USAMO/md/en-USAMO-1999-notes.md

USAMO 1999 Solution Notes

Evan ChEn《陳誼廷》

15 April 2024

This is a compilation of solutions for the 1999 USAMO. The ideas of the solution are a mix of my own work, the solutions provided by the competition organizers, and solutions found by the community. However, all the writing is maintained by me.

These notes will tend to be a bit more advanced and terse than the "official" solutions from the organizers. In particular, if a theorem or technique is not known to beginners but is still considered "standard", then I often prefer to use this theory anyways, rather than try to work around or conceal it. For example, in geometry problems I typically use directed angles without further comment, rather than awkwardly work around configuration issues. Similarly, sentences like "let $\mathbb{R}$ denote the set of real numbers" are typically omitted entirely.

Corrections and comments are welcome!

Contents

0 Problems ..... 2 1 Solutions to Day 1 ..... 3 1.1 USAMO 1999/1 ..... 3 1.2 USAMO 1999/2 ..... 4 1.3 USAMO 1999/3 2 Solutions to Day 2 ..... 8 2.1 USAMO 1999/4 ..... 8 2.2 USAMO 1999/5 ..... 9 2.3 USAMO 1999/6 ..... 10

§0 Problems

1. Some checkers placed on an $n \times n$ checkerboard satisfy the following conditions: (a) every square that does not contain a checker shares a side with one that does; (b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side. Prove that at least $\left(n^{2}-2\right) / 3$ checkers have been placed on the board.
2. Let $A B C D$ be a convex cyclic quadrilateral. Prove that

$$\mathrm{\mid }AB-CD\mathrm{\mid }+\mathrm{\mid }AD-BC\mathrm{\mid }\ge 2\mathrm{\mid }AC-BD\mathrm{\mid }|A\; B-C\; D|+|A\; D-B\; C|\; \backslash geq\; 2|A\; C-B\; D|$$

3. Let $p>2$ be a prime and let $a, b, c, d$ be integers not divisible by $p$, such that

$$\left\{\frac{ra}{p}\right\}+\left\{\frac{rb}{p}\right\}+\left\{\frac{rc}{p}\right\}+\left\{\frac{rd}{p}\right\}=2\backslash left\backslash \{\backslash frac\{r\; a\}\{p\}\backslash right\backslash \}+\backslash left\backslash \{\backslash frac\{r\; b\}\{p\}\backslash right\backslash \}+\backslash left\backslash \{\backslash frac\{r\; c\}\{p\}\backslash right\backslash \}+\backslash left\backslash \{\backslash frac\{r\; d\}\{p\}\backslash right\backslash \}=2$$

for any integer $r$ not divisible by $p$. (Here, ${t}=t-\lfloor t\rfloor$ is the fractional part.) Prove that at least two of the numbers $a+b, a+c, a+d, b+c, b+d, c+d$ are divisible by $p$. 4. Let $a_{1}, a_{2}, \ldots, a_{n}$ be a sequence of $n>3$ real numbers such that

$$a_1+\cdots +a_n\ge n\text{ and }{a}_1^2+\cdots +a_n^2\ge n^{2}a\_\{1\}+\backslash cdots+a\_\{n\}\; \backslash geq\; n\; \backslash quad\; \backslash text\; \{\; and\; \}\; \backslash quad\; a\_\{1\}^\{2\}+\backslash cdots+a\_\{n\}^\{2\}\; \backslash geq\; n^\{2\}$$

Prove that $\max \left(a_{1}, \ldots, a_{n}\right) \geq 2$. 5. The Y2K Game is played on a $1 \times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy. 6. Let $A B C D$ be an isosceles trapezoid with $A B | C D$. The inscribed circle $\omega$ of triangle $B C D$ meets $C D$ at $E$. Let $F$ be a point on the (internal) angle bisector of $\angle D A C$ such that $E F \perp C D$. Let the circumscribed circle of triangle $A C F$ meet line $C D$ at $C$ and $G$. Prove that the triangle $A F G$ is isosceles.

§1 Solutions to Day 1

§1.1 USAMO 1999/1

Available online at https://aops.com/community/p340035.

Problem statement

Some checkers placed on an $n \times n$ checkerboard satisfy the following conditions: (a) every square that does not contain a checker shares a side with one that does; (b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.

Prove that at least $\left(n^{2}-2\right) / 3$ checkers have been placed on the board.

