below are in PST):
4:00 - 4:15 PM: Introduction
4:15 - 5:30 PM: OMC 10/12 Competition
5:30 - 6:15 PM: Problem Review Session
6:15 - 6:30 PM: Awards and Closing Ceremony
As for sample problems, here are a few problems from our problem writers that came up on previous contests! Feel free to discuss any of them in this thread.
More Questions? Contact us at info@steamforall.org or ask in this thread!
We hope that you join us for the upcoming tournament!
Test Links:
OMC 10: https://drive.google.com/file/d/1UVlvprEDwY_l9zTGV7VqW_Ymz1FYQ7Uk/view?usp=sharing
OMC 12: https://drive.google.com/file/d/1SVWk3bJUsMUQR9RUQ1P6YBsIXcHS1uYn/view?usp=sharing
Statistics attached.
After a successful mock contest with the OMC 8, STEAM For All is excited to announce its first ever
OMC 10/12
The virtual Orange Mathematics Competition (OMC) 10/12, is a Mock American Mathematics Competition (AMC) 10/12 with questions written by STEAM for All volunteers! Our problem writers and testsolvers consist of a two-time MOPper, a two-time USAJMO qualifier, and plenty of AIME qualifiers.
The mock contest offers two entirely free online math contests, the OMC 10 and the OMC 12. Students may only take one exam, and students in grades 10 and below are eligible for the OMC 10 while students in grades 12 and below are eligible for the OMC 12. It is a wonderful opportunity for students to engage in a practice test with fun, original questions, listen to experienced competitors explain solutions to the most difficult questions on the exam, and finish off with an awards ceremony to recognize some of the top scoring students!
Register at https://www.steamforall.org/omc-10-12. Registration ends on Thursday, January 21st, 2021 at 12 PM PST.
The competition will be held on Saturday, January 23rd, 2021 from 4 - 6:30PM PST. The general schedule will be as follows (note that all times below are in PST):
4:00 - 4:15 PM: Introduction
4:15 - 5:30 PM: OMC 10/12 Competition
5:30 - 6:15 PM: Problem Review Session
6:15 - 6:30 PM: Awards and Closing Ceremony
As for sample problems, here are a few problems from our problem writers that came up on previous contests! Feel free to discuss any of them in this thread.
More Questions? Contact us at info@steamforall.org or ask in this thread!
We hope that you join us for the upcoming tournament!
Test Links:
OMC 10: https://drive.google.com/file/d/1UVlvprEDwY_l9zTGV7VqW_Ymz1FYQ7Uk/view?usp=sharing
OMC 12: https://drive.google.com/file/d/1SVWk3bJUsMUQR9RUQ1P6YBsIXcHS1uYn/view?usp=sharing
Statistics attached.
This post has been edited 9 times. Last edited by ilovemath04, Jan 25, 2021, 11:37 PM
that divides the numerator. Thus, we would group the numbers as follows:
powers of 
power of 
powers of 
,
.
