USAMO/md/en-USAMO-2012-notes.md

USAMO 2012 Solution Notes

Evan ChEn《陳誼廷》

15 April 2024

This is a compilation of solutions for the 2012 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 2012/1, proposed by Titu Andreescu ..... 3 1.2 USAMO 2012/2, proposed by Gregory Galperin ..... 5 1.3 USAMO 2012/3, proposed by Gabriel Carroll ..... 6 2 Solutions to Day 2 ..... 8 2.1 USAMO 2012/4, proposed by Gabriel Dospinescu ..... 8 2.2 USAMO 2012/5, proposed by Titu Andreescu, Cosmin Pohoata ..... 9 2.3 USAMO 2012/6, proposed by Gabriel Carroll ..... 11

§0 Problems

1. Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_{1}, a_{2}, \ldots$, $a_{n}$ with

$$\max (a_1,a_2,\dots ,a_n)\le n\cdot \min (a_1,a_2,\dots ,a_n),\backslash max\; \backslash left(a\_\{1\},\; a\_\{2\},\; \backslash ldots,\; a\_\{n\}\backslash right)\; \backslash leq\; n\; \backslash cdot\; \backslash min\; \backslash left(a\_\{1\},\; a\_\{2\},\; \backslash ldots,\; a\_\{n\}\backslash right),$$

there exist three that are the side lengths of an acute triangle. 2. A circle is divided into 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. 3. Determine which integers $n>1$ have the property that there exists an infinite sequence $a_{1}, a_{2}, a_{3}, \ldots$ of nonzero integers such that the equality

$$a_k+2a_{2k}+\cdots +n{a}_{nk}=0a\_\{k\}+2\; a\_\{2\; k\}+\backslash cdots+n\; a\_\{n\; k\}=0$$

holds for every positive integer $k$. 4. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ 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 $m, n$. 5. Let $P$ be a point in the plane of $\triangle A B C$, and $\gamma$ a line through $P$. Let $A^{\prime}, B^{\prime}, C^{\prime}$ be the points where the reflections of lines $P A, P B, P C$ with respect to $\gamma$ intersect lines $B C, C A, A B$ respectively. Prove that $A^{\prime}, B^{\prime}, C^{\prime}$ are collinear. 6. For integer $n \geq 2$, let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers satisfying

$$x_1+x_2+\cdots +x_n=0\text{ and }{x}_1^2+x_2^2+\cdots +x_n^2=1x\_\{1\}+x\_\{2\}+\backslash cdots+x\_\{n\}=0\; \backslash quad\; \backslash text\; \{\; and\; \}\; \backslash quad\; x\_\{1\}^\{2\}+x\_\{2\}^\{2\}+\backslash cdots+x\_\{n\}^\{2\}=1$$

For each subset $A \subseteq{1,2, \ldots, n}$, define $S_{A}=\sum_{i \in A} x_{i}$. (If $A$ is the empty set, then $S_{A}=0$.) Prove that for any positive number $\lambda$, the number of sets $A$ satisfying $S_{A} \geq \lambda$ is at most $2^{n-3} / \lambda^{2}$. For which choices of $x_{1}, x_{2}, \ldots, x_{n}, \lambda$ does equality hold?

§1 Solutions to Day 1

§1.1 USAMO 2012/1, proposed by Titu Andreescu

Available online at https://aops.com/community/p2669112.

Problem statement

Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_{1}, a_{2}, \ldots, a_{n}$ with

$$\max (a_1,a_2,\dots ,a_n)\le n\cdot \min (a_1,a_2,\dots ,a_n)\backslash max\; \backslash left(a\_\{1\},\; a\_\{2\},\; \backslash ldots,\; a\_\{n\}\backslash right)\; \backslash leq\; n\; \backslash cdot\; \backslash min\; \backslash left(a\_\{1\},\; a\_\{2\},\; \backslash ldots,\; a\_\{n\}\backslash right)$$

there exist three that are the side lengths of an acute triangle.

The answer is all $n \geq 13$. Define $\left(F_{n}\right)$ as the sequence of Fibonacci numbers, by $F_{1}=F_{2}=1$ and $F_{n+1}=$ $F_{n}+F_{n-1}$. We will find that Fibonacci numbers show up naturally when we work through the main proof, so we will isolate the following calculation now to make the subsequent solution easier to read.

Claim - For positive integers $m$, we have $F_{m} \leq m^{2}$ if and only if $m \leq 12$.

Proof. A table of the first 14 Fibonacci numbers is given below.

By examining the table, we see that $F_{m} \leq m^{2}$ is true for $m=1,2, \ldots 12$, and in fact $F_{12}=12^{2}=144$. However, $F_{m}>m^{2}$ for $m=13$ and $m=14$.

Now it remains to prove that $F_{m}>m^{2}$ for $m \geq 15$. The proof is by induction with base cases $m=13$ and $m=14$ being checked already. For the inductive step, if $m \geq 15$ then we have

as desired. We now proceed to the main problem. The hypothesis $\max \left(a_{1}, a_{2}, \ldots, a_{n}\right) \leq n$. $\min \left(a_{1}, a_{2}, \ldots, a_{n}\right)$ will be denoted by $(\dagger)$.

Proof that all $n \geq 13$ have the property. We first show now that every $n \geq 13$ has the desired property. Suppose for contradiction that no three numbers are the sides of an acute triangle. Assume without loss of generality (by sorting the numbers) that $a_{1} \leq a_{2} \leq \cdots \leq a_{n}$. Then since $a_{i-1}, a_{i}, a_{i+1}$ are not the sides of an acute triangle for each $i \geq 2$, we have that $a_{i+1}^{2} \geq a_{i}^{2}+a_{i-1}^{2}$; writing this out gives

and so on. The Fibonacci numbers appear naturally and by induction, we conclude that $a_{i}^{2} \geq F_{i} a_{1}^{2}$. In particular, $a_{n}^{2} \geq F_{n} a_{1}^{2}$.

However, we know $\max \left(a_{1}, \ldots, a_{n}\right)=a_{n}$ and $\min \left(a_{1}, \ldots, a_{n}\right)=a_{1}$, so ( $\dagger$ ) reads $a_{n} \leq n \cdot a_{1}$. Therefore we have $F_{n} \leq n^{2}$, and so $n \leq 12$, contradiction!

Proof that no $n \leq 12$ have the property. Assume that $n \leq 12$. The above calculation also suggests a way to pick the counterexample: we choose $a_{i}=\sqrt{F_{i}}$ for every $i$. Then $\min \left(a_{1}, \ldots, a_{n}\right)=a_{1}=1$ and $\max \left(a_{1}, \ldots, a_{n}\right)=\sqrt{F_{n}}$, so ( $\dagger$ ) is true as long as $n \leq 12$. And indeed no three numbers form the sides of an acute triangle: if $i<j<k$, then $a_{k}^{2}=F_{k}=F_{k-1}+F_{k-2} \geq F_{j}+F_{i}=a_{j}^{2}+a_{i}^{2}$.

§1.2 USAMO 2012/2, proposed by Gregory Galperin

Available online at https://aops.com/community/p2669115.

Problem statement

A circle is divided into 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.

First, consider the 431 possible non-identity rotations of the red points, and count overlaps with green points. If we select a rotation randomly, then each red point lies over a green point with probability $\frac{108}{431}$; hence the expected number of red-green incidences is

$$\frac{108}{431}\cdot 108>27\backslash frac\{108\}\{431\}\; \backslash cdot\; 108>27$$

and so by pigeonhole, we can find a red 28 -gon and a green 28 -gon which are rotations of each other.

Now, look at the 430 rotations of this 28 -gon (that do not give the all-red or all-green configuration) and compare it with the blue points. The same approach gives

$$\frac{108}{430}\cdot 28>7\backslash frac\{108\}\{430\}\; \backslash cdot\; 28>7$$

incidences, so we can find red, green, blue 8 -gons which are similar under rotation. Finally, the 429 nontrivial rotations of this 8 -gon expect

$$\frac{108}{429}\cdot 8>2\backslash frac\{108\}\{429\}\; \backslash cdot\; 8>2$$

incidences with yellow. So finally we have four monochromatic 3 -gons, one of each color, which are rotations of each other.

§1.3 USAMO 2012/3, proposed by Gabriel Carroll

Available online at https://aops.com/community/p2669119.

Problem statement

Determine which integers $n>1$ have the property that there exists an infinite sequence $a_{1}, a_{2}, a_{3}, \ldots$ of nonzero integers such that the equality

$$a_k+2a_{2k}+\cdots +n{a}_{nk}=0a\_\{k\}+2\; a\_\{2\; k\}+\backslash cdots+n\; a\_\{n\; k\}=0$$

holds for every positive integer $k$.

