MathDB

2

IMC 2006, problem 6, day 1

a_{0}, a_{1},\ldots, a_{n}

of real numbers such that

a_{n}\neq 0

, for which the following statement is true: If

f: \mathbb{R}\to\mathbb{R}

is an

n

times differentiable function and

x_{0}<x_{1}<\ldots <x_{n}

are real numbers such that

f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0

then there is

h\in (x_{0}, x_{n})

for which

a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.

IMC 2006 / B6 a.k.a. The Monster

The scores of this problem were: one time 17/20 (by the runner-up) one time 4/20 (by Andrei Negut) one time 1/20 (by the winner) the rest had zero... just to give an idea of the difficulty. Let

A_{i},B_{i},S_{i}

(

i=1,2,3

) be invertible real

2\times 2

matrices such that [*]not all

A_{i}

have a common real eigenvector, [*]

A_{i}=S_{i}^{-1}B_{i}S_{i}

for

i=1,2,3

, [*]

A_{1}A_{2}A_{3}=B_{1}B_{2}B_{3}=I

. Prove that there is an invertible

2\times 2

matrix

S

such that

A_{i}=S^{-1}B_{i}S

for all

i=1,2,3

.

4

2

IMC 2006, problem 4, day 1

Let f be a rational function (i.e. the quotient of two real polynomials) and suppose that

f(n)

is an integer for infinitely many integers n. Prove that f is a polynomial.

IMC 2006 / B4

Let

v_{0}

be the zero ector and let

v_{1},...,v_{n+1}\in\mathbb{R}^{n}

such that the Euclidian norm

|v_{i}-v_{j}|

is rational for all

0\le i,j\le n+1

. Prove that

v_{1},...,v_{n+1}

are linearly dependent over the rationals.

3

2

IMC 2006, problem 3, day 1

Let

A

be an

n

x

n

matrix with integer entries and

b_{1},b_{2},...,b_{k}

be integers satisfying

detA=b_{1}\cdot b_{2}\cdot ...\cdot b_{k}

. Prove that there exist

n

x

n

-matrices

B_{1},B_{2},...,B_{k}

with integers entries such that

A=B_{1}\cdot B_{2}\cdot ...\cdot B_{k}

and

detB_{i}=b_{i}

for all

i=1,...,k

.

IMC 2006, problem 3, day 2

Compare

\tan(\sin x)

with

\sin(\tan x)

, for

x\in \left]0,\frac{\pi}{2}\right[

.

2

IMC 2006, problem 2, day 1

Find the number of positive integers x satisfying the following two conditions: 1.

x<10^{2006}

2.

x^{2}-x

is divisible by

10^{2006}

IMC 2006, problem 2, day 2

Find all functions

f: \mathbb{R}\to{R}

such that for any

a<b

,

f([a,b])

is an interval of length

b-a

5

2

IMC 2006, problem 5, day 1

Let

a, b, c, d

three strictly positive real numbers such that

a^{2}+b^{2}+c^{2}=d^{2}+e^{2},

a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.

Compare

a^{3}+b^{3}+c^{3}

with

d^{3}+e^{3},

IMC 2006, problem 5, day 2

Show that there are an infinity of integer numbers

m,n

, with

gcd(m,n)=1

such that the equation

(x+m)^{3}=nx

has 3 different integer sollutions.

1

2

IMC 2006, problem 1, day 1

Let

f: \mathbb{R}\to \mathbb{R}

be a real function. Prove or disprove each of the following statements. (a) If f is continuous and range(f)=

\mathbb{R}

then f is monotonic (b) If f is monotonic and range(f)=

\mathbb{R}

then f is continuous (c) If f is monotonic and f is continuous then range(f)=

\mathbb{R}

IMC 2006, problem 1, day 2

Let

V

be a convex polygon. (a) Show that if

V

has

3k

vertices, then

V

can be triangulated such that each vertex is in an odd number of triangles. (b) Show that if the number of vertices is not divisible with 3, then

V

can be triangulated such that exactly 2 vertices have an even number of triangles.

2006 IMC

Subcontests

IMC 2006, problem 6, day 1

IMC 2006 / B6 a.k.a. The Monster

IMC 2006, problem 4, day 1

IMC 2006 / B4

IMC 2006, problem 3, day 1

IMC 2006, problem 3, day 2

IMC 2006, problem 2, day 1

IMC 2006, problem 2, day 2

IMC 2006, problem 5, day 1

IMC 2006, problem 5, day 2

IMC 2006, problem 1, day 1

IMC 2006, problem 1, day 2