MathDB

1

ABC is equidistant from the circumcenter O

A

of the acute triangle

ABC

is equidistant from the circumcenter

O

and the orthocenter

H.

Determine all possible values for the measure of angle

A.

31

1

Number of solutions bounded from above

Let a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^\plus{} be given and let N

(a_1, a_2, a_3)

be the number of solutions

(x_1, x_2, x_3)

of the equation \sum^3_{k\equal{}1} \frac{a_k}{x_k} \equal{} 1. where

x_1, x_2,

and

x_3

are positive integers. Prove that N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 \plus{} ln(2 a_1)).

29

1

155 birds are sitting down on the boundary of a circle

155 birds

P_1, \ldots, P_{155}

are sitting down on the boundary of a circle

C.

Two birds

P_i, P_j

are mutually visible if the angle at centre

m(\cdot)

of their positions

m(P_iP_j) \leq 10^{\circ}.

Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs

\{x,y\}

of mutually visible pairs of birds with

x,y \in \{P_1, \ldots, P_{155}\}.

One assumes that a position (point) on

C

can be occupied simultaneously by several birds, e.g. all possible birds.

28

1

Area of triangle >= SQRT(2)

Consider in a plane

P

the points

O,A_1,A_2,A_3,A_4

such that \sigma(OA_iA_j) \geq 1 \forall i, j \equal{} 1, 2, 3, 4, i \neq j. where

\sigma(OA_iA_j)

is the area of triangle

OA_iA_j.

Prove that there exists at least one pair

i_0, j_0 \in \{1, 2, 3, 4\}

such that

\sigma(OA_iA_j) \geq \sqrt{2}.

27

1

2^{1989} divides m^n - 1

Let

m

be a positive odd integer,

m > 2.

Find the smallest positive integer

n

such that

2^{1989}

divides m^n \minus{} 1.

26

1

Determine the range of X_0

Let n \in \mathbb{Z}^\plus{} and let

a, b \in \mathbb{R}.

Determine the range of

x_0

for which \sum^n_{i\equal{}0} x_i \equal{} a \text{ and } \sum^n_{i\equal{}0} x^2_i \equal{} b, where

x_0, x_1, \ldots , x_n

are real variables.

25

1

x^2 - ay^2 - bz^2 + abw^2 = 0

Let

a, b \in \mathbb{Z}

which are not perfect squares. Prove that if x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0 has a nontrivial solution in integers, then so does x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.

24

1

Configurations for which this maximum is attained

For points

A_1, \ldots ,A_5

on the sphere of radius 1, what is the maximum value that

min_{1 \leq i,j \leq 5} A_iA_j

can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set

\{A_1, \ldots ,A_5\}

for which this maximum is attained.)

21

1

Regular tetrahedron can be an obtuse-angled triangle

Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than

120^{\circ}.

19

1

All the numbers to be zero after finitely many operations

A natural number is written in each square of an

m \times n

chess board. The allowed move is to add an integer

k

to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.

18

1

Image of M by the rotation

Given a convex polygon

A_1A_2 \ldots A_n

with area

S

and a point

M

in the same plane, determine the area of polygon

M_1M_2 \ldots M_n,

where

M_i

is the image of

M

under rotation

R^{\alpha}_{A_i}

around

A_i

by \alpha_i, i \equal{} 1, 2, \ldots, n.

17

1

How many segments does a figure have?

Given seven points in the plane, some of them are connected by segments such that: (i) among any three of the given points, two are connected by a segment; (ii) the number of segments is minimal. How many segments does a figure satisfying (i) and (ii) have? Give an example of such a figure.

16

1

Prove that c <= 1/4

The set

\{a_0, a_1, \ldots, a_n\}

of real numbers satisfies the following conditions: (i) a_0 \equal{} a_n \equal{} 0, (ii) for 1 \leq k \leq n \minus{} 1, a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right) Prove that

c \leq \frac{1}{4n}.

15

1

a^2+b^2+c^2+d^2 = 1989

Let a, b, c, d,m, n \in \mathbb{Z}^\plus{} such that a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989, a\plus{}b\plus{}c\plus{}d \equal{} m^2, and the largest of

a, b, c, d

is

n^2.

Determine, with proof, the values of

m

and

n.

14

1

Both inscribable in and circumscribable about a circle

A bicentric quadrilateral is one that is both inscribable in and circumscribable about a circle, i.e. both the incircle and circumcircle exists. Show that for such a quadrilateral, the centers of the two associated circles are collinear with the point of intersection of the diagonals.

11

1

Show that n|a_n

Define sequence

(a_n)

by \sum_{d|n} a_d \equal{} 2^n. Show that

n|a_n.

