2011 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 2011 B6

p

be an odd prime. Show that for at least

(p+1)/2

n

\{0,1,2,\dots,p-1\},

\sum_{k=0}^{p-1}k!n^k \text{is not divisible by }p.

1

Putnam 2011 B5

a_1,a_2,\dots

be real numbers. Suppose there is a constant

A

such that for all

n,

\int_{-\infty}^{\infty}\left(\sum_{i=1}^n\frac1{1+(x-a_i)^2}\right)^2\,dx\le An.

Prove there is a constant

B>0

such that for all

n,

\sum_{i,j=1}^n\left(1+(a_i-a_j)^2\right)\ge Bn^3.

1

Putnam 2011 B4

In a tournament, 2011 players meet 2011 times to play a multiplayer game. Every game is played by all 2011 players together and ends with each of the players either winning or losing. The standings are kept in two

2011\times 2011

T=(T_{hk})

W=(W_{hk}).

T=W=0.

After every game, for every

(h,k)

h=k),

h

k

tied (that is, both won or both lost), the entry

T_{hk}

is increased by

1,

while if player

h

k

lost, the entry

W_{hk}

is increased by

1

W_{kh}

is decreased by

1.

Prove that at the end of the tournament,

\det(T+iW)

is a non-negative integer divisible by

2^{2010}.

1

Putnam 2011 B3

f

g

be (real-valued) functions defined on an open interval containing

0,

g

nonzero and continuous at

0.

fg

f/g

are differentiable at

0,

f

be differentiable at

0?

1

Putnam 2011 B2

S

be the set of all ordered triples

(p,q,r)

of prime numbers for which at least one rational number

x

px^2+qx+r=0.

Which primes appear in seven or more elements of

S?

1

Putnam 2011 B1

h

k

be positive integers. Prove that for every

\varepsilon >0,

there are positive integers

m

n

\varepsilon < \left|h\sqrt{m}-k\sqrt{n}\right|<2\varepsilon.

1

Putnam 2011 A6

G

be an abelian group with

n

elements, and let

\{g_1=e,g_2,\dots,g_k\}\subsetneq G

be a (not necessarily minimal) set of distinct generators of

G.

A special die, which randomly selects one of the elements

g_1,g_2,\dots,g_k

with equal probability, is rolled

m

times and the selected elements are multiplied to produce an element

g\in G.

Prove that there exists a real number

b\in(0,1)

\lim_{m\to\infty}\frac1{b^{2m}}\sum_{x\in G}\left(\mathrm{Prob}(g=x)-\frac1n\right)^2

is positive and finite.

1

Putnam 2011 A5

F:\mathbb{R}^2\to\mathbb{R}

g:\mathbb{R}\to\mathbb{R}

be twice continuously differentiable functions with the following properties:
•

F(u,u)=0

u\in\mathbb{R};

x\in\mathbb{R},g(x)>0

x^2g(x)\le 1;

(u,v)\in\mathbb{R}^2,

\nabla F(u,v)

\mathbf{0}

or parallel to the vector

\langle g(u),-g(v)\rangle.

Prove that there exists a constant

C

such that for every

n\ge 2

x_1,\dots,x_{n+1}\in\mathbb{R},

\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.

1

Putnam 2011 A4

For which positive integers

n

n\times n

matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd?

1

Putnam 2011 A3

Find a real number

c

and a positive number

L

\lim_{r\to\infty}\frac{r^c\int_0^{\pi/2}x^r\sin x\,dx}{\int_0^{\pi/2}x^r\cos x\,dx}=L.

1

Putnam 2011 A2

a_1,a_2,\dots

b_1,b_2,\dots

be sequences of positive real numbers such that

a_1=b_1=1

b_n=b_{n-1}a_n-2

n=2,3,\dots.

Assume that the sequence

(b_j)

is bounded. Prove that

S=\sum_{n=1}^{\infty}\frac1{a_1\cdots a_n}

converges, and evaluate

S.

1

Putnam 2011 A1

Define a growing spiral in the plane to be a sequence of points with integer coordinates

P_0=(0,0),P_1,\dots,P_n

n\ge 2

and:
• The directed line segments

P_0P_1,P_1P_2,\dots,P_{n-1}P_n

are in successive coordinate directions east (for

P_0P_1

), north, west, south, east, etc.
• The lengths of these line segments are positive and strictly increasing.
\begin{picture}(200,180)
\put(20,100){\line(1,0){160}} \put(100,10){\line(0,1){170}}
\put(0,97){West} \put(180,97){East} \put(90,0){South} \put(90,180){North}
\put(100,100){\circle{1}}\put(100,100){\circle{2}}\put(100,100){\circle{3}} \put(115,100){\circle{1}}\put(115,100){\circle{2}}\put(115,100){\circle{3}} \put(115,130){\circle{1}}\put(115,130){\circle{2}}\put(115,130){\circle{3}} \put(40,130){\circle{1}}\put(40,130){\circle{2}}\put(40,130){\circle{3}} \put(40,20){\circle{1}}\put(40,20){\circle{2}}\put(40,20){\circle{3}} \put(170,20){\circle{1}}\put(170,20){\circle{2}}\put(170,20){\circle{3}}
\multiput(100,99.5)(0,.5){3}{\line(1,0){15}} \multiput(114.5,100)(.5,0){3}{\line(0,1){30}} \multiput(40,129.5)(0,.5){3}{\line(1,0){75}} \multiput(39.5,20)(.5,0){3}{\line(0,1){110}} \multiput(40,19.5)(0,.5){3}{\line(1,0){130}}
\put(102,90){P0} \put(117,90){P1} \put(117,132){P2} \put(28,132){P3} \put(30,10){P4} \put(172,10){P5}
\end{picture}
How many of the points

(x,y)

with integer coordinates

0\le x\le 2011,0\le y\le 2011

cannot be the last point,

P_n,

of any growing spiral?