MathDB

1

Partition of 3D space

3

-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every

a \in \mathbb R^+

, there are points

M

and

N

inside that subset such that distance between

M

and

N

is exactly

a.

24

1

Prove that d_n is aperiodic

Let

d_n

be the last nonzero digit of the decimal representation of

n!

. Prove that

d_n

is aperiodic; that is, there do not exist

T

and

n_0

such that for all

n \geq n_0, d_{n+T} = d_n.

22

1

Show that there exists a permutation

Let

n

be a positive integer having at least two different prime factors. Show that there exists a permutation

a_1, a_2, \dots , a_n

of the integers

1, 2, \dots , n

such that

\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.

21

1

Greatest integer less than the sum

Find the greatest integer less than or equal to

\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}.

19

1

Polynomial of degree 990

Let

(F_n)_{n\geq 1}

be the Fibonacci sequence

F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),

and

P(x)

the polynomial of degree

990

satisfying

P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.

Prove that

P(1983) = F_{1983} - 1.

17

1

Distance of points

Let

P_1, P_2, \dots , P_n

be distinct points of the plane,

n \geq 2

. Prove that

\max_{1\leq i<j\leq n} P_iP_j > \frac{\sqrt 3}{2}(\sqrt n -1) \min_{1\leq i<j\leq n} P_iP_j

16

1

Set of polynomials

Let

F(n)

be the set of polynomials

P(x) = a_0+a_1x+\cdots+a_nx^n

, with

a_0, a_1, . . . , a_n \in \mathbb R

and

0 \leq a_0 = a_n \leq a_1 = a_{n-1 } \leq \cdots \leq a_{[n/2] }= a_{[(n+1)/2]}.

Prove that if

f \in F(m)

and

g \in F(n)

, then

fg \in F(m + n).

15

1

Is there a set of positive integers ?

Decide whether there exists a set

M

of positive integers satisfying the following conditions:
(i) For any natural number

m>1

there exist

a, b \in M

such that

a+b = m.

(ii) If

a, b, c, d \in M

,

a, b, c, d > 10

and

a + b = c + d

, then

a = c

or

a = d.

13

1

Set of 1983^3 points of the space R^3

Let

E

be the set of

1983^3

points of the space

\mathbb R^3

all three of whose coordinates are integers between

0

and

1982

(including

0

and

1982

). A coloring of

E

is a map from

E

to the set {red, blue}. How many colorings of

E

are there satisfying the following property: The number of red vertices among the

8

vertices of any right-angled parallelepiped is a multiple of

4

?

11

1

Show that 0 < f(x)-x < c

Let

f : [0, 1] \to \mathbb R

be continuous and satisfy: \begin{cases}bf(2x) = f(x), &\mbox{ if } 0 \leq x \leq 1/2,\\ f(x) = b + (1 - b)f(2x - 1), &\mbox{ if } 1/2 \leq x \leq 1,\end{cases} where

b = \frac{1+c}{2+c}

,

c > 0

. Show that

0 < f(x)-x < c

for every

x, 0 < x < 1.

10

1

There exists an interval I

Let

p

and

q

be integers. Show that there exists an interval

I

of length

1/q

and a polynomial

P

with integral coefficients such that

\left|P(x)-\frac pq \right| < \frac{1}{q^2}

for all

x \in I.

8

1

3n students participate in a test

In a test,

3n

students participate, who are located in three rows of

n

students in each. The students leave the test room one by one. If

N_1(t), N_2(t), N_3(t)

denote the numbers of students in the first, second, and third row respectively at time

t

, find the probability that for each t during the test,

|N_i(t) - N_j(t)| < 2, i \neq j, i, j = 1, 2, \dots .

7

1

Each term of the sequence is an integer

Let

a

be a positive integer and let

\{a_n\}

be defined by

a_0 = 0

and

a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ).

Show that for each positive integer

n

,

a_n

is a positive integer.

6

1

Existence of r

Suppose that

{x_1, x_2, \dots , x_n}

are positive integers for which

x_1 + x_2 + \cdots+ x_n = 2(n + 1)

. Show that there exists an integer

r

with

0 \leq r \leq n - 1

for which the following

n - 1

inequalities hold:

x_{r+1} + \cdots + x_{r+i} \leq 2i+ 1, \qquad \qquad \forall i, 1 \leq i \leq n - r;

x_{r+1} + \cdots + x_n + x_1 + \cdots+ x_i \leq 2(n - r + i) + 1, \qquad \qquad \forall i, 1 \leq i \leq r - 1.

Prove that if all the inequalities are strict, then

r

is unique and that otherwise there are exactly two such

r.

5

1

Strictly decreasing sequences

Consider the set of all strictly decreasing sequences of

n

natural numbers having the property that in each sequence no term divides any other term of the sequence. Let

A = (a_j)

and

B = (b_j)

be any two such sequences. We say that

A

precedes

B

if for some

k

,

a_k < b_k

and

a_i = b_i

for

i < k

. Find the terms of the first sequence of the set under this ordering.

4

1

Prove that APRQ is a parallegoram

On the sides of the triangle

ABC

, three similar isosceles triangles

ABP \ (AP = PB)

,

AQC \ (AQ = QC)

, and

BRC \ (BR = RC)

are constructed. The first two are constructed externally to the triangle

ABC

, but the third is placed in the same half-plane determined by the line

BC

as the triangle

ABC

. Prove that

APRQ

is a parallelogram.

2

1

Superabundant Numbers

Let

n

be a positive integer. Let

\sigma(n)

be the sum of the natural divisors

d

of

n

(including

1

and

n

). We say that an integer

m \geq 1

is superabundant (P.Erdos,

1944

) if

\forall k \in \{1, 2, \dots , m - 1 \}

,

\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.

Prove that there exists an infinity of superabundant numbers.

1

Airlines

The localities

P_1, P_2, \dots, P_{1983}

are served by ten international airlines

A_1,A_2, \dots , A_{10}

. It is noticed that there is direct service (without stops) between any two of these localities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings.

1983 IMO Shortlist

Subcontests

Partition of 3D space

Prove that d_n is aperiodic

Show that there exists a permutation

Greatest integer less than the sum

Polynomial of degree 990

Distance of points

Set of polynomials

Is there a set of positive integers ?

Set of 1983^3 points of the space R^3

Show that 0 < f(x)-x < c

There exists an interval I

3n students participate in a test

Each term of the sequence is an integer

Existence of r

Strictly decreasing sequences

Prove that APRQ is a parallegoram

Superabundant Numbers

Airlines

1983 IMO Shortlist

Subcontests

Partition of 3D space

Prove that d_n is aperiodic

Show that there exists a permutation

Greatest integer less than the sum

Polynomial of degree 990

Distance of points

Set of polynomials

Is there a set of positive integers ?

Set of 1983^3 points of the space R^3

Show that 0 &lt; f(x)-x &lt; c

There exists an interval I

3n students participate in a test

Each term of the sequence is an integer

Existence of r

Strictly decreasing sequences

Prove that APRQ is a parallegoram

Superabundant Numbers

Airlines

Show that 0 < f(x)-x < c