MathDB

1

It is impossible to draw a closed broken line

Prove that on the coordinate plane it is impossible to draw a closed broken line such that (i) the coordinates of each vertex are rational; (ii) the length each of its edges is 1; (iii) the line has an odd number of vertices.

27

1

Decimal expansion has n - 1 digits 1, one digit 7 is prime

Find all natural numbers

n

for which every natural number whose decimal representation has n \minus{} 1 digits

1

and one digit

7

is prime.

22

1

Graph K_m (m = 1990 > 2^10) has mono. odd circle length

Ten localities are served by two international airlines such that there exists a direct service (without stops) between any two of these localities and all airline schedules offer round-trip service between the cities they serve. Prove that at least one of the airlines can offer two disjoint round trips each containing an odd number of landings.

21

1

[(1 + 2^p + 2^{n-p})N - 1] / 2^n

Let

n

be a composite natural number and

p

a proper divisor of

n.

Find the binary representation of the smallest natural number

N

such that \frac{(1 \plus{} 2^p \plus{} 2^{n\minus{}p})N \minus{} 1}{2^n} is an integer.

20

1

In the decimal system with at most four different digits

Prove that every integer

k

greater than 1 has a multiple that is less than

k^4

and can be written in the decimal system with at most four different digits.

19

1

A point inside a regular tetrahedron T of unit volume

Let

P

be a point inside a regular tetrahedron

T

of unit volume. The four planes passing through

P

and parallel to the faces of

T

partition

T

into 14 pieces. Let

f(P)

be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for

f(P)

as

P

varies over

T.

18

1

Find smallest natural number k such that f^k(0) = 0

Let

a, b \in \mathbb{N}

with

1 \leq a \leq b,

and M \equal{} \left[\frac {a \plus{} b}{2} \right]. Define a function

f: \mathbb{Z} \mapsto \mathbb{Z}

by f(n) \equal{} \begin{cases} n \plus{} a, & \text{if } n \leq M, \\ n \minus{} b, & \text{if } n >M. \end{cases} Let f^1(n) \equal{} f(n), f_{i \plus{} 1}(n) \equal{} f(f^i(n)), i \equal{} 1, 2, \ldots Find the smallest natural number

k

such that f^k(0) \equal{} 0.

17

1

Unit cubes are made into beads by drilling a hole

Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let

A

be the beginning vertex and

B

be the end vertex. Let there be

p \times q \times r

cubes on the string

(p, q, r \geq 1).

(a) Determine for which values of

p, q,

and

r

it is possible to build a block with dimensions

p, q,

and

r.

Give reasons for your answers. (b) The same question as (a) with the extra condition that A \equal{} B.

15

1

1990, 1990 + 1, 1990 + 2, ..., 1990 + k

Determine for which positive integers

k

the set X \equal{} \{1990, 1990 \plus{} 1, 1990 \plus{} 2, \ldots, 1990 \plus{} k\} can be partitioned into two disjoint subsets

A

and

B

such that the sum of the elements of

A

is equal to the sum of the elements of

B.

14

1

Two triangles in partition each of which has two good sides

In the coordinate plane a rectangle with vertices

(0, 0),

(m, 0),

(0, n),

(m, n)

is given where both

m

and

n

are odd integers. The rectangle is partitioned into triangles in such a way that (i) each triangle in the partition has at least one side (to be called a “good” side) that lies on a line of the form x \equal{} j or y \equal{} k, where

j

and

k

are integers, and the altitude on this side has length 1; (ii) each “bad” side (i.e., a side of any triangle in the partition that is not a “good” one) is a common side of two triangles in the partition. Prove that there exist at least two triangles in the partition each of which has two good sides.

13

1

An eccentric mathematician has a ladder with n rungs

An eccentric mathematician has a ladder with

n

rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers

a

rungs of the ladder, and when he descends, each step he takes covers

b

rungs of the ladder, where

a

and

b

are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of

n,

expressed in terms of

a

and

b.

9

1

Measure of angle CAB

The incenter of the triangle

ABC

is

K.

The midpoint of

AB

is

C_1

and that of

AC

is

B_1.

The lines

C_1K

and

AC

meet at

B_2,

the lines

B_1K

and

AB

at

C_2.

If the areas of the triangles

AB_2C_2

and

ABC

are equal, what is the measure of angle

\angle CAB?

8

1

Square of the sum of its digits

For a given positive integer

k

denote the square of the sum of its digits by

f_1(k)

and let f_{n\plus{}1}(k) \equal{} f_1(f_n(k)). Determine the value of

f_{1991}(2^{1990}).

7

1

Numbers f(1989), f(1990), f(1991) are divisible by 13

Let f(0) \equal{} f(1) \equal{} 0 and f(n\plus{}2) \equal{} 4^{n\plus{}2} \cdot f(n\plus{}1) \minus{} 16^{n\plus{}1} \cdot f(n) \plus{} n \cdot 2^{n^2}, n \equal{} 0, 1, 2, \ldots Show that the numbers

f(1989), f(1990), f(1991)

are divisible by

13.

2

1

Is a cycle of 1990 committees possible with 11 countries?

