Lcz Notes!

Introduction

So I failed the AOIME. I don't really want to do any more AIME prep, so I have decided to go do some Oly prep :). Here's some oly notes/favorite problems.

ALGEBRA

Algebra is cool. I'm pretty good at it (by my standards shup smh).

.

Before oly...

  B. Problem 1:

Find $x$ if $x+2=5$ .

  B. Problem 2:

Find the sum of all $x$ such that $x^2-12x+234=0$ .

  B. Problem 3 (troll):

Find the sum of all $x$ in $\mathbb{C}$ such that $x^3+3x^2+3x+1=729$

  B. Problem 4 (2018 I/6)

Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$ .

  B. Problem 5 (AOIME/8 sigh...)

Define a sequence recursively by $f_1(x)=|x-1|$ and $f_n(x)=f_{n-1}(|x-n|)$ for integers $n>1$ . Find the least value of $n$ such that the sum of the zeros of $f_n$ exceeds $500,000$ .

  B. Problem 6 (AOIME/11 bash-ish?)

Let $P(x) = x^2 - 3x - 7$ , and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$ . David computes each of the three sums $P + Q$ , $P + R$ , and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$ , then $R(0) = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

  B. Problem 7 (1984/15):

Determine $w^2+x^2+y^2+z^2$ if

$\frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1$ $\frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1$ $\frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1$ $\frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1$

  B. Problem 8 (2020 I/14):

Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1.$ Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b.$ Find the sum of all possible values of $(a+b)^2.$

  B. Problem 9 (HINT: THIS IS GEOMETRY 2006 II/15):

Given that $x, y,$ and $z$ are real numbers that satisfy:

$x = \sqrt{y^2-\frac{1}{16}}+\sqrt{z^2-\frac{1}{16}}$ $y = \sqrt{z^2-\frac{1}{25}}+\sqrt{x^2-\frac{1}{25}}$ $z = \sqrt{x^2 - \frac 1{36}}+\sqrt{y^2-\frac 1{36}}$ and that $x+y+z = \frac{m}{\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime, find $m+n.$

  B. Problem 10 (2011 I/15):

For some integer $m$ , the polynomial $x^3 - 2011x + m$ has the three integer roots $a$ , $b$ , and $c$ . Find $|a| + |b| + |c|$ .

  B. Problem 11 (2014 I/13):

The polynomial $P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17}$ has $34$ complex roots of the form $z_k = r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34,$ with $0 < a_1 \le a_2 \le a_3 \le \cdots \le a_{34} < 1$ and $r_k>0.$ Given that $a_1 + a_2 + a_3 + a_4 + a_5 = m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

  B. Problem 12 (2007 I/14):

A sequence is defined over non-negative integral indexes in the following way: $a_{0}=a_{1}=3$ , $a_{n+1}a_{n-1}=a_{n}^{2}+2007$ .

Find the greatest integer that does not exceed $\frac{a_{2006}^{2}+a_{2007}^{2}}{a_{2006}a_{2007}}.$

  B. Problem 13 (2007 II/14):

Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$ , $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$

  B. Problem 14 (Gotta save the easy ones for the end, 2010 I/14):

For each positive integer n, let $f(n) = \sum_{k = 1}^{100} \lfloor \log_{10} (kn) \rfloor$ . Find the largest value of $n$ for which $f(n) \le 300$ .

  B. Problem 15 (2021 usamo/6):

Dad and mom are playing outside while you are in a conference call discussing serious matters.

What is $1+2+3+ . . . \infty$ ?

  B. Problem 16 (For difficulty of 16, 2016 II/15):

For $1 \leq i \leq 215$ let $a_i = \dfrac{1}{2^{i}}$ and $a_{216} = \dfrac{1}{2^{215}}$ . Let $x_1, x_2, ..., x_{216}$ be positive real numbers such that $\sum_{i=1}^{216} x_i=1$ and $\sum_{1 \leq i < j \leq 216} x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}$ . The maximum possible value of $x_2=\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Inequalities

Yay! I love inequalities. Clever algebraic manipulation+thereoms is all you need. It all comes from experience though...

Basics

Note that the set $(a_n)$ is $(a_1, a_2, a_3, . . . a_n)$ .

