2008 IMS

Part of IMS

Subcontests

(9)

1

Even subest of the table

n\times n

table is called even if it contains even elements of each row and each column. Find the minimum

k

such that each subset of this table with

k

elements contains an even subset

1

A map from interval to square

\gamma: [0,1]\rightarrow [0,1]\times [0,1]

be a mapping such that for each

s,t\in [0,1]

|\gamma(s) \minus{} \gamma(t)|\leq M|s \minus{} t|^\alpha in which

\alpha,M

are fixed numbers. Prove that if

\gamma

is surjective, then

\alpha\leq\frac12

1

Find all numbers

Find all natural numbers such that

n\sigma(n)\equiv 2\pmod {\phi( n)}

1

Probabilty

In a contest there are

n

yes-no problems. We know that no two contestants have the same set of answers. To each question we give a random uniform grade of set

\{1,2,3,\dots,2n\}

. Prove that the probability that exactly one person gets first is at least

\frac12

1

Number 2

Let a_0,a_1,\dots,a_{n \plus{} 1} be natural numbers such that a_0 \equal{} a_{n \plus{} 1} \equal{} 1,

a_i>1

1\leq i \leq n

1\leq j\leq n

, a_i|a_{i \minus{} 1} \plus{} a_{i \plus{} 1}. Prove that there exist one

2

in the sequence.

1

Regular elements in ring [tough one]

Prove that there does not exist a ring with exactly 5 regular elements. (

a

is called a regular element if ax \equal{} 0 or xa \equal{} 0 implies x \equal{} 0.) A ring is not necessarily commutative, does not necessarily contain unity element, or is not necessarily finite.

1

Maximum area

A,B

be different points on a parabola. Prove that we can find

P_1,P_2,\dots,P_{n}

A,B

on the parabola such that area of the convex polygon

AP_1P_2\dots P_nB

is maximum. In this case prove that the ratio of

S(AP_1P_2\dots P_nB)

to the sector between

A

B

doesn't depend on

A

B

, and only depends on

n

1

Entire function

f

be an entire function on

\mathbb C

\omega_1,\omega_2

are complex numbers such that

\frac {\omega_1}{\omega_2}\in{\mathbb C}\backslash{\mathbb Q}

. Prove that if for each

z\in \mathbb C

, f(z) \equal{} f(z \plus{} \omega_1) \equal{} f(z \plus{} \omega_2) then

f

1

Idempotent matrices

A_1,A_2,\dots,A_n

be idempotent matrices with real entries. Prove that: \mbox{N}(A_1)\plus{}\mbox{N}(A_2)\plus{}\dots\plus{}\mbox{N}(A_n)\geq \mbox{rank}(I\minus{}A_1A_2\dots A_n) \mbox{N}(A) is \mbox{dim}(\mbox{ker(A)})