Answer: all $n>2$. For $n=2$, we have $a_{k}+2 a_{2 k}=0$, which is clearly not possible, since it implies $a_{2^{k}}=\frac{a_{1}}{2^{k-1}}$ for all $k \geq 1$.

For $n \geq 3$ we will construct a completely multiplicative sequence (meaning $a_{i j}=a_{i} a_{j}$ for all $i$ and $j$ ). Thus $\left(a_{i}\right)$ is determined by its value on primes, and satisfies the condition as long as $a_{1}+2 a_{2}+\cdots+n a_{n}=0$. The idea is to take two large primes and use Bezout's theorem, but the details require significant care.

We start by solving the case where $n \geq 9$. In that case, by Bertrand postulate there exists primes $p$ and $q$ such that

Clearly $p \neq q$, and $q \geq 7$, so $p>3$. Also, $p<q<n$ but $2 q>n$, and $4 p \geq 4\left(\frac{1}{2}(q+1)\right)>$ $n$. We now stipulate that $a_{r}=1$ for any prime $r \neq p, q$ (in particular including $r=2$ and $r=3$ ). There are now three cases, identical in substance.

• If $p, 2 p, 3 p \in[1, n]$ then we would like to choose nonzero $a_{p}$ and $a_{q}$ such that

$$6p\cdot a_p+q\cdot a_q=6p+q-\frac{1}{2}n(n+1)6\; p\; \backslash cdot\; a\_\{p\}+q\; \backslash cdot\; a\_\{q\}=6\; p+q-\backslash frac\{1\}\{2\}\; n(n+1)$$

which is possible by Bézout lemma, since $\operatorname{gcd}(6 p, q)=1$.

• Else if $p, 2 p \in[1, n]$ then we would like to choose nonzero $a_{p}$ and $a_{q}$ such that

$$3p\cdot a_p+q\cdot a_q=3p+q-\frac{1}{2}n(n+1)3\; p\; \backslash cdot\; a\_\{p\}+q\; \backslash cdot\; a\_\{q\}=3\; p+q-\backslash frac\{1\}\{2\}\; n(n+1)$$

which is possible by Bézout lemma, since $\operatorname{gcd}(3 p, q)=1$.

• Else if $p \in[1, n]$ then we would like to choose nonzero $a_{p}$ and $a_{q}$ such that

$$p\cdot a_p+q\cdot a_q=p+q-\frac{1}{2}n(n+1)p\; \backslash cdot\; a\_\{p\}+q\; \backslash cdot\; a\_\{q\}=p+q-\backslash frac\{1\}\{2\}\; n(n+1)$$

which is possible by Bézout lemma, since $\operatorname{gcd}(p, q)=1$. (This case is actually possible in a few edge cases, for example when $n=9, q=7, p=5$.)

It remains to resolve the cases where $3 \leq n \leq 8$. We enumerate these cases manually:

• For $n=3$, let $a_{n}=(-1)^{\nu_{3}(n)}$.
• For $n=4$, let $a_{n}=(-1)^{\nu_{2}(n)+\nu_{3}(n)}$.
• For $n=5$, let $a_{n}=(-2)^{\nu_{5}(n)}$.
• For $n=6$, let $a_{n}=5^{\nu_{2}(n)} \cdot 3^{\nu_{3}(n)} \cdot(-42)^{\nu_{5}(n)}$.
• For $n=7$, let $a_{n}=(-3)^{\nu_{7}(n)}$.
• For $n=8$, we can choose $(p, q)=(5,7)$ in the prior construction.

This completes the constructions for all $n>2$.

§2 Solutions to Day 2

§2.1 USAMO 2012/4, proposed by Gabriel Dospinescu

Available online at https://aops.com/community/p2669997.

Problem statement

Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ 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 $m, n$.

Answer: $f \equiv 1, f \equiv 2$, and $f$ the identity. As these obviously work, we prove these are the only ones. By putting $n=1$ and $n=2$ we give $f(1), f(2) \in{1,2}$. Also, we will use the condition

$$m!-n!\text{ divides }f(m)!-f(n)!m!-n!\backslash text\; \{\; divides\; \}\; f(m)!-f(n)!$$

We consider four cases on $f(1)$ and $f(2)$, and dispense with three of them.

• If $f(2)=1$ then for all $m \geq 3$ we have $m!$ - 2 divides $f(m)!-1$, so $f(m)=1$ for modulo 2 reasons. Then clearly $f(1)=1$.
• If $f(1)=f(2)=2$ we first obtain $3!-1 \mid f(3)!-2$, which implies $f(3)=2$. Then $m!-3 \mid f(m)!-2$ for $m \geq 4$ implies $f(m)=2$ for modulo 3 reasons.

