MathDB

1

Show that the number 1992 appears in the sequence

x

define

g(x)

as greatest odd divisor of

x,

and f(x) \equal{} \begin{cases} \frac {x}{2} \plus{} \frac {x}{g(x)} & \text{if \ $$x$$ is even}, \\ 2^{\frac {x \plus{} 1}{2}} & \text{if \ $$x$$ is odd}. \end{cases} Construct the sequence x_1 \equal{} 1, x_{n \plus{} 1} \equal{} f(x_n). Show that the number 1992 appears in this sequence, determine the least

n

such that x_n \equal{} 1992, and determine whether

n

is unique.

18

1

Continued sequence fraction

Let

\lfloor x \rfloor

denote the greatest integer less than or equal to

x.

Pick any

x_1

in

[0, 1)

and define the sequence

x_1, x_2, x_3, \ldots

by x_{n\plus{}1} \equal{} 0 if x_n \equal{} 0 and x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor otherwise. Prove that x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}}, where F_1 \equal{} F_2 \equal{} 1 and F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n for

n \geq 1.

19

1

Modulo Polynomial

Let f(x) \equal{} x^8 \plus{} 4x^6 \plus{} 2x^4 \plus{} 28x^2 \plus{} 1. Let

p > 3

be a prime and suppose there exists an integer

z

such that

p

divides

f(z).

Prove that there exist integers

z_1, z_2, \ldots, z_8

such that if g(x) \equal{} (x \minus{} z_1)(x \minus{} z_2) \cdot \ldots \cdot (x \minus{} z_8), then all coefficients of f(x) \minus{} g(x) are divisible by

p.

17

1

Number of digits equal to one in the binary representation

Let

\alpha(n)

be the number of digits equal to one in the binary representation of a positive integer

n.

Prove that: (a) the inequality

\alpha(n) (n^2 ) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1)

holds; (b) the above inequality is an equality for infinitely many positive integers, and (c) there exists a sequence

(n_i )^{\infty}_1

such that

\frac{\alpha ( n^2_i )}{\alpha (n_i }

goes to zero as

i

goes to

\infty.

Alternative problem: Prove that there exists a sequence a sequence

(n_i )^{\infty}_1

such that

\frac{\alpha ( n^2_i )}{\alpha (n_i )}

(d)

\infty;

(e) an arbitrary real number

\gamma \in (0,1)

; (f) an arbitrary real number

\gamma \geq 0

; as

i

goes to

\infty.

15

1

Every element in M, sum of any number with form m^k

Does there exist a set

M

with the following properties? (i) The set

M

consists of 1992 natural numbers. (ii) Every element in

M

and the sum of any number of elements have the form

m^k

(m, k \in \mathbb{N}, k \geq 2).

12

1

f(x) - f(y) = a(x, y)(g(x) - g(y))

Let

f, g

and

a

be polynomials with real coefficients,

f

and

g

in one variable and

a

in two variables. Suppose f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R} Prove that there exists a polynomial

h

with f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.

11

1

Angle BDE = 24 degrees, angle CED = 18 degrees

In a triangle

ABC,

let

D

and

E

be the intersections of the bisectors of

\angle ABC

and

\angle ACB

with the sides

AC,AB,

respectively. Determine the angles

\angle A,\angle B, \angle C

if \angle BDE \equal{} 24 ^{\circ}, \angle CED \equal{} 18 ^{\circ}.

9

1

[f(x)]^3 - f(x) = 33^1992

Let

f(x)

be a polynomial with rational coefficients and

\alpha

be a real number such that \alpha^3 \minus{} \alpha \equal{} [f(\alpha)]^3 \minus{} f(\alpha) \equal{} 33^{1992}. Prove that for each

n \geq 1,

\left [ f^{n}(\alpha) \right]^3 \minus{} f^{n}(\alpha) \equal{} 33^{1992}, where f^{n}(x) \equal{} f(f(\cdots f(x))), and

n

is a positive integer.

8

1

1992-gon with side lengths 1, 2, 3, ..., 1992 circumscribed

Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions: (i) its side lengths are

1, 2, 3, \ldots, 1992

in some order; (ii) the polygon is circumscribable about a circle. Alternative formulation: Does there exist a 1992-gon with side lengths

1, 2, 3, \ldots, 1992

circumscribed about a circle? Answer the same question for a 1990-gon.

3

1

Convex quadrilaterals are congruent

The diagonals of a quadrilateral

ABCD

are perpendicular:

AC \perp BD.

