1999 IMC

Part of IMC

Subcontests

(6)

2

ugly but hard

p>1

a real number. Find a real constant

c_p

for which the following statement holds: If

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

is a continuously differentiable function with

f(1)>f(-1)

|f'(y)|\le1 \forall y\in[-1,1]

\exists x\in[-1,1]: f'(x)>0

\forall y\in[-1,1]: |f(y)-f(x)|\le c_p\sqrt[p]{f'(x)}|y-x|

p=1

IMC 1999 Problem 12

A

\mathbb{Z}/n\mathbb{Z}

\frac{\ln(n)}{100}

elements. Define

f(r)=\sum_{s\in A} e^{\dfrac{2 \pi i r s}{n}}

. Show that for some

r \ne 0

|f(r)| \geq \frac{|A|}{2}

2

Very nice one

2n

n\times n

grid are marked. Show that for some

k > 1

2k

distinct marked points, say

a_1,...,a_{2k}

a_{2i-1}

a_{2i}

are in the same row,

a_{2i}

a_{2i+1}

are in the same column,

\forall i

, indices taken mod 2n.

IMC 1999 Problem 11

S

be the set of words made from the letters

a,b

c

. The equivalence relation

\sim

S

uu \sim u

u \sim v \Rightarrow uw \sim vw \; \text{and} \; wu \sim wv

u, v

w

. Prove that every word in

S

is equivalent to a word of length

\leq 8

2

functional equation

Find all strictly monotonic functions

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

f\left(\frac{x^2}{f(x)}\right)=x

x

there exists none

Prove that there's no function

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

f(x)^2\ge f(x+y)\left(f(x)+y\right)

x,y>0

2

Constant function

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

\left|\sum^n_{k=1}3^k\left(f(x+ky)-f(x-ky)\right)\right|\le1

n\in\mathbb{N},x,y\in\mathbb{R}

f

is a constant function.

very well-known

x_i\ge -1

\sum^n_{i=1}x_i^3=0

\sum^n_{i=1}x_i \le \frac{n}{3}

2

Easy one

\forall n \in \mathbb{N}_0, \exists A \in \mathbb{R}^{n\times n}: A^3=A+I

\det(A)>0, \forall A

fulfilling the above condition.

Trivial

R

be a ring where

\forall a\in R: a^2=0

abc+abc=0

a,b,c\in R

2

IMC 1999 / A2

Does there exist a bijective map

f:\mathbb{N} \rightarrow \mathbb{N}

\sum^{\infty}_{n=1}\frac{f(n)}{n^2}

Easy probability

We roll a regular 6-sided dice

n

times. What is the probabilty that the total number of eyes rolled is a multiple of 5?