Given

n

countries with three representatives each,

m

committees

A(1),A(2), \ldots, A(m)

are called a cycle if (i) each committee has

n

members, one from each country; (ii) no two committees have the same membership; (iii) for i \equal{} 1, 2, \ldots,m, committee

A(i)

and committee A(i \plus{} 1) have no member in common, where A(m \plus{} 1) denotes

A(1);

(iv) if 1 < |i \minus{} j| < m \minus{} 1, then committees

A(i)

and

A(j)

have at least one member in common. Is it possible to have a cycle of 1990 committees with 11 countries?

4

1

Natural numbers decomposed in r disjoint subsets

Assume that the set of all positive integers is decomposed into

r

(disjoint) subsets A_1 \cup A_2 \cup \ldots \cup A_r \equal{} \mathbb{N}. Prove that one of them, say

A_i,

has the following property: There exists a positive

m

such that for any

k

one can find numbers

a_1, a_2, \ldots, a_k

in

A_i

with 0 < a_{j \plus{} 1} \minus{} a_j \leq m, (1 \leq j \leq k \minus{} 1).

1

A integer can be written as the sum of 1990 integers

The integer

9

can be written as a sum of two consecutive integers: 9 \equal{} 4\plus{}5. Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: 9 \equal{} 4\plus{}5 \equal{} 2\plus{}3\plus{}4. Is there an integer that can be written as a sum of

1990

consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly

1990

ways?

24

1

Inequality with wx + xy + yz + zw = 1

Let

w, x, y, z

are non-negative reals such that wx \plus{} xy \plus{} yz \plus{} zw \equal{} 1. Show that \frac {w^3}{x \plus{} y \plus{} z} \plus{} \frac {x^3}{w \plus{} y \plus{} z} \plus{} \frac {y^3}{w \plus{} x \plus{} z} \plus{} \frac {z^3}{w \plus{} x \plus{} y}\geq \frac {1}{3}.

12

1

Show that $ca = cb$.

Let

ABC

be a triangle, and let the angle bisectors of its angles

CAB

and

ABC

meet the sides

BC

and

CA

at the points

D

and

F

, respectively. The lines

AD

and

BF

meet the line through the point

C

parallel to

AB

at the points

E

and

G

respectively, and we have FG \equal{} DE. Prove that CA \equal{} CB. Original formulation: Let

ABC

be a triangle and

L

the line through

C

parallel to the side

AB.

Let the internal bisector of the angle at

A

meet the side

BC

at

D

and the line

L

at

E

and let the internal bisector of the angle at

B

meet the side

AC

at

F

and the line

L

at

G.

If GF \equal{} DE, prove that AC \equal{} BC.

10

1

Find the volume of the smaller part

A plane cuts a right circular cone of volume

V

into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the volume of the smaller part. Original formulation: A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone.

5

1

GIH>90

Given a triangle

ABC

. Let

G

,

I

,

H

be the centroid, the incenter and the orthocenter of triangle

ABC

, respectively. Prove that

\angle GIH > 90^{\circ}

.

26

1

cubic polynomial and sequence

Let

p(x)

be a cubic polynomial with rational coefficients.

q_1

,

q_2

,

q_3

, ... is a sequence of rationals such that q_n \equal{} p(q_{n \plus{} 1}) for all positive

n

. Show that for some

k

, we have q_{n \plus{} k} \equal{} q_n for all positive

n

.

1990 IMO Shortlist

Subcontests

It is impossible to draw a closed broken line

Decimal expansion has n - 1 digits 1, one digit 7 is prime

Graph K_m (m = 1990 > 2^10) has mono. odd circle length

[(1 + 2^p + 2^{n-p})N - 1] / 2^n

In the decimal system with at most four different digits

A point inside a regular tetrahedron T of unit volume

Find smallest natural number k such that f^k(0) = 0

Unit cubes are made into beads by drilling a hole

1990, 1990 + 1, 1990 + 2, ..., 1990 + k

Two triangles in partition each of which has two good sides

An eccentric mathematician has a ladder with n rungs

Measure of angle CAB

Square of the sum of its digits

Numbers f(1989), f(1990), f(1991) are divisible by 13

Is a cycle of 1990 committees possible with 11 countries?

Natural numbers decomposed in r disjoint subsets

A integer can be written as the sum of 1990 integers

Inequality with wx + xy + yz + zw = 1

Show that $ca = cb$.

Find the volume of the smaller part

GIH>90

cubic polynomial and sequence

1990 IMO Shortlist

Subcontests

It is impossible to draw a closed broken line

Decimal expansion has n - 1 digits 1, one digit 7 is prime

Graph K_m (m = 1990 &gt; 2^10) has mono. odd circle length

[(1 + 2^p + 2^{n-p})N - 1] / 2^n

In the decimal system with at most four different digits

A point inside a regular tetrahedron T of unit volume

Find smallest natural number k such that f^k(0) = 0

Unit cubes are made into beads by drilling a hole

1990, 1990 + 1, 1990 + 2, ..., 1990 + k

Two triangles in partition each of which has two good sides

An eccentric mathematician has a ladder with n rungs

Measure of angle CAB

Square of the sum of its digits

Numbers f(1989), f(1990), f(1991) are divisible by 13

Is a cycle of 1990 committees possible with 11 countries?

Natural numbers decomposed in r disjoint subsets

A integer can be written as the sum of 1990 integers

Inequality with wx + xy + yz + zw = 1

Show that $ca = cb$.

Find the volume of the smaller part

GIH&gt;90

cubic polynomial and sequence

Graph K_m (m = 1990 > 2^10) has mono. odd circle length

GIH>90