$\sum_{\text{sym}} a^2b$ $=a^2b+a^2c+b^2a+b^2c+c^2a+c^2b$ . It is "symmetric".

$\sum_{\text{cyc}} a^2b$ $=a^2b+b^2c+c^2a$ . It is "cyclic".

  AM-GM says that for nonnegative reals , their Arithmetic Mean is  their Geometric Mean.
  .

  Cauchy's Inequality says that for nonnegative reals ,

  Titu's lemma follows directly from Cauchy, it says for the same variables,

  We say a set  majorizes  if

  (a) the sum of the elements are the same.

  (b) the elements are in decreasing order.

  (c) the leftmost sums of  are  those of .

For example, $(24,9,2)$ majorizes $(20, 8, 7)$ and $(5,1,1)$ majorizes $(4,3,0)$ .

  Muirhead's Inequality says that for reals  and nonnegative reals , where  majorizes ,

  Holder's Inequality (for , you'll probably only need it for , plus this is the "basics" chapter anyway...) says that
  for positive reals ,

These are cool, but remember that these should only be used when the inequality is homogenized already.. These are all pretty easy to prove as well (with the help of the trivial inequality, but because it is trivial, we didn't include it here.)

  Example 1:

(Evan Chen) Let $a,b,c>0$ with $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$ . Prove that $(a+1)(b+1)(c+1) \geq 64$

Solution: We need to try to homogenize this somehow. Plugging in the expression for on the LHS for $1$ won't work. If we try to do something on the left side, we'll still have a degree $3>-1$ . Wait a second, why are they all $a+1$ 's? Let's try to get rid of the $a+1$ 's first. Well, if we add $3$ to both sides of the given condition, we get

$\frac{a+1}{a}+\frac{b+1}{b}+\frac{c+1}{c}=4$ , $\frac{(a+1)(b+1)(c+1)}{abc} \geq \frac{64}{27}$ , So now what we are trying to prove is $abc \geq 27$

By AM-GM. Obviously the trivial solution $(3,3,3)$ satisfies this, so we haven't made any silly mistakes. We still haven't homogenized, but now the path is clear. Multiplying both sides by $(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^3=1$ , we get

$\frac{(ab+bc+ac)^3}{(abc)^2} \geq 27$ , $(\frac{ab+bc+ac}{3})^3 \geq (abc)^2$ , $(\frac{ab+bc+ac}{3}) \geq (abc)^{2/3}$

Which is true from AM-GM.

 Example 2:

Let $a,b,c>0$ (Again Evan Chen) and $abc=1$ . Prove that $a^2+b^2+c^2 \geq a+b+c$ .

Solution: First we homogenize:

$a^2+b^2+c^2 \geq (a+b+c)(abc)^{1/3}$

Which is true from Muirhead's because $(2,0,0)$ majorizes $(\frac{4}{3}, \frac{1}{3}, \frac{1}{3})$ (and dividing by 2 at the end...)

Cauchy:

  Problem 1: (2009 usamo/4):

For $n \ge 2$ let $a_1$ , $a_2$ , ..., $a_n$ be positive real numbers such that $(a_1+a_2+ ... +a_n)\left( {1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n} \right) \le \left(n+ {1 \over 2} \right) ^2$ Prove that $\text{max}(a_1, a_2, ... ,a_n) \le 4 \text{min}(a_1, a_2, ... , a_n)$ .

Try to solve this on your own! Very cute problem.

  Example 3 (2004 usamo/5)

Let $a$ , $b$ , and $c$ be positive real numbers. Prove that $(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \ge (a+b+c)^3$ .

Solution:

1. The $(a+b+c)$ is cubed, so we try to use Holder's. The simplest way to do this is just to use $(a^3+1+1) . . .$ on the LHS.

2. Now all we have to prove is that $a^5-a^3-a^2+1 \geq 0$ , or $(a^3-1)(a^2-1) \geq 0$ . Now note that if $a<1$ , this is true, if $a=1$ , this is true, and if $a>1$ , this is true as well, and as we have exhausted all cases, we are done.

More advanced stuff, learn some calculus

You will need to know derivatives for this part. It's actually quite simple. Derivative=Tangent.

  The derivative of  is

Adding and other stuff works in the same way. You also probably need to know

  -Product Rule:

  -Quotient Rule:

  -Summing:

The second derivative of a function is just applying the derivative twice. A function is convex on an interval if it's second derivative is always positive in that interval. A function is convex if it's second derivative is negative in that interval.

