[/quote]
,
be a square of center 
and let 
be the symmetric of the point 
with respect to point 
.
be the intersection of 
and 
,
be the intersection of 
and 
.
is the angle bisector of 
.
be a positive integer. For every positive integer 
the sequence 
is defined, where 
are integers. Among these 
and 
and 
intersect at 
,
and 
intersect at 
.
intersects line 
at 
,
intersects line 
and 
,
and 
are collinear in that order. Prove that if lines 
and 
intersect at 
.
?
Prove that
vertices, each edge has one of the colors 
,
,
.
,
groups such that any two vertices connected by an edge of color 
are in different groups, find the minimum possible value of 
.
such that, for all 
,
such that for all integer 
and a subset ![$[n],$](http://latex.artofproblemsolving.com/d/2/7/d27d2715f4f95c24a28b7c6371a006d4c803b2cf.png)
satisfy 
for all 
then 
be the set of positive integers. Find all functions 
such that for all 
,![\[m^2f(m) + n^2f(n) + 3mn(m + n)\]](http://latex.artofproblemsolving.com/0/f/f/0ff783fa217b7be4bda5006e417f1471ea3a6fca.png)
is a perfect cube.
and 
.
that satisfy![\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]](http://latex.artofproblemsolving.com/f/b/b/fbb693770cd90d8a144026608074122e54da5802.png)
for all 
.
![$\mathbb{R}[x]$](http://latex.artofproblemsolving.com/a/8/8/a88b9f4858016b1771635c611d44b56161a08009.png)
be the set of all polynomials with real coefficients. Find all functions ![$f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$](http://latex.artofproblemsolving.com/0/3/3/033be3ab043bd47e537f01791c813739ad5c7da5.png)
satisfying the following conditions:
maps the zero polynomial to itself,![$P \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/f/7/c/f7c4c970908aa5570a9afcc45fc16c1d36cc3d5d.png)
,
,![$P, Q \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/c/3/0/c301578bade5b93056d77454921cba0a9812212c.png)
,
and 
have the same set of real roots.
are there such that 
,
.
satisfying the following properties
for all 
.
for all 
and 
.
.
,
and 
is cyclic. Given 
,
and 
,
may be expressed as 
for positive integers 
and 
,
.
Y by son7, jacoporizzo, Pleaseletmewin, fukano_2, JustinLee2017, 606234, math31415926535, winnertakeover, Jerry122805, Monkey_king1, ab456, RedFireTruck, MathJams, ImSh95, RJ5303707, v4913, sotpidot, amazingxin777, vsamc, 618173, fidgetboss_4000, Ha_ha_ha, john0512, bowenying24, SuperJJ, superagh, centslordm, aops98888, russellk, skyscraper, nikitadas, megarnie, rayford, KLBBC, Mango247, Mango247
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 
,
.