Four squares,

ABEF,BCGH,CDIJ,DAKL,

are erected externally on its sides. The intersection points of the pairs of straight lines

CL, DF, AH, BJ

are denoted by

P_1,Q_1,R_1, S_1,

respectively (left figure), and the intersection points of the pairs of straight lines

AI, BK, CE DG

are denoted by

P_2,Q_2,R_2, S_2,

respectively (right figure). Prove that

P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2

where

P_1,Q_1,R_1, S_1

and

P_2,Q_2,R_2, S_2

are the two quadrilaterals. Alternative formulation: Outside a convex quadrilateral

ABCD

with perpendicular diagonals, four squares

AEFB, BGHC, CIJD, DKLA,

are constructed (vertices are given in counterclockwise order). Prove that the quadrilaterals

Q_1

and

Q_2

formed by the lines

AG, BI, CK, DE

and

AJ, BL, CF, DH,

respectively, are congruent.

2

1

f(f(x)) + af(x) = b(a + b)x

Let \mathbb{R}^\plus{} be the set of all non-negative real numbers. Given two positive real numbers

a

and

b,

suppose that a mapping f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{} satisfies the functional equation: f(f(x)) \plus{} af(x) \equal{} b(a \plus{} b)x. Prove that there exists a unique solution of this equation.

1

There exist an infinite number of pairs of integers (x, y)

Prove that for any positive integer

m

there exist an infinite number of pairs of integers

(x, y)

such that (i)

x

and

y

are relatively prime; (ii)

y

divides x^2 \plus{} m; (iii)

x

divides y^2 \plus{} m. (iv) x \plus{} y \leq m \plus{} 1\minus{} (optional condition)

16

1

E 5

Prove that

\frac{5^{125}-1}{5^{25}-1}

is a composite number.

7

1

Three circles.

Two circles

\Omega_{1}

and

\Omega_{2}

are externally tangent to each other at a point

I

, and both of these circles are tangent to a third circle

\Omega

which encloses the two circles

\Omega_{1}

and

\Omega_{2}

. The common tangent to the two circles

\Omega_{1}

and

\Omega_{2}

at the point

I

meets the circle

\Omega

at a point

A

. One common tangent to the circles

\Omega_{1}

and

\Omega_{2}

which doesn't pass through

I

meets the circle

\Omega

at the points

B

and

C

such that the points

A

and

I

lie on the same side of the line

BC

. Prove that the point

I

is the incenter of triangle

ABC

. Alternative formulation. Two circles touch externally at a point

I

. The two circles lie inside a large circle and both touch it. The chord

BC

of the large circle touches both smaller circles (not at

I

). The common tangent to the two smaller circles at the point

I

meets the large circle at a point

A

, where the points

A

and

I

are on the same side of the chord

BC

. Show that the point

I

is the incenter of triangle

ABC

.

5

1

Lines joining centers of equilateral triangles perpendicular

A convex quadrilateral has equal diagonals. An equilateral triangle is constructed on the outside of each side of the quadrilateral. The centers of the triangles on opposite sides are joined. Show that the two joining lines are perpendicular. Alternative formulation. Given a convex quadrilateral

ABCD

with congruent diagonals AC \equal{} BD. Four regular triangles are errected externally on its sides. Prove that the segments joining the centroids of the triangles on the opposite sides are perpendicular to each other. Original formulation: Let

ABCD

be a convex quadrilateral such that AC \equal{} BD. Equilateral triangles are constructed on the sides of the quadrilateral. Let

O_1,O_2,O_3,O_4

be the centers of the triangles constructed on

AB,BC,CD,DA

respectively. Show that

O_1O_3

is perpendicular to

O_2O_4.

1992 IMO Shortlist

Subcontests

Show that the number 1992 appears in the sequence

Continued sequence fraction

Modulo Polynomial

Number of digits equal to one in the binary representation

Every element in M, sum of any number with form m^k

f(x) - f(y) = a(x, y)(g(x) - g(y))

Angle BDE = 24 degrees, angle CED = 18 degrees

[f(x)]^3 - f(x) = 33^1992

1992-gon with side lengths 1, 2, 3, ..., 1992 circumscribed

Convex quadrilaterals are congruent

f(f(x)) + af(x) = b(a + b)x

There exist an infinite number of pairs of integers (x, y)

E 5

Three circles.

Lines joining centers of equilateral triangles perpendicular