Amount of work for future math (PLEASE HELP)

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seanmeng822

36 posts

seanmeng822

#1 h Jan 1, 2015, 2:57 AM • 5 Y

Y by mathmaster2012, spin8, onezero, Adventure10, Mango247

I pretty much started competition math at the beginning of freshman year. (I got 87 on AMC 10 in 8th grade, so that doesn't really count).

Last year I was around 15 points off the JMO cutoff.

Now, I'm a sophomore, and I think I can pretty comfortably pull off 135 on AMC 10 and 9ish on AIME.
(Those were the scores I got for mock contests that I recently took, but I should be better after a couple of months).

I want to know how much more additional effort I need to put in for a reasonably good performance in Olympiad level math (USAJMO in particular). I want to get more than a 7/42 on JMO (maybe even double digits), but I want to know what I'm getting myself into.

I saw the JMO questions before (I even did one), but most of them are like GG bro.

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jonyj1005

1705 posts

jonyj1005

#2 h Jan 1, 2015, 3:44 AM • 3 Y

Y by smoothtorus, Adventure10, Mango247

Just put effort whenever you have time.
And if you don't have time make some time.

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Wolstenholme

543 posts

Wolstenholme

#3 h Jan 1, 2015, 4:53 AM • 123 Y

Y by niraekjs, stan23456, blasterboy, miru99, Einstein314, tastymath75025, TheMaskedMagician, crastybow, seanmeng822, DrMath, ImpossibleCube, Not_a_Username, DigitalKing257, hamup1, Blockman123, DVA6102, zmyshatlp, IsabeltheCat, droid347, LarkaPies, jh235, fclvbfm934, TheCrafter, fireclaw105, chenjamin, blueflute19, PlatinumFalcon, rjiang16, 2015WOOTer, zacchro, hamhandedmathy, spartan168, EulerMacaroni, Tommy2000, BFYSharks, tau172, jam10307, dantx5, hwl0304, blippy1998, abishek99, Wave-Particle, wu2481632, 15Pandabears, Darn, bestwillcui1, MathematicalWays, LoneConquerorer, mrowhed, quangminhltv99, ingenio, azmath333, mathboxboro, 62861, kat123, MathAwesome123, thkim1011, huricane, Stens, ac_math, trumpeter, SHARKYBOY, mathmaster2012, Target_cmi, whiteawesomesun, hotstuffFTW, bearytasty, BOGTRO, dawbyrd, r31415, MathStudent2002, Magikarp1, ohmcfifth, Benq, Python54, shiningsunnyday, Royalreter1, eshan, champion999, arvind_r, speck, dhusb45, JasperL, nkim9005, hiabc, turbo300, Gibby, Iamawesome1, sophie8, mathisawesome2169, expiLnCalc, mathlogician, 277546, Stormersyle, sriraamster, mathleticguyyy, BXU65, Math-wiz, VipMath, SenorIncongito, magicarrow, Kanep, anonman, ThisUsernameIsTaken, samrocksnature, lethan3, tennisrules, michaelwenquan, Math4Life7, rayfish, Batsuh, mahaler, Adventure10, BestAOPS, LeonidasTheConquerer, Tem8, and 7 other users

I wrote a similar post here - http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=585590&p=3463818&hilit=mildorf#p3463818. However, in the months since, I have practiced lots of different things so will update my answer.

To start, let me say that you're not at too much of a disadvantage because of your relatively late start - I started essentially at the end of freshman year and some people (a good example is Holden Lee) started even later. My advice will be partitioned into three categories: Easy (as in, let's solve a problem or two), Medium (Going for HM), and Hardcore (let's get a score of 40+ without breaking a sweat)

Easy:

This shouldn't be too hard. It seems like you've been practicing, which means you've probably become more comfortable with thinking mathematically. Given that you seem confident on JMO qualification, you probably have solid fundamentals. At the JMO 1/4 level, it's really just about thinking about the problem for a long period of time rather than prior knowledge. Your best bet would be to familiarize yourself with the feel of the contest. I would recommend doing a bunch of past JMO's (and if you want more, JBMO's and CMO's) and reading solutions. I personally believe in doing each contest more than once - even if you're just getting problems because you remember the solution, that increases confidence and comfort, which is the goal here.