  Jensen's inequality says that if  is a convex function in the interval , for all  in ,

The opposite holds if $f(x)$ is concave.

  Karamata's inequality says that if  is convex in the interval , the sequence  majorizes ,, and all  are in ,

  TLT (Tangent Line Trick) is basically where you either

  a. take the derivative, and plug in the equality cases or

  b. plugging in both equality cases to form a line.

  Problem 2:

Show that $\frac{1}{\sqrt5}+\frac{1}{\sqrt4}+\frac{1}{\sqrt2}>\frac{1}{\sqrt4}+\frac{1}{\sqrt4}+\frac{1}{\sqrt3}$

  Problem 3: Using Jensen's and Holder's, solve 2001 IMO/2:

Let $a,b,c$ be positive real numbers. Prove $\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1$ .

  Problem 4: (2017 usamo/6)

Find the minimum possible value of $\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4},$ given that $a,b,c,d,$ are nonnegative real numbers such that $a+b+c+d=4$ .

  Problem 5: (Japanese MO 1997/6)

Prove that

$\frac{\left(b+c-a\right)^{2}}{\left(b+c\right)^{2}+a^{2}}+\frac{\left(c+a-b\right)^{2}}{\left(c+a\right)^{2}+b^{2}}+\frac{\left(a+b-c\right)^{2}}{\left(a+b\right)^{2}+c^{2}}\geq\frac35$

for any positive real numbers $a$ , $b$ , $c$ .

Problems

  Problem 6 (2011 usamo/1):

Let $a$ , $b$ , $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \le 4$ . Prove that $\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.$

  Problem 7 (1974 usamo/2):

Prove that if $a$ , $b$ , and $c$ are positive real numbers, then

$a^ab^bc^c\ge (abc)^{(a+b+c)/3}$

  Problem 8 (1995 imo/2):

Let $a, b, c$ be positive real numbers such that $abc = 1$ . Prove that $\frac{1}{a^3(b+c)} + \frac{1}{b^3(c+a)} + \frac{1}{c^3(a+b)} \geq \frac{3}{2}.$

  Problem 9 (2000 imo/2):

Let $a, b, c$ be positive real numbers so that $abc = 1$ . Prove that $(a-1+\frac{1}{b})(b-1+\frac{1}{c})(c-1+\frac{1}{a}) \leq -1$ .

  Problem 10 (2003 usamo/5):

Let $a$ , $b$ , $c$ be positive real numbers. Prove that

$\dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \le 8.$

  Problem 11 (2001 usamo/3):

Let $a, b, c \geq 0$ and satisfy

$a^2 + b^2 + c^2 + abc = 4.$ Show that

$0 \le ab + bc + ca - abc \leq 2.$

  Problem 12 (2012 usajmo/3):

Let $a$ , $b$ , $c$ be positive real numbers. Prove that $\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2).$

  Problem 13 (2005 imo/3):

Let $x,y,z$ be three positive reals such that $xyz\geq 1$ . Prove that $\frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 .$

  Problem 14 (2003 imo shortlist/A6. cough.):

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$ , $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$ .

Let $M=\max\{z_2,\ldots,z_{2n}\}$ . Prove that $\frac{M+z_2+\dots+z_{2n}}{2n})^2 \geq \frac{x_1+\dots+x_n}{n} (\frac{y_1+\dots+y_n}{n}).$

Extra

  Ravi Substitution for triangles. .

  Problem 15: (1983 imo/6, using Ravi!)

Let $a$ , $b$ and $c$ be the lengths of the sides of a triangle. Prove that

$a^2 b(a-b) + b^2 c(b-c) + c^2 (c-a) \geq 0$ .

Determine when equality occurs.

Inequality Problems:

https://artofproblemsolving.com/wiki/index.php/Category:Olympiad_Inequality_Problems

https://artofproblemsolving.com/wiki/index.php/Category:Olympiad_Algebra_Problems

Function Equations

Oops...I kind of suck at these :P

~Lcz 6/9/2020 at 12:49 CST

Basics

  injective: a-1, b-3, c-2, nothing-4

  surjective: a-1, b-1, b-2, c-3

  bijective: a-1, b-3, c-2

  A function  where its domain and range are same is an involution if  for every  in its range/domain. This function is bijective.