Take a spanning tree on the set $V$ of checkers where the $|V|-1$ edges of the tree are given by orthogonal adjacency. By condition (a) we have

$$\sum _{v\in V}(4-\mathrm{deg}v)\ge n^2-\mathrm{\mid }V\mathrm{\mid }\backslash sum\_\{v\; \backslash in\; V\}(4-\backslash operatorname\{deg\}\; v)\; \backslash geq\; n^\{2\}-|V|$$

and since $\sum_{v \in V} \operatorname{deg} v=2(|V|-1)$ we get

$$4\mathrm{\mid }V\mathrm{\mid }-(2\mathrm{\mid }V\mathrm{\mid }-2)\ge n^2-\mathrm{\mid }V\mathrm{\mid }4|V|-(2|V|-2)\; \backslash geq\; n^\{2\}-|V|$$

which implies $|V| \geq \frac{n^{2}-2}{3}$.

§1.2 USAMO 1999/2

Available online at https://aops.com/community/p340036.

Problem statement

Let $A B C D$ be a convex cyclic quadrilateral. Prove that

$$\mathrm{\mid }AB-CD\mathrm{\mid }+\mathrm{\mid }AD-BC\mathrm{\mid }\ge 2\mathrm{\mid }AC-BD\mathrm{\mid }\mathrm{.}|A\; B-C\; D|+|A\; D-B\; C|\; \backslash geq\; 2|A\; C-B\; D|\; .$$

Let the diagonals meet at $P$, and let $A P=p q, D P=p r, B P=q s, C P=r s$. Then set $A B=q x, C D=r x, A D=p y, B C=s y$.

In this way we compute

$$\mathrm{\mid }AC-BD\mathrm{\mid }=\mathrm{\mid }(p-s)(q-r)\mathrm{\mid }|A\; C-B\; D|=|(p-s)(q-r)|$$

and

$$\mathrm{\mid }AB-CD\mathrm{\mid }=\mathrm{\mid }q-r\mathrm{\mid }x|A\; B-C\; D|=|q-r|\; x$$

By triangle inequality on $\triangle A X B$, we have $x \geq|p-s|$. So $|A B-C D| \geq|A C-B D|$.

§1.3 USAMO 1999/3

Available online at https://aops.com/community/p340038.

Problem statement

Let $p>2$ be a prime and let $a, b, c, d$ be integers not divisible by $p$, such that

$$\left\{\frac{ra}{p}\right\}+\left\{\frac{rb}{p}\right\}+\left\{\frac{rc}{p}\right\}+\left\{\frac{rd}{p}\right\}=2\backslash left\backslash \{\backslash frac\{r\; a\}\{p\}\backslash right\backslash \}+\backslash left\backslash \{\backslash frac\{r\; b\}\{p\}\backslash right\backslash \}+\backslash left\backslash \{\backslash frac\{r\; c\}\{p\}\backslash right\backslash \}+\backslash left\backslash \{\backslash frac\{r\; d\}\{p\}\backslash right\backslash \}=2$$

for any integer $r$ not divisible by $p$. (Here, ${t}=t-\lfloor t\rfloor$ is the fractional part.) Prove that at least two of the numbers $a+b, a+c, a+d, b+c, b+d, c+d$ are divisible by $p$.

First of all, we apparently have $r(a+b+c+d) \equiv 0(\bmod p)$ for every prime $p$, so it automatically follows that $a+b+c+d \equiv 0(\bmod p)$. By scaling appropriately, and also replacing each number with its remainder modulo $p$, we are going to assume that

We are going to prove that $d=p-1$, which will solve the problem. Claim - For each integer $r=1,2, \ldots, p-1$ we have

$$2(r-1)=\lfloor \frac{rb}{p}\rfloor +\lfloor \frac{rc}{p}\rfloor +\lfloor \frac{rd}{p}\rfloor \mathrm{.}2(r-1)=\backslash left\backslash lfloor\backslash frac\{r\; b\}\{p\}\backslash right\backslash rfloor+\backslash left\backslash lfloor\backslash frac\{r\; c\}\{p\}\backslash right\backslash rfloor+\backslash left\backslash lfloor\backslash frac\{r\; d\}\{p\}\backslash right\backslash rfloor\; .$$

