[/quote]
,
be the set of positive real numbers. Find all functions 
such that, for all 
,
be a triangle, and let 
be a point on side 
.
at 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle 
distinct from point 
.
and 
intersect 
and 
respectively.
such that 
for all real number 
and 
and 
on the plane such that 
is an equilateral . Let 
and 
at 
and 
at 
,
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds.
be a positive integer such that 
.
.
be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let 
be the intersection point of lines 
.
on 
and 
on 
such that 
Interior points.
)
.
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
and 
then prove 
be the set of positive real numbers.
such that there exists a function 
with the property:![\[
f(y^2f(x) + y + c) = xf(x+y^2)
\]](http://latex.artofproblemsolving.com/e/3/0/e30ee6d9f69cc43422a5c2b9813192a632d359cb.png)
for all 
.
be non-negative real numbers such that 
Prove that
of positive integers less than or equal to 20 are there such that 
,
,
are all integers?
be the maximum number of distinct scores. What is the probability that there will be 
Y by mm999aops, mannshah1211, megarnie, kante314, Gumball, SK_pi3145, centslordm, MathWhiz214, Pleaseletmewin, jacoporizzo, tigerzhang, bfan0805, interactivemath, vsamc, mahaler, Jc426, jhu08, Geometry285, math31415926535, jehu26
Hello AoPS!
STEAM For All is excited to announce its second
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 three-time MOPper, multiple USA(J)MO qualifiers, 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 Friday, October 29th, 2021 at 12 PM PST.
The competition will be held on Saturday, October 30th, 2021 from 4 - 6:30 PM PDT. The general schedule will be as follows (note that all times below are in PDT):
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
For sample problems, here are a few problems from January's OMC 10/12! Feel free to discuss any of them in this thread (or any other problems from past SFA contests).
More Questions? Contact us at info@steamforall.org or ask in this thread!
We hope that you join us for the upcoming tournament!
STEAM For All is excited to announce its second
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 three-time MOPper, multiple USA(J)MO qualifiers, 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 Friday, October 29th, 2021 at 12 PM PST.
The competition will be held on Saturday, October 30th, 2021 from 4 - 6:30 PM PDT. The general schedule will be as follows (note that all times below are in PDT):
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
For sample problems, here are a few problems from January's OMC 10/12! Feel free to discuss any of them in this thread (or any other problems from past SFA contests).
![[asy]
size(5cm);
pair A, B, C;
A=(0,0);
B=(0,5);
C=(12,0);
draw(A--B--C--A);
filldraw(A--(2,0)--(2,2)--(0,2)--cycle,grey);
filldraw((2.698+12/13,3.876-5/13)--(0.852+12/13,4.645-5/13)--(0.0828+12/13,2.799-5/13)--(1.929+12/13,2.030-5/13)--cycle,grey);
label("$A$", A, SW);
label("$B$", B, NW);
label("$C$", C, SE);
[/asy]](http://latex.artofproblemsolving.com/6/a/2/6a2e0ce6d7373438eb8f60f58929e425aee915a2.png)
More Questions? Contact us at info@steamforall.org or ask in this thread!
We hope that you join us for the upcoming tournament!
.