Medium:

We're getting serious. At this point, you better be putting in hour(s) every day (it depends on how quick you pick things up). Logic isn't going to cut it anymore - you need theorems, motivation, and most importantly, experience. Here are some materials I would recommend - I'm going to split it up into various subjects.

General:

Engel's Problem Solving strategies is a good source. Also, try working through some USAMO's and ISL's.

Geometry:

The best way to get a solid foundation here (as in, Ceva, Menelaus, Power of a Point, Radical Axis, Trig Ceva/trig bashing, Angle Chasing, Spiral Similarity, Triangle Centers) is to either read something like Geometry Revisited or, frankly, get a friend to give you a packet to work through from AMSP/{insert other mathcamp}. A MUST-READ is Yufei Zhao's handout - http://yufeizhao.com/olympiad/geolemmas.pdf. You can also surf wikipedia and look these things up. This is probably all you need, however you might want to delve deeper, so....

As you get more advanced, you definitely want to learn about Projective Geometry. I would recommend looking at http://diendantoanhoc.net/forum/index.phpapp=core&module=attach&section=attach&attach_id=16951 and http://imomath.com/index.php?options=330&lmm=0

I don't know a good source, but try looking up pedal triangles and anti-steiner points (and the things that go with them, like the droz-farny theorem). This also includes isogonal conjugates.

I also recommend learning how to bary bash and complex bash. The best sources for these two are Evan Chen's bary handout - http://www.artofproblemsolving.com/resources/papers/bary_full.pdf - and http://hoaxung.files.wordpress.com/2010/04/marko-radovanovic-complex-numbers-in-geometry.pdf for complex numbers. For complex, you also want to look at http://web.mit.edu/yisun/www/notes/complex.pdf, however only the former source provides solutions.

Polynomials:

Learn irreducibly criterion like Extended Eisenstein (and maybe even Schonemann's criterion). imomath.com has great stuff here.

Functional Equations:

Again imomath.com is a good enough foundation. There are some good examples in Problems in Elementary Number Theory (PEN) by Hojoo Lee as well.

Inequalities:

Evan Chen's inequality handout - http://www.mit.edu/~evanchen/handouts/ineq/en.pdf - is a good place to start. For a more in depth look at basic inequalities, try and work through mildorf's handout - http://www.artofproblemsolving.com/resources/papers/mildorfinequalities.pdf

If you're the bashy type, try learning SOS (I can't find links right now, ask around)

Combinatorics

Uhhh... I'm pretty bad at this so I can't tell you much. You should know induction, Pigeonhole, the Principle of Inclusion Exclusion, maybe even tiling.

Number Theory

I'm going to get a ton of flack for this, but I stand by my theory - work through PEN. It's long. It's hard. In my opinion, it's WORTH IT. This gave me the confidence and foundation I needed to do NT. Just don't try and solve the first problem...

Hardcore:

Get ready. Sh*t just got real.

General

Read PFTB. I can't stress this enough. It's sooooooooo good.

Geometry:

Do problems. Do lots and lots and lots of problems. If you do synthetic, you're going to want a HUGE repertoire of configurations you know better than the back of your hand. If you bash, you better practice a ton, and you better know how to bash fast.

I would recommend reading short articles about cool configurations - darij grinberg's website has a lot of this. Something I recently had fun studying was the neuberg cubic - check it out. Basically just get way more specific - I personally read papers about things like mixtilinear incircles, isodynamic points, etc...

Polynomials:

For the very hardcore: http://yufeizhao.com/olympiad/intpoly.pdf. Maybe also look up stuff about Galois Theory - it gets you cool irreducibility stuff like Capelli's Criterion.

Functional Equations:

I don't really know.. they're pretty contrived.

Inequalities:

Learn specific methods. Evan Chen has a good handout on Lagrange Multipliers if you're the bashy type. Isolated fudging is magical. Smoothing is very important. Try reading books like Secrets in Inequalities. For the trolls among you, Harazi integration is great. One of my favorite papers is http://web.mit.edu/~darij/www/vornicus.pdf which is kind of like SOS but maybe even more general.

