MathDB

1

Finite set of unit circles

A finite set of unit circles is given in a plane such that the area of their union

U

is

S

. Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that

\frac{2S}{9}.

18

1

Sum of the areas is equal

Several equal spherical planets are given in outer space. On the surface of each planet there is a set of points that is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets is equal to the area of the surface of one planet.

16

1

Behavior of u_n

A sequence of real numbers

u_1, u_2, u_3, \dots

is determined by

u_1

and the following recurrence relation for

n \geq 1

:

4u_{n+1} = \sqrt[3]{ 64u_n + 15.}

Describe, with proof, the behavior of

u_n

as

n \to \infty.

14

1

Convex pentagon is a regular one

Prove that a convex pentagon (a five-sided polygon)

ABCDE

with equal sides and for which the interior angles satisfy the condition

\angle A \geq \angle B \geq \angle C \geq \angle D \geq \angle E

is a regular pentagon.

13

1

Determine P(n+1)

Let

P

be a polynomial of degree

n

satisfying

P(k) = \binom{n+1}{k}^{-1} \qquad \text{ for } k = 0, 1, . . ., n.

Determine

P(n + 1).

11

1

Geometric Inequality

On a semicircle with unit radius four consecutive chords

AB,BC, CD,DE

with lengths

a, b, c, d

, respectively, are given. Prove that

a^2 + b^2 + c^2 + d^2 + abc + bcd < 4.

10

1

Smallest n with the property

Determine the smallest natural number

n

having the following property: For every integer

p, p \geq n

, it is possible to subdivide (partition) a given square into

p

squares (not necessarily equal).

9

1

Explicit formula for a_n

A sequence

(a_n)

is defined by means of the recursion

a_1 = 1, a_{n+1} = \frac{1 + 4a_n +\sqrt{1+ 24a_n}}{16}.

Find an explicit formula for

a_n.

6

1

Prove that P(z) ≡ Q(z)

Let

P(z)

and

Q(z)

be complex-variable polynomials, with degree not less than

1

. Let

P_k = \{z \in \mathbb C | P(z) = k \}, Q_k = \{ z \in \mathbb C | Q(z) = k \}.

Let also

P_0 = Q_0

and

P_1 = Q_1

. Prove that

P(z) \equiv Q(z).

5

1

Probablity in a cube

A cube is assembled with

27

white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer.

4

1

On Fibonacci Sequence

Let

\{fn\}

be the Fibonacci sequence

\{1, 1, 2, 3, 5, \dots.\}.

(a) Find all pairs

(a, b)

of real numbers such that for each

n

,

af_n +bf_{n+1}

is a member of the sequence.
(b) Find all pairs

(u, v)

of positive real numbers such that for each

n

,

uf_n^2 +vf_{n+1}^2

is a member of the sequence.

3

1

7 variables inequality

Find the minimum value of

\max(a + b + c, b + c + d, c + d + e, d + e + f, e + f + g)

subject to the constraints
(i)

a, b, c, d, e, f, g \geq 0,

(ii)

a + b + c + d + e + f + g = 1.

2

1

Sphere tangent to a tetrahedron

A sphere

S

is tangent to the edges

AB,BC,CD,DA

of a tetrahedron

ABCD

at the points

E,F,G,H

respectively. The points

E,F,G,H

are the vertices of a square. Prove that if the sphere is tangent to the edge

AC

, then it is also tangent to the edge

BD.

1981 IMO Shortlist

Subcontests

Finite set of unit circles

Sum of the areas is equal

Behavior of u_n

Convex pentagon is a regular one

Determine P(n+1)

Geometric Inequality

Smallest n with the property

Explicit formula for a_n

Prove that P(z) ≡ Q(z)

Probablity in a cube

On Fibonacci Sequence

7 variables inequality

Sphere tangent to a tetrahedron