Proof. By plugging in $r=1$ to the given we have $a+b+c+d=2 p$. Now, we have

$$2=\sum _{\mathrm{c}\mathrm{y}\mathrm{c}}(\frac{ra}{p}-\lfloor \frac{ra}{p}\rfloor )2=\backslash sum\_\{\backslash mathrm\{cyc\}\}\backslash left(\backslash frac\{r\; a\}\{p\}-\backslash left\backslash lfloor\backslash frac\{r\; a\}\{p\}\backslash right\backslash rfloor\backslash right)$$

and since $a+b+c+d=2 p$ the conclusion follows. We vaguely outline the approach now, before giving a formalization. Imagine the interval $[0,1]$. One by one, for each $r=1,2,3, \ldots, p-1$, we mark the fractions with denominator $r$ on this number line; the resulting pictures may be better known as Farey fractions. At each step, we can place the three numbers $b / p, c / p, d / p$ into one of the resulting sub-intervals. Our goal is to show that $d / p$ is always in the rightmost interval, while $b / p$ and $c / p$ are always to the right of symmetrically mirrored points. An example of a possible diagram is shown below (not to scale).

In symbols, it will be enough to prove the following.

Claim - For each $r=1,2, \ldots, p-2$ we have $\frac{r-1}{r}<\frac{d}{p}<1$. Equivalently, for each $r=1,2, \ldots, p-2$ we have $\left\lfloor\frac{r b}{p}\right\rfloor+\left\lfloor\frac{r c}{p}\right\rfloor=r-1$. Proof. Assume this is not true and take the minimal counterexample $r>1$. Then evidently

$$r-1>\lfloor \frac{rd}{p}\rfloor \ge \lfloor \frac{(r-1)d}{p}\rfloor =r-2r-1>\backslash left\backslash lfloor\backslash frac\{r\; d\}\{p\}\backslash right\backslash rfloor\; \backslash geq\backslash left\backslash lfloor\backslash frac\{(r-1)\; d\}\{p\}\backslash right\backslash rfloor=r-2$$

Now, we have that

$$2(r-1)=\lfloor \frac{rb}{p}\rfloor +\lfloor \frac{rc}{p}\rfloor +\underset{=r-2}{\underbrace{\lfloor \frac{rd}{p}\rfloor }}\mathrm{.}2(r-1)=\backslash left\backslash lfloor\backslash frac\{r\; b\}\{p\}\backslash right\backslash rfloor+\backslash left\backslash lfloor\backslash frac\{r\; c\}\{p\}\backslash right\backslash rfloor+\backslash underbrace\{\backslash left\backslash lfloor\backslash frac\{r\; d\}\{p\}\backslash right\backslash rfloor\}\_\{=r-2\}\; .$$

Thus $\left\lfloor\frac{r b}{p}\right\rfloor>\left\lfloor\frac{(r-1) b}{p}\right\rfloor$, and $\left\lfloor\frac{r c}{p}\right\rfloor>\left\lfloor\frac{(r-1) b}{p}\right\rfloor$. An example of this situation is illustrated below with $r=7$ (not to scale).

Right now, $\frac{b}{p}$ and $\frac{c}{p}$ are just to the right of $\frac{u}{r}$ and $\frac{v}{r}$ for some $u$ and $v$ with $u+v=r$. The issue is that the there is some fraction just to the right of $\frac{b}{p}$ and $\frac{c}{p}$ from an earlier value of $r$, and by hypothesis its denominator is going to be strictly greater than 1 .

It is at this point we are going to use the properties of Farey sequences. When we consider the fractions with denominator $r+1$, they are going to lie outside of the interval they we have constrained $\frac{b}{p}$ and $\frac{c}{p}$ to lie in.

Indeed, our minimality assumption on $r$ guarantees that there is no fraction with denominator less than $r$ between $\frac{u}{r}$ and $\frac{b}{p}$. So if $\frac{u}{r}<\frac{b}{p}<\frac{s}{t}$ (where $\frac{u}{r}$ and $\frac{s}{t}$ are the closest fractions with denominator at most $r$ to $\frac{b}{p}$ ) then Farey theory says the next fraction inside the interval $\left[\frac{u}{r}, \frac{s}{t}\right]$ is $\frac{u+s}{r+t}$, and since $t>1$, we have $r+t>r+1$. In other words, we get an inequality of the form

The same holds for $\frac{c}{p}$ as

Finally,

So now we have that

$$\lfloor \frac{(r+1)b}{p}\rfloor +\lfloor \frac{(r+1)c}{p}\rfloor +\lfloor \frac{(r+1)d}{p}\rfloor \le u+v+(r-1)=2r-1\backslash left\backslash lfloor\backslash frac\{(r+1)\; b\}\{p\}\backslash right\backslash rfloor+\backslash left\backslash lfloor\backslash frac\{(r+1)\; c\}\{p\}\backslash right\backslash rfloor+\backslash left\backslash lfloor\backslash frac\{(r+1)\; d\}\{p\}\backslash right\backslash rfloor\; \backslash leq\; u+v+(r-1)=2\; r-1$$

which is a contradiction.