Combinatorics

Graph Theory, posets, set theory. Read up on Hall's Marriage Theorem, Dilworth's Theorem, Gallai's Theorem. The de Bruijn-Tengbergen-Kruyswijk Theorem is pretty cool. Know Turan's Theorem by heart. Know what independent sets, dominating sets are. Sperner's Theorem is nice, so is the LYM inequality. Definitely learn the Probabilistic Method - http://www.math.cmu.edu/~ploh/docs/math/mop2009/prob-comb.pdf. Maybe read Pearls in Graph Theory or Diestel's Graph Theory. On a different note, looking up things like Van der Waerdan's Theorem (and it's generalization, Szemeredi's Theorem) will probably teach you a lot.

Number Theory

Learn about Cyclotomic Polynomials - http://lessol.w.staszic.waw.pl/pdfy/h.pdf. Read Victor Wang's handout (it's on his AoPS blog, too lazy to go find it). Learn about finite fields. Some personal favorite theorems of mine are Zsigmondy's Theorem and Kobayashi's Theorem. There is some nice stuff you should know about uniformity and rational approximation (Van der Corput's Theorem, Dirichlet's Theorem). Quadratic Reciprocity is nice (you should probably be able to prove it).

A lot of the stuff above is also found in PFTB.

To actually answer your question, the amount of effort you put in depends wholly on you. Can you get through long books? How many times do you need to read a solution before you get it? While I can't answer that for you, I can give you advice about how to USE your time, so that's what I did

.

This post has been edited 3 times. Last edited by Wolstenholme, Jan 7, 2015, 2:21 AM

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seanmeng822

36 posts

seanmeng822

#4 h Jan 1, 2015, 9:37 PM • 1 Y

Y by Adventure10

Quote:

I'm going to get a ton of flack for this, but I stand by my theory - work through PEN.

What's PEN?

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MathSlayer4444

1631 posts

MathSlayer4444

#5 h Jan 1, 2015, 9:39 PM • 3 Y

Y by Adventure10, Mango247, and 1 other user

Probably this

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Royalreter1

1913 posts

Royalreter1

#6 h Jan 1, 2015, 11:06 PM • 3 Y

Y by MathSlayer4444, Adventure10, Mango247

There's a whole forumhere. This is the book itself.

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DrMath

2130 posts

DrMath

#7 h Jan 2, 2015, 2:26 PM • 2 Y

Y by Adventure10, Mango247

I looked but couldn't find anything on Harazi integration - does anyone know what it is?

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Wolstenholme

543 posts

Wolstenholme

#8 h Jan 2, 2015, 7:29 PM • 9 Y

Y by zacchro, jh235, huricane, champion999, mrowhed, 62861, Adventure10, and 2 other users

Harazi integration is something discussed in a late chapter of Problems From The Book. I think it's easiest to show with an example: say you wanted to prove the inequality $\frac{1}{3a} + \frac{1}{3b} + \frac{1}{3c} + \frac{3}{a + b + c} \ge \sum_{cyc}\left(\frac{1}{2a + b} + \frac{1}{2b + a}\right)$ for $a, b, c > 0.$

This inequality is relatively difficult - however, one quickly notices the similarity to Schur's Inequality $x^3 + y^3 + z^3 + 3xyz \ge x^2y + x^2z + y^2z + y^2x + z^2x + z^2y.$ Plugging in $x = t^{a - \frac{1}{3}}$ and $y = t^{b - \frac{1}{3}}$ and $z = t^{c - \frac{1}{3}}$ we have that $t^{3a - 1} + t^{3b - 1} + t^{3c - 1} + 3t^{a + b + c - 1} \ge \sum_{cyc}\left(t^{2a + b - 1} + t^{2b + a - 1}\right).$ This inequality holds for all $t$ between $0$ and $1$ so we can say that $\int_0^1{t^{3a - 1} + t^{3b - 1} + t^{3c - 1} + 3t^{a + b + c - 1}}dt \ge \int_0^1{\sum_{cyc}\left(t^{2a + b - 1} + t^{2b + a - 1}\right)}dt$ which is equivalent to our original inequality!

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LarkaPies

29 posts

LarkaPies

#9 h Jan 6, 2015, 5:56 AM • 1 Y

Y by Adventure10

Hey, Wolstenholme, I hope it wouldn't be too rude of me to ask how much math you did on average every day for the past year. I'm really curious.