-Function=anything. (HINT: IT DOESNT HAVE TO BE A POLYNOMIAL)

-Symmetry (switch x and y)

-Plug in $(0,0)$ , $(1,1)$ , $(0,y)$ , $(x,0)$ , $(1,y)$ , $(x,1)$ , $f(x),x)$ , etc.. first

-Check for linear/constant solutions first. They are usually the only ones.

- $fff$ trick: if $f(f(x))=x+1$ , $f(f(f(x)))=f(x+1)=f(x)+1$ by using the inner two $f$ 's, and the outer two.

-Pointwise trap!!!!!! If $x(f(x))=x^2$ , this does NOT mean that $f(x)=x, -x$ for all $x$ . Remember, function=anything and you could have $f(a)=a$ , $f(b)=-b$ for some $(a,b)$ . The most common way to get rid of this is just by doing $f(a)=a$ , $f(b)=-b$ , and then show that this leads to a contradiction.

-Be aware of your domain/range. ( $\mathbb{R}, \mathbb{Q}, \mathbb{Z}, \mathbb{C}$ (Probably not)). For example, "Cauchy Functional Equation" Changes a LOT when we go from $\mathbb{Q} -> \mathbb{Q}$ to $\mathbb{R} -> \mathbb{R}$ .

Ok these things are sort of cool

  Example 1 (2002 usamo/4):

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that

$f(x^2 - y^2) = xf(x) - yf(y)$

for all pairs of real numbers $x$ and $y$ .

Solution:

I claim that the sole solution is $f(x)=cx$ for some constant $c$ .

First note that setting $x$ and $y$ as zero respectively yields $f(x^2)=xf(x)$ and $f(-y^2)=-yf(y)$ . Letting $y=x$ in the second equation means that $f$ is odd.

Now, we can substitute into the original equation our findings:

$f(x^2-y^2)=f(x^2)-f(y^2) \implies f(x^2-y^2)+f(y^2)=f(x^2) \implies f(a)+f(b)=f(a+b)$ where $a=x^2-y^2, b=y^2$ .

Finally,

$f(x^2+2x+1)=(x+1)(f(x+1))$

$\implies f(x^2)+f(2x)+f(1)=(x+1)(f(1))+(x+1)(f(x))$

$\implies xf(x)+2f(x)+f(1)=(x+1)(f(1))+(x+1)(f(x))$

$\implies f(x)=xf(1)$ . Setting $f(1)=c$ , we get the desired. $\blacksquare$

  Problem 1 (2009 imo/5):

Determine all functions $f$ from the set of positive integers to the set of positive integers such that, for all positive integers $a$ and $b$ , there exists a non-degenerate triangle with sides of lengths

$a,f(b)$ and $f(b+f(a)-1)$ .

This question...is...cool...?

  Example 2 (ISL/2015):

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$

with the property that

$f(x-f(y))=f(f(x))-f(y)-1$

holds for all $x,y\in\mathbb{Z}$ .

BOGUS Solution:

The answer is $f(x)=-1$ or $f(x)=x+1$ . These clearly work.

Claim: there exists a $y$ such that $f(y)=-1$ .

Proof: set $(x,y)=(x, f(x))$ .

Then $f(x+1)=f(f(x))$ (*) where $f(y)=-1$ . Now $f(x)=c$ for some constant $c$ , or $f(x)=x+1$ .

The former yields $c=-1 \implies \boxed{f(x)=-1}$ , the latter works and yields $\boxed{f(x)=x+1}$ .

Why this is wrong: You can't just assume because $f(x)=f(y), x=y$ or $f(x)=c$ . Remember, a function is anything. Maybe this function is surjective.

Solution (after that step):

Now we plug in $(f(x-1), x)$ (because we know that the $f(f(f(x)-1)))$ will turn into $f(x+1)$ , and therefore will be proving that $f(x)$ is linear).

$f(-1)+1= f(f(f(x)-1)))-f(x)$

$= f(f(x))-f(x)$ (from (*) on where $x$ is actually $f(x)-1$ .

$= f(x+1)-f(x)$ (follows directly from (*)).

Therefore, this equation is linear.