Now, since

which for $p>2$ gives $d=p-1$.

§2 Solutions to Day 2

§2.1 USAMO 1999/4

Available online at https://aops.com/community/p63591.

Problem statement

Let $a_{1}, a_{2}, \ldots, a_{n}$ be a sequence of $n>3$ real numbers such that

$$a_1+\cdots +a_n\ge n\text{ and }{a}_1^2+\cdots +a_n^2\ge n^{2}a\_\{1\}+\backslash cdots+a\_\{n\}\; \backslash geq\; n\; \backslash quad\; \backslash text\; \{\; and\; \}\; \backslash quad\; a\_\{1\}^\{2\}+\backslash cdots+a\_\{n\}^\{2\}\; \backslash geq\; n^\{2\}$$

Prove that $\max \left(a_{1}, \ldots, a_{n}\right) \geq 2$.

Proceed by contradiction, assuming $a_{i}<2$ for all $i$. If all $a_{i} \geq 0$, then $n^{2} \leq \sum_{i} a_{i}^{2}<n \cdot 2^{2}$, contradiction. Otherwise, assume at least one $a_{i}$ is negative. Note that if $-x$ and $-y$ are both present in the sequence $(x, y>0)$, then we can replace them with $-(x+y)$ and 0 . So we may assume that there is exactly one negative term, say $a_{n}=-M$.

Now, smooth all the nonnegative $a_{i}$ to be 2 , making all inequalities strict. Now, we have that

This gives $n-2<M<n-2$, contradiction. Equality in the original occurs when $n-1$ of the $a_{i}$ are equal to 2 and the last one is equal to $-(n-2)$.

§2.2 USAMO 1999/5

Available online at https://aops.com/community/p340040.

Problem statement

The Y2K Game is played on a $1 \times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.

The main insight is that a construct of the form

$$S\mathrm{\square }\mathrm{\square }SS\; \backslash square\; \backslash square\; S$$

(here the $\square$ is blank) will kill any player which plays inside it. We call this a trap accordingly.

Claim - The second player can force a trap to exist; in this case the game will never end in a draw.

Proof. Actually the second player can construct a trap on her second turn by playing an $S$ far enough away from the edges of the board and the first player's initial move.

Claim - The second player always has a move which prevents her from losing. Proof. Since there are an odd number of empty squares at the start of the second player's turn, there must be a square which is bordered by either two filled or two empty squares. The second player can then play $O$ in this square, which is always safe.

Together these two claims finish the problem. Remark. Actually, one can show that the "only" way to lose is to be forced to play inside a trap. Indeed, suppose playing in a certain cell $c$ loses. If we wrote $O$, that means $c$ is bordered by exactly one $S$, with a blank cell on the neighbor. But we could also write $S$; checking cases we find $c$ is part of a trap.

Thus a player can lose only if all blank cells are in traps; ergo, the number of blank cells is even. This never happens for the second player. Thus this gives an alternative solution, and moreover a reason to believe that all correct solutions must involve traps.

A similar proof shows that for large $n$, with a $1 \times n$ board, the first player has a winning strategy for odd $n$, and the second player has a winning strategy for even $n$.

§2.3 USAMO 1999/6

Available online at https://aops.com/community/p340041.

Problem statement

Let $A B C D$ be an isosceles trapezoid with $A B | C D$. The inscribed circle $\omega$ of triangle $B C D$ meets $C D$ at $E$. Let $F$ be a point on the (internal) angle bisector of $\angle D A C$ such that $E F \perp C D$. Let the circumscribed circle of triangle $A C F$ meet line $C D$ at $C$ and $G$. Prove that the triangle $A F G$ is isosceles.

Note $E$ is contact point of $A$-excircle of $\triangle A C D$, so $F$ is $A$-excenter. Hence $C F$ is external angle bisector of $\angle A C G$ which implies $F A=F G$ (since $F$ is the arc midpoint on the circumcircle of $A F G)$.