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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.

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