9

1

Prove that c_n = 0 implies n = 0

\forall n > 0, n \in \mathbb{Z},

there exists uniquely determined integers

a_n, b_n, c_n \in \mathbb{Z}

such \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}. Prove that c_n \equal{} 0 implies n \equal{} 0.

8

1

Rectangle that is the disjoint union of a finite number of r

Let

R

be a rectangle that is the union of a finite number of rectangles

R_i,

1 \leq i \leq n,

satisfying the following conditions: (i) The sides of every rectangle

R_i

are parallel to the sides of

R.

(ii) The interiors of any two different rectangles

R_i

are disjoint. (iii) Each rectangle

R_i

has at least one side of integral length. Prove that

R

has at least one side of integral length. Variant: Same problem but with rectangular parallelepipeds having at least one integral side.

7

1

Regular n-gon E can be joined by two circular arcs

Show that any two points lying inside a regular n\minus{}gon

E

can be joined by two circular arcs lying inside

E

and meeting at an angle of at least \left(1 \minus{} \frac{2}{n} \right) \cdot \pi.

6

1

16Q^3 >= 27r^4P

For a triangle

ABC,

let

k

be its circumcircle with radius

r.

The bisectors of the inner angles

A, B,

and

C

of the triangle intersect respectively the circle

k

again at points

A', B',

and

C'.

Prove the inequality

16Q^3 \geq 27 r^4 P,

where

Q

and

P

are the areas of the triangles

A'B'C'

and

ABC

respectively.

5

1

Find roots satisfying power of 16 sum that equals n

Find the roots

r_i \in \mathbb{R}

of the polynomial p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n satisfying \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.

4

1

Sum x^k/k! + 1 = 0 has no rational roots

Prove that

\forall n > 1, n \in \mathbb{N}

the equation \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0 has no rational roots.

3

1

Ali Barber II

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?

2

1

Ali Barber I

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?

10

1

Functional equation

Let

g: \mathbb{C} \rightarrow \mathbb{C}

,

\omega \in \mathbb{C}

,

a \in \mathbb{C}

, \omega^3 \equal{} 1, and

\omega \ne 1

. Show that there is one and only one function

f: \mathbb{C} \rightarrow \mathbb{C}

such that f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C}

12

1

Beautiful problem (at least for me)

There are n cars waiting at distinct points of a circular race track. At the starting signal each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet they both change direction (but not speed). Show that at some time each car is back at its starting point.

1989 IMO Shortlist

Subcontests

ABC is equidistant from the circumcenter O

Number of solutions bounded from above

155 birds are sitting down on the boundary of a circle

Area of triangle >= SQRT(2)

2^{1989} divides m^n - 1

Determine the range of X_0

x^2 - ay^2 - bz^2 + abw^2 = 0

Configurations for which this maximum is attained

Regular tetrahedron can be an obtuse-angled triangle

All the numbers to be zero after finitely many operations

Image of M by the rotation

How many segments does a figure have?

Prove that c <= 1/4

a^2+b^2+c^2+d^2 = 1989

Both inscribable in and circumscribable about a circle

Show that n|a_n

Prove that c_n = 0 implies n = 0

Rectangle that is the disjoint union of a finite number of r

Regular n-gon E can be joined by two circular arcs

16Q^3 >= 27r^4P

Find roots satisfying power of 16 sum that equals n

Sum x^k/k! + 1 = 0 has no rational roots

Ali Barber II

Ali Barber I

Functional equation

Beautiful problem (at least for me)

1989 IMO Shortlist

Subcontests

ABC is equidistant from the circumcenter O

Number of solutions bounded from above

155 birds are sitting down on the boundary of a circle

Area of triangle &gt;= SQRT(2)

2^{1989} divides m^n - 1

Determine the range of X_0

x^2 - ay^2 - bz^2 + abw^2 = 0

Configurations for which this maximum is attained

Regular tetrahedron can be an obtuse-angled triangle

All the numbers to be zero after finitely many operations

Image of M by the rotation

How many segments does a figure have?

Prove that c &lt;= 1/4

a^2+b^2+c^2+d^2 = 1989

Both inscribable in and circumscribable about a circle

Show that n|a_n

Prove that c_n = 0 implies n = 0

Rectangle that is the disjoint union of a finite number of r

Regular n-gon E can be joined by two circular arcs

16Q^3 &gt;= 27r^4P

Find roots satisfying power of 16 sum that equals n

Sum x^k/k! + 1 = 0 has no rational roots

Ali Barber II

Ali Barber I

Functional equation

Beautiful problem (at least for me)

Area of triangle >= SQRT(2)

Prove that c <= 1/4

16Q^3 >= 27r^4P