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Wolstenholme

543 posts

Wolstenholme

#10 h Jan 6, 2015, 3:28 PM • 5 Y

Y by LarkaPies, zmyshatlp, champion999, Adventure10, Mango247

Well it definitely varied - some days I didn't do ANY math (although I did think about problems) while others I went on 6-8 hour marathons

. Overall I think I averaged maybe 4 hours a day last year, which was mostly made possible by my school allowing me to have a Chromebook out in class (which meant I could read articles and look at problems for hours during boring classes). This allowed me to not sacrifice time outside of school.

However, this doesn't at all mean that amount of time is right for someone else. On a lot of topics, I really have trouble understanding the main ideas and spend hours rereading/working through them, while other people might just need a couple of minutes before they have it down. I personally know many people who get more out of thirty minutes of math than I get out of three hours of math.

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jh235

910 posts

jh235

#11 h Jan 7, 2015, 1:44 AM • 2 Y

Y by Adventure10, Mango247

Wolstelhome: some/a lot of the links are broken.

Do you have fixed links or no?

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Wolstenholme

543 posts

Wolstenholme

#12 h Jan 7, 2015, 2:06 AM • 2 Y

Y by Adventure10, Mango247

^ Sorry about that... which links? Some of the links include the right parentheses I put at the end of them so I'll edit that... tell me if the links work now.

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jh235

910 posts

jh235

#13 h Jan 7, 2015, 2:26 AM • 2 Y

Y by Adventure10, Mango247

for example, the probabalistic method (ploh), inequalities (vornicu) but someone on the woot classroom told me that a correct link is this, and the cyclotomic polynomials.

Thanks.

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Wolstenholme

543 posts

Wolstenholme

#14 h Jan 7, 2015, 3:08 AM • 2 Y

Y by jh235, Adventure10

Hmm I see what you mean about Darij Grinberg's article (the Vornicu-Schur Inequality one). To access it, go to http://web.mit.edu/~darij/www/ and the link should work from that page. However, Po-Shen's article about the Probabilistic Method seems to work for me. If you can't access it, I actually remembered another really really good link - http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1943887#p1943887

The Cyclotomic Polynomials link also works for me but if you can't find it search up "Lawrence Sun Cyclotomic Polynomials"

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aravindsidd

369 posts

aravindsidd

#15 h Jan 7, 2015, 1:52 PM • 2 Y

Y by Adventure10, Mango247

This should be stickied.

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jh235

910 posts

jh235

#16 h Jan 7, 2015, 2:09 PM • 2 Y

Y by Adventure10, Mango247

Thanks Wolstenholme.

Also, what exactly is SOS? And, do you have any other problems for Harazi integration? (If you give solutions, could you hide them please.)

( Also, I agree that this should be stickied.)

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Wolstenholme

543 posts

Wolstenholme

#17 h Jan 7, 2015, 5:12 PM • 3 Y

Y by jh235, Adventure10, and 1 other user

SOS is a powerful, but bashy, method of solving inequalities. A good article on it is found on https://sites.google.com/site/imocanada/2009-summer-camp (it's written by David Arthur). Essentially you take a three-variable inequality in $a, b, c$ and put it in the form $S_a(b - c)^2 + S_b(c - a)^2 + S_c(a - b)^2$ and if the $S_a, S_b, S_c$ satisfy some simple properties then that equation is always equal to or greater than $0.$

Here's another problem to do with Harazi integration: for $a, b, c > 0$ prove that $\frac{1}{2a} + \frac{1}{2b} + \frac{1}{2c} \ge \frac{1}{b + c} + \frac{1}{c + a} + \frac{1}{a + b}$

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zacchro

179 posts

zacchro

#18 h Jan 11, 2015, 5:23 PM • 3 Y

Y by Adventure10, Mango247, and 1 other user

Solution

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MSTang

6012 posts

MSTang

#19 h Apr 19, 2015, 5:58 AM • 2 Y

Y by Adventure10, Mango247

Just wanted to point out a pretty simple, non-calculus-y solution to that problem. So advanced tools like Harazi integration aren't exactly necessary, but they can be helpful if you've already mastered the basics/intermediates.

By Cauchy, $(a+b)\left(\dfrac1a+\dfrac1b\right) \ge (1+1)^2 = 4,$ so $\dfrac{1}{4a} + \dfrac{1}{4b} \ge \dfrac{1}{a+b}.$ Take the similar inequaliies in $a, c$ and $b, c$ and add the three together to get the requested result. $\square$

(That solution was motivated for me by wanting to achieve terms in exactly two of the variables on the right; I realized that I should focus on two of the terms on the LHS at a time, and try to work with them separately. Fractions also suggested Cauchy-Schwarz.)

This post has been edited 1 time. Last edited by MSTang, Apr 19, 2015, 5:59 AM

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hwl0304

1840 posts

hwl0304

#20 h Apr 19, 2015, 6:08 AM • 3 Y

Y by AstrapiGnosis, Adventure10, Mango247

We could also use AM-HM to get $\dfrac{a+b}{2}\ge\dfrac{2ab}{a+b}$ , or $\dfrac{a+b}{4ab}\ge\dfrac{1}{a+b}$ . We get the same result as MSTang. $\square$

Simpler! Lol, i dont even know what harazi integration is

This post has been edited 1 time. Last edited by hwl0304, Apr 19, 2015, 6:11 AM

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ABCDE

1963 posts

ABCDE

#21 h Apr 19, 2015, 2:00 PM • 2 Y

Y by Adventure10, Mango247

Using AM-HM/Cauchy like that is basically using Karamata but writing out the Jensen equivalent just like writing out the AM-GM equivalent of Muirhead.

Karamata says that the sum of the values of a convex function is bigger than another sum if the values of the first one majorize the second one (pretty bad explanation so look it up if you want to learn it)

So for this one if wlog a>b>c then the denominators of the LHS majorize the denominators of the RHS so we can apply Karamata to $f(x)=1/x$ .

I'm pretty sure the other one has a Karamata solution, so it has a weighted AM-HM solution too.

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wu2481632

4233 posts

wu2481632

#22 h Sep 26, 2015, 9:55 PM • 2 Y

Y by Adventure10, Mango247

Hi,

"- and
://hoaxung.files.wordpress.com/2010/04/marko-radovanovic-complex-numbers-in-geometry.pdf
for complex numbers. For complex, you also want to look at"

The link in Wolstenholme's post is broken; would someone please provide a link that works? Sorry about the revive.

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Eugenis

2404 posts

Eugenis

#23 h Sep 26, 2015, 9:57 PM • 3 Y

Y by wu2481632, MathPanda1, Adventure10

https://uqu.edu.sa/files2/tiny_mce/plugins/filemanager/files/4041834/cnum_mr.pdf

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muti66

931 posts

muti66

#24 h Oct 2, 2015, 4:20 AM • 1 Y

Y by Adventure10

I feel like I should klnow what PFTB is, but I don't. What is it?

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trumpeter

3332 posts

trumpeter

#25 h Oct 2, 2015, 4:22 AM • 1 Y

Y by Adventure10

Problems from the Book

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muti66

931 posts

muti66

#26 h Oct 2, 2015, 4:31 AM • 1 Y

Y by Adventure10

Which ones? XYZ press?

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librian2000

126 posts

librian2000

#27 h Oct 2, 2015, 4:32 AM • 1 Y

Y by Adventure10

@muti66. https://www.awesomemath.org/product/problems-from-the-book/
This book can be reedemed for points on Clevermath as well

However, it is known for being very challenging and geared towards olympiad preparation.

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trumpeter

3332 posts

trumpeter

#28 h Oct 2, 2015, 4:32 AM • 2 Y

Y by JustinHerchel, Adventure10

Yes, the book from XYZ press

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Delray

348 posts

Delray

#29 h Mar 20, 2017, 2:38 PM • 2 Y

Y by Adventure10, Mango247

What is PFTB?