Now, letting $f(x)=kx+c$ ,

From (*), we get

$k(x+1)+c = k(kx+c)+c$

$k(x+1)=k(kx+c).$

If $k\not=0$ , then

$kx+c=x+1$

So $\boxed{f(x)=x+1}$ . Otherwise, $f(x)$ is constant and plugging back into (*), we get the only solution as $\boxed{f(x)=-1}$

Ay, ok. Heres a semi-recent one.

  Problem 2 (2016 usamo/4):

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$ , $(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$

  Problem 3(One of my favorites, found it in Inter. Alg I think; 1981 imo/6)

The function $f(x,y)$ satisfies

(1) $f(0,y)=y+1,$

(2) $f(x+1,0)=f(x,1),$

(3) $f(x+1,y+1)=f(x,f(x+1,y)),$

for all non-negative integers $x,y$ . Determine $f(4,1981)$ .

  Problem 4(cute, 1983 imo/1):

Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy the conditions:

(i) $f(xf(y))=yf(x)$ for all $x,y$ ;

(ii) $f(x)\to0$ as $x\to \infty$ .

  Problem 5 (1986 imo/5, easy but lots of pitfalls oops):

Find all (if any) functions $f$ taking the non-negative reals onto the non-negative reals, such that

(a) $f(xf(y))f(y) = f(x+y)$ for all non-negative $x$ , $y$ ;

(b) $f(2) = 0$ ;

(c) $f(x) \neq 0$ for every $0 \leq x < 2$ .

Problems

  Problem 6 (1993 usamo/3):

Consider functions $f : [0, 1] \rightarrow \mathbb{R}$ which satisfy

(i) $f(x)\ge0$ for all $x$ in $[0, 1]$ , (ii) $f(1) = 1$ , (iii) $f(x) + f(y) \le f(x + y)$ whenever $x$ , $y$ , and $x + y$ are all in $[0, 1]$ . Find, with proof, the smallest constant $c$ such that

$f(x) \le cx$ for every function $f$ satisfying (i)-(iii) and every $x$ in $[0, 1]$ .

  Problem 7 (2015 usajmo/4):

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that $f(x)+f(t)=f(y)+f(z)$ for all rational numbers $x<y<z<t$ that form an arithmetic progression. ( $\mathbb{Q}$ is the set of all rational numbers.)

  Problem 8 (2015 imo/5):

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f$ : $\mathbb{R}\rightarrow\mathbb{R}$ satisfying the equation

$f(x+f(x+y))+f(xy) = x+f(x+y)+yf(x)$

for all real numbers $x$ and $y$ .

  Problem 9 (POINTWISE TRAP!!! 2014 usamo/2):

Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))$ for all $x, y \in \mathbb{Z}$ with $x \neq 0$ .

  Problem 10 (woah casework, 2012 usamo/4):

Find all functions $f : \mathbb{Z}^+ \to \mathbb{Z}^+$ (where $\mathbb{Z}^+$ is the set of positive integers) such that $f(n!) = f(n)!$ for all positive integers $n$ and such that $m - n$ divides $f(m) - f(n)$ for all distinct positive integers $m$ , $n$ .

  Problem 11 (2012 imo/4, misplaced i think):

Find all functions $f: \mathbb{Z} \to \mathbb{Z}$ such that, for all integers $a, b,$ and $c$ that satisfy $a + b+ c = 0$ , the following equality holds: $f(a)^2 + f(b)^2 + f(c)^2 = 2f(a)f(b) + 2f(b)f(c) + 2f(c)f(a).$

  Problem 12 (2011 imo/3):

Let $f: \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies $f(x + y) \le yf(x) + f(f(x))$ for all real numbers $x$ and $y$ . Prove that $f(x) = 0$ for all $x \le 0$ .

  Problem 13 (This was the first oly FE question I solved. That's why it's last :P 2010 imo/1):

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds

$f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor$

where $\left\lfloor a\right\rfloor$ is greatest integer not greater than $a.$

Extra

Nothin much

https://artofproblemsolving.com/wiki/index.php/Category:Functional_Equation_Problems

MONSTROUS Functional Equations

Ok we're not going to talk about these (The pro ETHAN LIU says I don't even have to learn these :D)

Hopefully they don't appear on usa(j)mo next year

otherwise i'm screwed lol

.

COMBINATORICS

.

