1998 IMC

Part of IMC

Subcontests

(6)

2

integral inequality

f: [0,1]\rightarrow\mathbb{R}

be a continuous function satisfying

xf(y)+yf(x)\le 1

x,y\in[0,1]

. (a) Show that

\int^1_0 f(x)dx \le \frac{\pi}4

. (b) Find such a funtion for which equality occurs.

IMC 1998 Problem 12

f: (0,1) \rightarrow [0, \infty)

is zero except at a countable set of points

a_{1}, a_2, a_3, ...

b_n = f(a_n)

\sum b_{n}

converges, then

f

is differentiable at at least one point. Show that for any sequence

b_{n}

of non-negative reals with

\sum b_{n} =\infty

, we can find a sequence

a_{n}

such that the function

f

defined as above is nowhere differentiable.

2

polynomial with real coefficients

P

be a polynomial of degree

n

with only real zeros and real coefficients. Prove that for every real

x

(n-1)(P'(x))^2\ge nP(x)P''(x)

. When does equality occur?

IMC 1998 Problem 11

S

is a family of balls in

\mathbb{R}^{n}

n > 1

) such that the intersection of any two contains at most one point. Show that the set of points belonging to at least two members of

S

2

prove that there exists \xi

f: \mathbb{R}\rightarrow\mathbb{R}

is twice differentiable and satisfies

f(0)=2,f'(0)=-2,f(1)=1

. Prove that there is a

\xi \in ]0,1[

for which we have

f(\xi)\cdot f'(\xi)+f''(\xi)=0

IMC 1998 Problem 10

S_{n}=\{1,2,...,n\}

. How many functions

f:S_{n} \rightarrow S_{n}

f(k) \leq f(k+1)

f(k)=f(f(k+1))

k <n?

2

f(x)=2x(1-x)

f(x)=2x(1-x), x\in\mathbb{R}

f_n=f\circ f\circ ... \circ f

n

times. (a) Find

\lim_{n\rightarrow\infty} \int^1_0 f_n(x)dx

. (b) Now compute

\int^1_0 f_n(x)dx

IMC 1998 Problem 9

0< c< 1

f(x) = \begin{cases} \frac{x}{c} & x \in [0,c] \\ \frac{1-x}{1-c} & x \in [c, 1] \end{cases}

f^{n}(x)=f(f(...f(x)))

. Show that for each positive integer

n

f^{n}

has a non-zero finite nunber of fixed points which aren't fixed points of

f^k

k< n

2

composition of permutations

Consider the following statement: for any permutation

\pi_1\not=\mathbb{I}

\{1,2,...,n\}

there is a permutation

\pi_2

such that any permutation on these numbers can be obtained by a finite compostion of

\pi_1

\pi_2

. (a) Prove the statement for

n=3

n=5

. (b) Disprove the statement for

n=4

IMC 1998 Problem 8

S

ist the set of all cubic polynomials

f

|f(\pm 1)| \leq 1

|f(\pm \frac{1}{2})| \leq 1

\sup_{f \in S} \max_{-1 \leq x \leq 1} |f''(x)|

and all members of

f

which give equality.

2

real vector space with subspaces

V

be a 10-dimensional real vector space and

U_1,U_2

two linear subspaces such that

U_1 \subseteq U_2, \dim U_1 =3, \dim U_2=6

\varepsilon

be the set of all linear maps

T: V\rightarrow V

T(U_1)\subseteq U_1, T(U_2)\subseteq U_2

. Calculate the dimension of

\varepsilon

. (again, all as real vector spaces)

IMC 1998 Problem 7

V

is a real vector space and

f, f_{i}: V \rightarrow \mathbb{R}

i = 1, 2, ... , k.

f

is zero at all points for which all of

f_{i }

are zero. Show that

f

is a linear combination of the

f_{i}