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Rubaiya

125 posts

Rubaiya

#30 h Mar 20, 2017, 7:46 PM • 3 Y

Y by wu2481632, Adventure10, Mango247

trumpeter wrote:

Problems from the Book

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Delray

348 posts

Delray

#31 h Mar 20, 2017, 8:40 PM • 1 Y

Y by Adventure10

@above
Thanks

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N0NY0

372 posts

N0NY0

#32 h Mar 20, 2017, 10:02 PM • 1 Y

Y by Adventure10

revive

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Hopeooooo

819 posts

Hopeooooo

#33 h Oct 8, 2022, 10:22 PM • 1 Y

Y by JJaganBeaster

Wolstenholme wrote:

I wrote a similar post here - http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=585590&p=3463818&hilit=mildorf#p3463818. However, in the months since, I have practiced lots of different things so will update my answer.

To start, let me say that you're not at too much of a disadvantage because of your relatively late start - I started essentially at the end of freshman year and some people (a good example is Holden Lee) started even later. My advice will be partitioned into three categories: Easy (as in, let's solve a problem or two), Medium (Going for HM), and Hardcore (let's get a score of 40+ without breaking a sweat)

Easy:

This shouldn't be too hard. It seems like you've been practicing, which means you've probably become more comfortable with thinking mathematically. Given that you seem confident on JMO qualification, you probably have solid fundamentals. At the JMO 1/4 level, it's really just about thinking about the problem for a long period of time rather than prior knowledge. Your best bet would be to familiarize yourself with the feel of the contest. I would recommend doing a bunch of past JMO's (and if you want more, JBMO's and CMO's) and reading solutions. I personally believe in doing each contest more than once - even if you're just getting problems because you remember the solution, that increases confidence and comfort, which is the goal here.

Medium:

We're getting serious. At this point, you better be putting in hour(s) every day (it depends on how quick you pick things up). Logic isn't going to cut it anymore - you need theorems, motivation, and most importantly, experience. Here are some materials I would recommend - I'm going to split it up into various subjects.

General:

Engel's Problem Solving strategies is a good source. Also, try working through some USAMO's and ISL's.

Geometry:

The best way to get a solid foundation here (as in, Ceva, Menelaus, Power of a Point, Radical Axis, Trig Ceva/trig bashing, Angle Chasing, Spiral Similarity, Triangle Centers) is to either read something like Geometry Revisited or, frankly, get a friend to give you a packet to work through from AMSP/{insert other mathcamp}. A MUST-READ is Yufei Zhao's handout - http://yufeizhao.com/olympiad/geolemmas.pdf. You can also surf wikipedia and look these things up. This is probably all you need, however you might want to delve deeper, so....

As you get more advanced, you definitely want to learn about Projective Geometry. I would recommend looking at http://diendantoanhoc.net/forum/index.phpapp=core&module=attach&section=attach&attach_id=16951 and http://imomath.com/index.php?options=330&lmm=0

I don't know a good source, but try looking up pedal triangles and anti-steiner points (and the things that go with them, like the droz-farny theorem). This also includes isogonal conjugates.

I also recommend learning how to bary bash and complex bash. The best sources for these two are Evan Chen's bary handout - http://www.artofproblemsolving.com/resources/papers/bary_full.pdf - and http://hoaxung.files.wordpress.com/2010/04/marko-radovanovic-complex-numbers-in-geometry.pdf for complex numbers. For complex, you also want to look at http://web.mit.edu/yisun/www/notes/complex.pdf, however only the former source provides solutions.

Polynomials:

Learn irreducibly criterion like Extended Eisenstein (and maybe even Schonemann's criterion). imomath.com has great stuff here.

Functional Equations:

Again imomath.com is a good enough foundation. There are some good examples in Problems in Elementary Number Theory (PEN) by Hojoo Lee as well.

Inequalities:

Evan Chen's inequality handout - http://www.mit.edu/~evanchen/handouts/ineq/en.pdf - is a good place to start. For a more in depth look at basic inequalities, try and work through mildorf's handout - http://www.artofproblemsolving.com/resources/papers/mildorfinequalities.pdf

If you're the bashy type, try learning SOS (I can't find links right now, ask around)

Combinatorics

Uhhh... I'm pretty bad at this so I can't tell you much. You should know induction, Pigeonhole, the Principle of Inclusion Exclusion, maybe even tiling.

Number Theory

I'm going to get a ton of flack for this, but I stand by my theory - work through PEN. It's long. It's hard. In my opinion, it's WORTH IT. This gave me the confidence and foundation I needed to do NT. Just don't try and solve the first problem...

