Pigeonhole Principle

$latex \pi = 3.14159265358979323846264 $
$latex \text{Let } a_1, a_2,\dots a_{24} \text{ represent the first 24 digits of } \pi$
Prove:
$latex (a_1 - a_2)(a_3 - a_4) \dots (a_{23} - a_{24}) \text{ is even}$

Proof:

13 Odd digits = {3.14159265358979323846264 }

11 Even digits

$latex \text {12 brackets :}(a_1 - a_2)(a_3 - a_4) \dots (a_{23} - a_{24})$

Put 13 odds into 12 brackets, by Pigeonhole Principle, there is certainly one bracket where
$latex (a_j - a_k) \text{ is a difference of 2 odds, which is an even = 2n}$

2n multiplies with any number will always give even.
The product of 2n with the other 11 brackets will always be even.

Therefore
$latex (a_1 - a_2)(a_3 - a_4) \dots (a_{23} - a_{24}) \text { is even}$

IMO Technique

(a+b)³ = a³ + 3a²b+ 3ab² + b³

Different equivalent forms:
(1)：(a+b)³ = a³ + b³+3ab(a+b)
(2)：a³ + b³ = (a+b)³ - 3ab(a+b)
(3): a³ + b³ = (a+b)(a² -ab + b²)
(4)：(a+b)³ - ( a³ + b³ ) = 3ab(a+b)

1997 USAMO Q5:
Prove:
$latex \frac{1}{a^{3}+b^{3}+abc} +
\frac{1}{b^{3}+c^{3}+abc} +
\frac{1}{c^{3}+a^{3}+abc} \leq
\frac{1}{abc}$

Proof:
Apply (3):
a³ + b³ = (a+b)(a² -ab + b²) ≥ (a+b)ab

Note:
a² -ab + b²= (a-b)² + ab ≥ ab
since (a-b)² ≥ 0

$latex \frac{abc}{a^{3}+b^{3}+abc}
\leq \frac{abc}{(a+b)ab + abc}
= \frac{c}{a+b+c}$


Symmetrically,
$latex \frac{abc}{b^{3}+c^{3}+abc}
\leq \frac{a}{a+b+c}$

$latex \frac{abc}{c^{3}+a^{3}+abc}
\leq \frac{b}{a+b+c}$

Add 3 RHS:
$latex \frac{a+b+c}{a+b+c} = 1$

$latex \frac{abc}{a^{3}+b^{3}+abc} +
\frac{abc}{b^{3}+c^{3}+abc} +
\frac{abc}{c^{3}+a^{3}+abc} \leq 1$

$latex \frac{1}{a^{3}+b^{3}+abc} +
\frac{1}{b^{3}+c^{3}+abc} +
\frac{1}{c^{3}+a^{3}+abc} \leq
\frac{1}{abc}$

[QED]

IMO Super-coach: Rukshin

Rukshin at 15 was a troubled russian kid with drink and violence, then a miracle happened: He fell in love with Math and turned all his creative, aggressive, and competitive energies toward it.

He tried to compete in Math olympiads, but outmatched by peers. Still he believed he knew how to win; he just could not do it himself.

He formed a team of schoolchildren a year younger than he and trained them.
At 19 he became an IMO coach who produced Perelman (Gold IMO & Fields/Clay Poincare Conjecture). In the decades since, his students took 70 IMO, include > 40 Golds.

Rukshin's thoughts on IMO:

1. IMO is more like a sport. It has its coaches, clubs, practice sessions, competitions.

2. Natural ability is necessary but NOT sufficient for success: The talented kid needs to have the right coach, the right team, the right kind of family support, and, most important, the WILL to win.

3. At the beginning, it is nearly impossible to tell the difference between future (Math) stars and those who will be good (at IMO) but never great (Mathematician).