Home > Olympiad, Problem Solving > Putnam 2012 Day 1

## Putnam 2012 Day 1

1. Let ${d_1,d_2,\dots,d_{12}}$ be real numbers in the open interval ${(1,12).}$ Show that there exist distinct indices ${i,j,k}$ such that ${d_i,d_j,d_k}$ are the side lengths of an acute triangle.

2. Let ${*}$ be a commutative and associative binary operation on a set ${S.}$ Assume that for every ${x}$ and ${y}$ in ${S,}$ there exists ${z}$ in ${S}$ such that ${x*z=y.}$ (This ${z}$ may depend on ${x}$ and ${y.}$) Show that if ${a,b,c}$ are in ${S}$ and ${a*c=b*c,}$ then ${a=b.}$

3. Let ${f:[-1,1]\rightarrow\mathbb{R}}$ be a continuous function such that

(i) ${f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right)}$ for every ${x}$ in ${[-1,1],}$

(ii) ${ f(0)=1,}$ and

(iii) ${\lim_{x\rightarrow 1^-}\frac{f(x)}{\sqrt{1-x}}}$ exists and is finite.

Prove that ${f}$ is unique, and express ${f(x)}$ in closed form.

4. Let ${q}$ and ${r}$ be integers with ${q>0,}$ and let ${A}$ and ${B}$ be intervals on the real line. Let ${T}$ be the set of all ${b+mq}$ where ${b}$ and ${m}$ are integers with ${b}$ in ${B,}$ and let ${S}$ be the set of all integers ${a}$ in ${A}$ such that ${ra}$ is in ${T.}$ Show that if the product of the lengths of ${A}$ and ${B}$ is less than ${q,}$ then ${S}$ is the intersection of ${A}$ with some arithmetic progression.

5. Let ${\mathbb{F}_p}$ denote the field of integers modulo a prime ${p,}$ and let ${n}$ be a positive integer. Let ${v}$ be a fixed vector in ${\mathbb{F}_p^n,}$ let ${M}$ be an ${n\times n}$ matrix with entries in ${\mathbb{F}_p,}$ and define ${G:\mathbb{F}_p^n\rightarrow \mathbb{F}_p^n}$ by ${G(x)=v+Mx.}$ Let ${G^{(k)}}$ denote the ${k}$-fold composition of ${G}$ with itself, that is, ${G^{(1)}(x)=G(x)}$ and ${G^{(k+1)}(x)=G(G^{(k)}(x)).}$ Determine all pairs ${p,n}$ for which there exist ${v}$ and ${M}$ such that the ${p^n}$ vectors ${G^{(k)}(0),}$ ${k=1,2,\dots,p^n}$ are distinct.

6. Let ${f(x,y)}$ be a continuous, real-valued function on ${\mathbb{R}^2.}$ Suppose that, for every rectangular region ${R}$ of area ${1,}$ the double integral of ${f(x,y)}$ over ${R}$ equals ${0.}$ Must ${f(x,y)}$ be identically ${0}$?

Hints: 1. Note that $F_{12}=144$ (I’m talking here about the Fibonacci sequence).

2. You can prove that the law has an identity element, and then it is almost finished.