Modular Arithmetic and Introduction to Spiral Similarity
by eashang1, Jun 13, 2018, 2:30 AM
Geometry 2 has now covered the basics of various topics that are common in olympiad geometry - namely spiral similarity, the Miquel point, the Simson line, and angle chasing with cyclic quadrilaterals. Though we haven't gone very in depth with many of these concepts yet, the class has been getting increasingly more interesting (I'm especially looking forward to the day after tomorrow's class, which will cover the nine-point circle) :p
IN NT2 we got to modular arithmetic and using FLT. Though I'm very familiar with both these concepts, the problems apply them in so many different ways that they are very difficult to master. I will never understand how some people are able to come up with such outlandish solutions to olympiad number theory...
I forgot to mention: tomorrow is test day
I'll have two tests each with five proofs and two hours. I haven't done a math competition in months, so I'm really looking forward to them, but I'm worried I'll fail miserably. I'm fairly certain each test will range from easy AIME level to USAMO level. Fortunately, people supposedly usually get a 40-60 on them, and I'm fairly confident I can get at least that...?
Here are a bunch of the geo/NT problems that I thought were nice from the past two days:
IN NT2 we got to modular arithmetic and using FLT. Though I'm very familiar with both these concepts, the problems apply them in so many different ways that they are very difficult to master. I will never understand how some people are able to come up with such outlandish solutions to olympiad number theory...
I forgot to mention: tomorrow is test day
I'll have two tests each with five proofs and two hours. I haven't done a math competition in months, so I'm really looking forward to them, but I'm worried I'll fail miserably. I'm fairly certain each test will range from easy AIME level to USAMO level. Fortunately, people supposedly usually get a 40-60 on them, and I'm fairly confident I can get at least that...?Here are a bunch of the geo/NT problems that I thought were nice from the past two days:
for 
.
and 
?
.
to a power, multiplies it by some constant, and adds multiple such terms. All of these operations (multiplication, addition, and subtraction) preserve the modulo. Thus 
.
,
is chosen on the arc 
of the circumcircle of a square 
.
![[asy]
size (5cm);
pair A = (5,5);
pair B = (0,5);
pair C = (0,0);
pair D = (5,0);
pair P = dir(285);
draw (B--P--C);
draw(A--B--C--D--cycle);
draw(circumcircle(A,B,C));
dot("A", A);
dot("B", B);
dot("C", C);
dot("D", D);
dot ("P", P);
[/asy]](http://latex.artofproblemsolving.com/e/5/c/e5cd3373b351f5d36b969464f54ebbdbfbe53157.png)
about point 
such that 
maps to 
.
.
and 
(since rotations maintain congruency), 
.
by construction, 
is the diagonal of the square with side length 
.
.This post has been edited 8 times. Last edited by eashang1, Jun 16, 2018, 11:03 PM
January 2022
October 2021