Hence we are left with the case where $f(1)=1$ and $f(2)=2$. Continuing, we have

$$3!-1\mid f(3)!-1\text{ and }3!-2\mid f(3)!-2⟹f(3)=33!-1\; \backslash mid\; f(3)!-1\; \backslash quad\; \backslash text\; \{\; and\; \}\; \backslash quad\; 3!-2\; \backslash mid\; f(3)!-2\; \backslash Longrightarrow\; f(3)=3$$

Continuing by induction, suppose $f(1)=1, \ldots, f(k)=k$.

$$k!\cdot k=(k+1)!-k!\mid f(k+1)!-k!k!\backslash cdot\; k=(k+1)!-k!\backslash mid\; f(k+1)!-k!$$

and thus we deduce that $f(k+1) \geq k$, and hence

$$k\mid \frac{f(k+1)!}{k!}-1k\; \backslash left\backslash lvert\backslash ,\; \backslash frac\{f(k+1)!\}\{k!\}-1\backslash right.$$

Then plainly $f(k+1) \leq 2 k$ for $\bmod k$ reasons, but also $f(k+1) \equiv 1(\bmod k)$ so we conclude $f(k+1)=k+1$.

Remark. Shankar Padmanabhan gives the following way to finish after verifying that $f(3)=3$. Note that if

$$M=((((3!)!)!)!\dots )!M=((((3!)!)!)!\backslash ldots)!$$

for any number of iterated factorials then $f(M)=M$. Thus for any $n$, we have

$$M-n\mathrm{\mid }f(M)-f(n)=M-f(n)⟹M-n\mathrm{\mid }n-f(n)M-n|f(M)-f(n)=M-f(n)\; \backslash Longrightarrow\; M-n|\; n-f(n)$$

and so taking $M$ large enough implies $f(n)=n$.

§2.2 USAMO 2012/5, proposed by Titu Andreescu, Cosmin Pohoata

Available online at https://aops.com/community/p2669960.

Problem statement

Let $P$ be a point in the plane of $\triangle A B C$, and $\gamma$ a line through $P$. Let $A^{\prime}, B^{\prime}, C^{\prime}$ be the points where the reflections of lines $P A, P B, P C$ with respect to $\gamma$ intersect lines $B C, C A, A B$ respectively. Prove that $A^{\prime}, B^{\prime}, C^{\prime}$ are collinear.

We present three solutions. 【 First solution (complex numbers). Let $p=0$ and set $\gamma$ as the real line. Then $A^{\prime}$ is the intersection of $b c$ and $p \bar{a}$. So, we get

$$a^{\mathrm{\prime }}=\frac{\bar{a}(\bar{b}c-b\bar{c})}{(\bar{b}-\bar{c})\bar{a}-(b-c)a}\mathrm{.}a^\{\backslash prime\}=\backslash frac\{\backslash bar\{a\}(\backslash bar\{b\}\; c-b\; \backslash bar\{c\})\}\{(\backslash bar\{b\}-\backslash bar\{c\})\; \backslash bar\{a\}-(b-c)\; a\}\; .$$

Note that

$${\bar{a}}^{\mathrm{\prime }}=\frac{a(b\bar{c}-\bar{b}c)}{(b-c)a-(\bar{b}-\bar{c})\bar{a}}\mathrm{.}\backslash bar\{a\}^\{\backslash prime\}=\backslash frac\{a(b\; \backslash bar\{c\}-\backslash bar\{b\}\; c)\}\{(b-c)\; a-(\backslash bar\{b\}-\backslash bar\{c\})\; \backslash bar\{a\}\}\; .$$

Thus it suffices to prove

This is equivalent to

This determinant has the property that the rows sum to zero, and we're done. Remark. Alternatively, if you don't notice that you could just blindly expand:

\I Second solution (Desargues involution). We let $C^{\prime \prime}=\overline{A^{\prime} B^{\prime}} \cap \overline{A B}$. Consider complete quadrilateral $A B C A^{\prime} B^{\prime} C^{\prime \prime} C$. We see that there is an involutive pairing $\tau$ at $P$ swapping $\left(P A, P A^{\prime}\right),\left(P B, P B^{\prime}\right),\left(P C, P C^{\prime \prime}\right)$. From the first two, we see $\tau$ coincides with reflection about $\ell$, hence conclude $C^{\prime \prime}=C$.