Hardcore:

Get ready. Sh*t just got real.

General

Read PFTB. I can't stress this enough. It's sooooooooo good.

Geometry:

Do problems. Do lots and lots and lots of problems. If you do synthetic, you're going to want a HUGE repertoire of configurations you know better than the back of your hand. If you bash, you better practice a ton, and you better know how to bash fast.

I would recommend reading short articles about cool configurations - darij grinberg's website has a lot of this. Something I recently had fun studying was the neuberg cubic - check it out. Basically just get way more specific - I personally read papers about things like mixtilinear incircles, isodynamic points, etc...

Polynomials:

For the very hardcore: http://yufeizhao.com/olympiad/intpoly.pdf. Maybe also look up stuff about Galois Theory - it gets you cool irreducibility stuff like Capelli's Criterion.

Functional Equations:

I don't really know.. they're pretty contrived.

Inequalities:

Learn specific methods. Evan Chen has a good handout on Lagrange Multipliers if you're the bashy type. Isolated fudging is magical. Smoothing is very important. Try reading books like Secrets in Inequalities. For the trolls among you, Harazi integration is great. One of my favorite papers is http://web.mit.edu/~darij/www/vornicus.pdf which is kind of like SOS but maybe even more general.

Combinatorics

Graph Theory, posets, set theory. Read up on Hall's Marriage Theorem, Dilworth's Theorem, Gallai's Theorem. The de Bruijn-Tengbergen-Kruyswijk Theorem is pretty cool. Know Turan's Theorem by heart. Know what independent sets, dominating sets are. Sperner's Theorem is nice, so is the LYM inequality. Definitely learn the Probabilistic Method - http://www.math.cmu.edu/~ploh/docs/math/mop2009/prob-comb.pdf. Maybe read Pearls in Graph Theory or Diestel's Graph Theory. On a different note, looking up things like Van der Waerdan's Theorem (and it's generalization, Szemeredi's Theorem) will probably teach you a lot.

Number Theory

Learn about Cyclotomic Polynomials - http://lessol.w.staszic.waw.pl/pdfy/h.pdf. Read Victor Wang's handout (it's on his AoPS blog, too lazy to go find it). Learn about finite fields. Some personal favorite theorems of mine are Zsigmondy's Theorem and Kobayashi's Theorem. There is some nice stuff you should know about uniformity and rational approximation (Van der Corput's Theorem, Dirichlet's Theorem). Quadratic Reciprocity is nice (you should probably be able to prove it).

A lot of the stuff above is also found in PFTB.

To actually answer your question, the amount of effort you put in depends wholly on you. Can you get through long books? How many times do you need to read a solution before you get it? While I can't answer that for you, I can give you advice about how to USE your time, so that's what I did

.

What is PFTB?

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tennisrules

139 posts

tennisrules

#34 h Oct 8, 2022, 10:29 PM

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Delray wrote:

What is PFTB?

Rubaiya wrote:

trumpeter wrote:

Problems from the Book

@above maybe you should read the posts above...

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scannose

963 posts

scannose

#35 h Oct 8, 2022, 10:30 PM

Y by

lol don't quote the whole thing; delete most of the stuff from the quote and only leave the parts that have to do with what you're going to say
also potb is problems from the book. Someone answered the same question earlier in this topic so you probably would want to scroll up to see if it's mentioned in earlier quotes before asking

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Sleepy_Head

560 posts

Sleepy_Head

#36 h Oct 8, 2022, 11:11 PM

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scannose wrote:

you probably would want to scroll up to see if it's mentioned in earlier quotes before asking

~~ironic~~

(this is a joke im not tryna flame anyone)

This post has been edited 1 time. Last edited by Sleepy_Head, Oct 8, 2022, 11:11 PM

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scannose

963 posts

scannose

#37 h Oct 8, 2022, 11:21 PM • 3 Y

Y by Mango247, Mango247, Mango247

lol i was sniped and the popup didn't come out yet

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Sleepy_Head

560 posts

Sleepy_Head

#38 h Oct 8, 2022, 11:33 PM

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scannose wrote:

lol i was sniped and the popup didn't come out yet

yeah thats what i thought

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