2002 IMC

Part of IMC

Subcontests

(12)

1

IMC 2002 Problem 6

n\times n

matrix with real entries let

||M||=\sup_{x\in \mathbb{R}^{n}\setminus\{0\}}\frac{||Mx||_{2}}{||x||_{2}}

||\cdot||_{2}

denotes the Euclidean norm on

\mathbb{R}^{n}

. Assume that an

n\times n

A

with real entries satisfies

||A^{k}-A^{k-1}||\leq\frac{1}{2002k}

for all positive integers

k

||A^{k}||\leq 2002

for all positive integers

k

1

IMC 2002 Problem 12

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

be a convex function whose gradient

\nabla f

exists at every point of

\mathbb{R}^{n}

and satisfies the condition

\exists L>0\; \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||\leq L||x_{1}-x_{2}||.

\forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||^{2}\leq L\langle\nabla f(x_{1})-\nabla f(x_{2}), x_{1}-x_{2}\rangle.

1

IMC 2002 Problem 9

n\geq 1

a_{n}=\sum_{k=0}^{\infty}\frac{k^{n}}{k!}, \;\; b_{n}=\sum_{k=0}^{\infty}(-1)^{k}\frac{k^{n}}{k!}.

a_{n}\cdot b_{n}

1

IMC 2002 Problem 7

Compute the determinant of the

n\times n

A=(a_{ij})_{ij}

a_{ij}=\begin{cases} (-1)^{|i-j|} & \text{if}\, i\ne j,\\ 2 & \text{if}\, i= j. \end{cases}

1

IMC 2002 Problem 10

OABC

be a tetrahedon with

\angle BOC=\alpha,\angle COA =\beta

\angle AOB =\gamma

. The angle between the faces

OAB

OAC

\sigma

and the angle between the faces

OAB

OBC

\rho

\gamma > \beta \cos\sigma + \alpha \cos\rho

1

IMC 2002 Problem 11

A

n \times n

n >1

A^{H}

be the conjugate transpose of

A

A\cdot A^{H} =I_{n}

A=S\cdot (S^{H})^{-1}

for some complex Matrix

S

1

Uncountably much solutions

Prove or disprove the following statements: (a) There exists a monotone function

f : [0, 1] \rightarrow [0, 1]

such that for each

y \in [0, 1]

f(x) = y

has uncountably many solutions

x

. (b) There exists a continuously differentiable function

f : [0, 1] \rightarrow [0, 1]

such that for each

y \in [0, 1]

f(x) = y

has uncountably many solutions

x

1

Closed set

f : [a, b] \rightarrow [a, b]

be a continuous function and let

p \in [a, b]

p_0 = p

p_{n+1} = f(p_n)

n = 0, 1, 2,...

. Suppose that the set

T_p = \{p_n : n = 0, 1, 2,...\}

is closed, i.e., if

x \not\in T_p

\exists \delta > 0

such that for all

x' \in T_p

|x'-x|\ge\delta

T_p

has finitely many elements.

1

sum equals zero

n

be a positive integer and let

a_k = \dfrac{1}{\binom{n}{k}}, b_k = 2^{k-n},\ (k=1..n)

\sum_{k=1}^n \dfrac{a_k-b_k}{k} = 0

1

Well-known

Does there exist a continuously differentiable function

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

such that for every

x \in \mathbb{R}

f(x) > 0

f'(x) = f(f(x))

1

Parabolas

A standard parabola is the graph of a quadratic polynomial

y = x^2 + ax + b

with leading co\"efficient 1. Three standard parabolas with vertices

V1, V2, V3

intersect pairwise at points

A1, A2, A3

A \mapsto s(A)

be the reflection of the plane with respect to the

x

-axis. Prove that standard parabolas with vertices

s (A1), s (A2), s (A3)

intersect pairwise at the points

s (V1), s (V2), s (V3)

1

IMC 2002 / B2

200 students participated in a math contest. They had 6 problems to solve. Each problem was correctly solved by at least 120 participants. Prove that there must be 2 participants such that every problem was solved by at least one of these two students.