-;

This will be a big chapter because I have no idea what the categories of combo are.

Yo I'm trash at this stuff. I'm trash at everything tbh. Let's give it a try, shall we?

Before oly

  B. Problem 1

How many ways can $5$ people stand in a line?

  B. Problem 2

How many ways can $5$ people stand in a circle?

  B. Problem 3

How many ways can you choose $3$ indistinguishable cupcakes from $7$ of them?

  B. Problem 4

How many ways can you choose $3$ distinguishable cupcakes from $7$ of them?

  B. Problem 5 (2015 I/5):

In a drawer Sandy has $5$ pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer. On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random. The probability that Wednesday is the first day Sandy selects matching socks is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, Find $m+n$

  B. Problem 6 (2015 10A/22):

Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?

$\textbf{(A)}\dfrac{47}{256}\qquad\textbf{(B)}\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256}$

  B. Problem 7 (2013 10A/24):

Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?

$\textbf{(A)}\ 540\qquad\textbf{(B)}\ 600\qquad\textbf{(C)}\ 720\qquad\textbf{(D)}\ 810\qquad\textbf{(E)}\ 900$

  B. Problem 8 (1988/10):

A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face?

  B. Problem 9 (1997/10):

Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true:

i. Either each of the three cards has a different shape or all three of the card have the same shape.

ii. Either each of the three cards has a different color or all three of the cards have the same color.

iii. Either each of the three cards has a different shade or all three of the cards have the same shade.

How many different complementary three-card sets are there?

  B. Problem 10 (2008 II/10):

In a $4 \times 4$ graph of points each a unit away from it's nearest neighbor, define a growing path to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let $m$ be the maximum possible number of points in a growing path, and let $r$ be the number of growing paths consisting of exactly $m$ points. Find $mr$ .

  B. Problem 11 (2020 I/9):

Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random with replacement from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m.$

  B. Problem 12 (2011 II/12):

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

  B. Problem 13 (2020 I/7):

A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as $1$ member or as many as $23$ members. Let $N$ be the number of such committees that can be formed. Find the sum of the prime numbers that divide $N.$

  B. Problem 14 (this question sucks, that's why it's here. AOIME/9):

While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.

  B. Problem 15 (2012 II/14):

In a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when $N$ is divided by $1000$ .

  B. Problem 16 (All combo is easy, sorry. 2011 II/14):

There are $N$ permutations $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \ldots, 30$ such that for $m \in \left\{{2, 3, 5}\right\}$ , $m$ divides $a_{n+m} - a_{n}$ for all integers $n$ with $1 \leq n < n+m \leq 30$ . Find the remainder when $N$ is divided by $1000$ .

  B. Problem 17 (2017 I/7, the only combo question I've ever liked on the AIME):

For nonnegative integers $a$ and $b$ with $a + b \leq 6$ , let $T(a, b) = \binom{6}{a} \binom{6}{b} \binom{6}{a + b}$ . Let $S$ denote the sum of all $T(a, b)$ , where $a$ and $b$ are nonnegative integers with $a + b \leq 6$ . Find the remainder when $S$ is divided by $1000$ .

  B. CHALLENGE PROBLEM 18 (2021 usajmo/6):

Find $(\binom{33478293652874789324082988729347895769465312438698074}{174073247580238490758958739601257694})(107918163081-69^6)$

Basics

We're pretty much only going to be working with Expected value and pigeonhole principle here, which corresponds to about 25%(a lot less) of olympiad combo problems.

First of all, we will always denote expected value as $\mathbb{E}$ . Denote several events $e_1, e_2, e_3, . . . e_n$ and their probabilities as $p_1, p_2, p_3, . . . p_n$ . Then

Next, there's this cool (I think so) property of the expected value.

Say the average number of cookies someone in this world has ate is $9.7823489138728394$ . Then there exists a person that has ate at least $10$ cookies, and also a person who ate at most $9$ cookies. It seems simple, but it's pretty powerful.

  Pigeonhole principle says that if you have  pigeonholes and  pigeons, there must exist a pigeonhole with at least ceiling() pigeons.

This follows directly from our previous claim.

This stuff is pretty easy

  Example 1 (AoPS):

In his spare time, Richard Rusczyk shuffles a standard deck of 52 playing cards. He then turns the cards up one by one from the top of the deck until the third ace appears. If the expected (average) number of cards Richard will turn up is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Solution:

The four aces divide the deck into $5$ parts, and the expected number of cards in each of them is $\frac{48}{5}$ . Therefore the answer is $\frac{48}{5}+3=\frac{63}{5} \implies \boxed{068}$

  Example 2 (1987 imo/1):

Let $p_{n}(k)$ be the number of permutations of the set $\{1,2,3,\ldots,n\}$ which have exactly $k$ fixed points. Prove that $\sum_{k=0}^{n}k p_{n}(k)=n!$ .

Solution:

Look at the summation. It's actually just the expected number of fixed points, multiplied by the number of permutations, or $n!$ .

Because every integer has a $\frac{(n-1)!}{n!}=\frac{1}{n}$ chance of being a fixed point, the expected number of fixed points is $1$ , and multiplying by $n!$ , we see that the requested sum is indeed $n!$ .

  Problem 1 (Easy Pigeonhole! 1972 imo/1):

Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same sum.

  Problem 2 (another easy Pigeonhole, 1976 usamo/1):

(a) Suppose that each square of a $4\times 7$ chessboard is colored either black or white. Prove that with any such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board such as the one outlined in the figure) whose four distinct unit corner squares are all of the same color.

(b) Exhibit a black-white coloring of a $4\times 6$ board in which the four corner squares of every rectangle, as described above, are not all of the same color.

  Problem 3 (Pigeonhole mastery lol, 2012 usamo/2):

A circle is divided into $432$ congruent arcs by $432$ points. The points are colored in four colors such that some $108$ points are colored Red, some $108$ points are colored Green, some $108$ points are colored Blue, and the remaining $108$ points are colored Yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.

  Problem 4 (Get out your combinatorics identities, 1981 imo/2):

Let $1 \le r \le n$ and consider all subsets of $r$ elements of the set $\{ 1, 2, \ldots , n \}$ . Each of these subsets has a smallest member. Let $F(n,r)$ denote the arithmetic mean of these smallest numbers; prove that

$F(n,r) = \frac{n+1}{r+1}.$

  Problem 5 (Well, it's trivial. Prove it! 2003 imo/1):

$S$ is the set $\{1, 2, 3, \dots ,1000000\}$ . Show that for any subset $A$ of $S$ with $101$ elements we can find $100$ distinct elements $x_i$ of $S$ , such that the sets $\{a + x_i \mid a \in A\}$ are all pairwise disjoint.

  Problem 6 (1998 imo/2):

In a competition, there are $a$ contestants and $b$ judges, where $b \geq 3$ is an odd integer. Each judge rates each contestant as either “pass” or “fail”. Suppose $k$ is a number such that, for any two judges, their ratings coincide for at most $k$ contestants. Prove that $\frac{k}{a} \geq \frac{b-1}{2b}$

oof

Ok. More advanced stuff, a.k.a. "Greedy" Algorithm:

REPEAT. UNTIL. YOU CANT. ANYMORE.

(Well, that's basically it. Just do a thing like "If this thing is $3m$ , make it into a $2m$ and a $m$ . Repeat until you can't anymore.)

  Problem 6 (Construction! 2014 imo/5):

For each positive integer $n$ , the Bank of Cape Town issues coins of denomination $\tfrac{1}{n}$ . Given a finite collection of such coins (of not necessarily different denominations) with total value at most $99+\tfrac{1}{2}$ , prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$ .

  Problem 7 (cough. 2016 imo/2):

Find all integers $n$ for which each cell of $n \times n$ table can be filled with one of the letters $I,M$ and $O$ in such a way that:

in each row and each column, one third of the entries are $I$ , one third are $M$ and one third are $O$ ; and

in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are $I$ , one third are $M$ and one third are $O$ .

  Problem 8 (I don't feel like putting this farther up. 2017 IMO/1):

For each integer $a_0 > 1$ , define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases}$ Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$ .

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Lcz Notes!

Contents

Introduction

ALGEBRA

Before oly...

Inequalities

Basics

More advanced stuff, learn some calculus

Problems

Extra

Function Equations

Basics

Ok these things are sort of cool

Problems

Extra

MONSTROUS Functional Equations

COMBINATORICS

Before oly

Basics

This stuff is pretty easy

oof