ब Third solution (barycentric), by Catherine $\mathbf{X u}$. We will perform barycentric coordinates on the triangle $P C C^{\prime}$, with $P=(1,0,0), C^{\prime}=(0,1,0)$, and $C=(0,0,1)$. Set $a=C C^{\prime}, b=C P, c=C^{\prime} P$ as usual. Since $A, B, C^{\prime}$ are collinear, we will define $A=(p: k: q)$ and $B=(p: \ell: q)$.

Claim - Line $\gamma$ is the angle bisector of $\angle A P A^{\prime}, \angle B P B^{\prime}$, and $\angle C P C^{\prime}$.

Proof. Since $A^{\prime} P$ is the reflection of $A P$ across $\gamma$, etc. Thus $B^{\prime}$ is the intersection of the isogonal of $B$ with respect to $\angle P$ with the line $C A$; that is,

$$B^{\mathrm{\prime }}=(\frac{p}{k}\frac{b^2}{\mathrm{\ell }}:\frac{b^2}{\mathrm{\ell }}:\frac{c^2}{q})B^\{\backslash prime\}=\backslash left(\backslash frac\{p\}\{k\}\; \backslash frac\{b^\{2\}\}\{\backslash ell\}:\; \backslash frac\{b^\{2\}\}\{\backslash ell\}:\; \backslash frac\{c^\{2\}\}\{q\}\backslash right)$$

Analogously, $A^{\prime}$ is the intersection of the isogonal of $A$ with respect to $\angle P$ with the line $C B$; that is,

$$A^{\mathrm{\prime }}=(\frac{p}{\mathrm{\ell }}\frac{b^2}{k}:\frac{b^2}{k}:\frac{c^2}{q})\mathrm{.}A^\{\backslash prime\}=\backslash left(\backslash frac\{p\}\{\backslash ell\}\; \backslash frac\{b^\{2\}\}\{k\}:\; \backslash frac\{b^\{2\}\}\{k\}:\; \backslash frac\{c^\{2\}\}\{q\}\backslash right)\; .$$

The ratio of the first to third coordinate in these two points is both $b^{2} p q: c^{2} k \ell$, so it follows $A^{\prime}, B^{\prime}$, and $C^{\prime}$ are collinear.

Remark (Problem reference). The converse of this problem appears as problem 1052 attributed S. V. Markelov in the book Geometriya: 9-11 Klassy: Ot Uchebnoy Zadachi k Tvorcheskoy, 1996, by I. F. Sharygin.

§2.3 USAMO 2012/6, proposed by Gabriel Carroll

Available online at https://aops.com/community/p2670037.

Problem statement

For integer $n \geq 2$, let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers satisfying

$$x_1+x_2+\cdots +x_n=0\text{ and }{x}_1^2+x_2^2+\cdots +x_n^2=1x\_\{1\}+x\_\{2\}+\backslash cdots+x\_\{n\}=0\; \backslash quad\; \backslash text\; \{\; and\; \}\; \backslash quad\; x\_\{1\}^\{2\}+x\_\{2\}^\{2\}+\backslash cdots+x\_\{n\}^\{2\}=1$$

For each subset $A \subseteq{1,2, \ldots, n}$, define $S_{A}=\sum_{i \in A} x_{i}$. (If $A$ is the empty set, then $S_{A}=0$.) Prove that for any positive number $\lambda$, the number of sets $A$ satisfying $S_{A} \geq \lambda$ is at most $2^{n-3} / \lambda^{2}$. For which choices of $x_{1}, x_{2}, \ldots, x_{n}, \lambda$ does equality hold?

Let $\varepsilon_{i}$ be a coin flip of 0 or 1 . Then we have

In other words, $\sum_{A} S_{A}^{2}=2^{n-2}$. Since can always pair $A$ with its complement, we conclude

$$\sum _{S_A>0}{S}_A^2=2^{n-3}\backslash sum\_\{S\_\{A\}>0\}\; S\_\{A\}^\{2\}=2^\{n-3\}$$

Equality holds iff $S_{A} \in{ \pm \lambda, 0}$ for every $A$. This occurs when $x_{1}=1 / \sqrt{2}, x_{2}=-1 / \sqrt{2}$, $x_{3}=\cdots=0$ (or permutations), and $\lambda=1 / \sqrt{2}$.