MathDB

2012 Putnam

Part of Putnam

Subcontests

(6)
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Putnam 2012 A5

Let Fp\mathbb{F}_p denote the field of integers modulo a prime p,p, and let nn be a positive integer. Let vv be a fixed vector in Fpn,\mathbb{F}_p^n, let MM be an n×nn\times n matrix with entries in Fp,\mathbb{F}_p, and define G:Fpn→FpnG:\mathbb{F}_p^n\to \mathbb{F}_p^n by G(x)=v+Mx.G(x)=v+Mx. Let G(k)G^{(k)} denote the kk-fold composition of GG with itself, that is, G(1)(x)=G(x)G^{(1)}(x)=G(x) and G(k+1)(x)=G(G(k)(x)).G^{(k+1)}(x)=G(G^{(k)}(x)). Determine all pairs p,np,n for which there exist vv and MM such that the pnp^n vectors G(k)(0),G^{(k)}(0), k=1,2,…,pnk=1,2,\dots,p^n are distinct.

Putnam 2012 B5

Prove that, for any two bounded functions g1,g2:R→[1,∞),g_1,g_2 : \mathbb{R}\to[1,\infty), there exist functions h1,h2:R→Rh_1,h_2 : \mathbb{R}\to\mathbb{R} such that for every x∈R,x\in\mathbb{R},sup⁡s∈R(g1(s)xg2(s))=max⁡t∈R(xh1(t)+h2(t)).\sup_{s\in\mathbb{R}}\left(g_1(s)^xg_2(s)\right)=\max_{t\in\mathbb{R}}\left(xh_1(t)+h_2(t)\right).
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Putnam 2012 A3

Let f:[−1,1]→Rf:[-1,1]\to\mathbb{R} be a continuous function such that
(i) f(x)=2−x22f(x22−x2)f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right) for every xx in [−1,1],[-1,1],
(ii) f(0)=1, f(0)=1, and
(iii) lim⁡x→1−f(x)1−x\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}} exists and is finite.
Prove that ff is unique, and express f(x)f(x) in closed form.

Putnam 2012 B3

A round-robin tournament among 2n2n teams lasted for 2n−12n-1 days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the nn games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one winning team from each day without choosing any team more than once?
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Putnam 2012 A1

Let d1,d2,…,d12d_1,d_2,\dots,d_{12} be real numbers in the open interval (1,12).(1,12). Show that there exist distinct indices i,j,ki,j,k such that di,dj,dkd_i,d_j,d_k are the side lengths of an acute triangle.

Putnam 2012 B1

Let SS be a class of functions from [0,∞)[0,\infty) to [0,∞)[0,\infty) that satisfies:
(i) The functions f1(x)=ex−1f_1(x)=e^x-1 and f2(x)=ln⁡(x+1)f_2(x)=\ln(x+1) are in S;S;
(ii) If f(x)f(x) and g(x)g(x) are in S,S, the functions f(x)+g(x)f(x)+g(x) and f(g(x))f(g(x)) are in S;S;
(iii) If f(x)f(x) and g(x)g(x) are in SS and f(x)≥g(x)f(x)\ge g(x) for all x≥0,x\ge 0, then the function f(x)−g(x)f(x)-g(x) is in S.S.
Prove that if f(x)f(x) and g(x)g(x) are in S,S, then the function f(x)g(x)f(x)g(x) is also in S.S.