2001 Putnam

Part of Putnam

Subcontests

(6)

2

Putnam 2001 A6

Can an arc of a parabola inside a circle of radius

1

have a length greater than

4

Putnam 2001 B6

(a_n)_{n \ge 1}

is an increasing sequence of positive real numbers such that

\lim \tfrac{a_n}{n}=0

. Must there exist infinitely many positive integers

n

a_{n-i}+a_{n+i}<2a_n

i=1,2,\cdots,n-1

2

Putnam 2001 A5

Prove that there are unique positive integers

a

n

a^{n+1}-(a+1)^n=2001

Putnam 2001 B5

a

b

be real numbers in the interval

\left(0,\tfrac{1}{2}\right)

g

be a continuous real-valued function such that

g(g(x))=ag(x)+bx

x

g(x)=cx

for some constant

c

2

Putnam 2001 A4

ABC

1

E

F

G

lie, respectively, on sides

BC

CA

AB

AE

BF

R

BF

CG

S

CG

AE

T

. Find the area of the triangle

RST

Putnam 2001 B4

S

denote the set of rational numbers different from

\{ -1, 0, 1 \}

f: S \rightarrow S

f(x)=x-1/x

. Prove or disprove that

\cap_{n=1}^{\infty} f^{(n)} (S) = \emptyset

f^{(n)}

f

composed with itself

n

2

Putnam 2001 A3

For each integer

m

, consider the polynomial

P_m(x)=x^4-(2m+4)x^2+(m-2)^2.

For what values of

m

P_m(x)

the product of two non-consant polynomials with integer coefficients?

Putnam 2001 B3

For any positive integer

n

\left< n \right>

denote the closest integer to

\sqrt {n}

\displaystyle\sum_{n=1}^{\infty} \dfrac {2^{\left< n \right>} + 2^{- \left< n \right>}}{2^n}

2

Putnam 2001 A2

k

\mathcal{C}_k

is biased so that, when tossed, it has probability

\tfrac{1}{(2k+1)}

of falling heads. If the

n

coins are tossed, what is the probability that the number of heads is odd? Express the answer as a rational function

n

Putnam 2001 B2

Find all pairs of real numbers

(x,y)

satisfying the system of equations: \begin{align*}\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2)\\ \frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4)\end{align*}

2

Putnam 2001 A1

S

and a binary operation

*

, i.e. for each

a,b\in S

a*b\in S

(a*b)*a=b

a,b\in S

a*(b*a)=b

a,b \in S

Putnam 2001 B1

n

be an even positive integer. Write the numbers

1, 2, \cdots, n^2

in the squares of an

n \times n

grid so that the

k

th row, from left to right, is

(k-1)n + 1, \ (k-1)n + 2, \ \cdots, \ (k-1)n + n.

Color the squares of the grid so that half of the squares in each row and in each column are red and the other half are black (a checkerboard coloring is one possibility). Prove that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares.