MathDB

1

fn(X) becomes more circular - IMO LongList 1992 CAN4

X

be a bounded, nonempty set of points in the Cartesian plane. Let

f(X)

be the set of all points that are at a distance of at most

1

from some point in

X

. Let

f_n(X) = f(f(\cdots(f(X))\cdots))

(

n

times). Show that

f_n(X)

becomes “more circular” as

n

gets larger. In other words, if

r_n = \sup\{\text{radii of circles contained in } f_n(X) \}

and

R_n = \inf \{\text{radii of circles containing } f_n(X)\}

, then show that

R_n/r_n

gets arbitrarily close to

1

as

n

becomes arbitrarily large.
I'm not sure that I'm posting this in a right forum. If it's in a wrong forum, please mods move it.

82

1

Polynomials - IMO LongList 1992 VIE1 (Last Problem of ILL)

Let

f(x) = x^m + a_1x^{m-1} + \cdots+ a_{m-1}x + a_m

and

g(x) = x^n + b_1x^{n-1} + \cdots + b_{n-1}x + b_n

be two polynomials with real coefficients such that for each real number

x, f(x)

is the square of an integer if and only if so is

g(x)

. Prove that if

n +m > 0

, then there exists a polynomial

h(x)

with real coefficients such that

f(x) \cdot g(x) = (h(x))^2.

[hide="Remark."]Remark. The original problem stated

g(x) = x^n + b_1x^{n-1} + \cdots + {\color{red}{ b_{n-1}}} + b_n

, but I think the right form of the problem is what I wrote.

81

1

orthocenter and circumcenter - IMO LongList 1992 USA4

Suppose that points

X, Y,Z

are located on sides

BC, CA

, and

AB

, respectively, of triangle

ABC

in such a way that triangle

XY Z

is similar to triangle

ABC

. Prove that the orthocenter of triangle

XY Z

is the circumcenter of triangle

ABC.

80

1

Graph and degree of vertices - IMO LongList 1992 USA3

Given a graph with

n

vertices and a positive integer

m

that is less than

n

, prove that the graph contains a set of

m+1

vertices in which the difference between the largest degree of any vertex in the set and the smallest degree of any vertex in the set is at most

m-1.

78

1

Sequence with Fibonacci numbers - IMO LongList 1992 USA1

Let

F_n

be the nth Fibonacci number, defined by

F_1 = F_2 = 1

and

F_n = F_{n-1} + F_{n-2}

for

n > 2

. Let

A_0, A_1, A_2,\cdots

be a sequence of points on a circle of radius

1

such that the minor arc from

A_{k-1}

to

A_k

runs clockwise and such that

\mu(A_{k-1}A_k)=\frac{4F_{2k+1}}{F_{2k+1}^2+1}

for

k \geq 1

, where

\mu(XY )

denotes the radian measure of the arc

XY

in the clockwise direction. What is the limit of the radian measure of arc

A_0A_n

as

n

approaches infinity?

77

1

Choosing numbers - IMO LongList 1992 TWN3

Show that if

994

integers are chosen from

1, 2,\cdots , 1992

and one of the chosen integers is less than

64

, then there exist two among the chosen integers such that one of them is a factor of the other.

76

1

Decomposable numbers - IMO LonList 1992 TWN2

Given any triangle

ABC

and any positive integer

n

, we say that

n

is a decomposable number for triangle

ABC

if there exists a decomposition of the triangle

ABC

into

n

subtriangles with each subtriangle similar to

\triangle ABC

. Determine the positive integers that are decomposable numbers for every triangle.

75

1

Positive integers in the sequence - IMO LongList 1992 TWN1

A sequence

\{an\}

of positive integers is defined by

a_n=\left[ n +\sqrt n + \frac 12 \right] , \qquad \forall n \in \mathbb N

Determine the positive integers that occur in the sequence.

74

1

Every real number - IMO LongList 1992 TUR3

Let

S = \{\frac{\pi^n}{1992^m} | m,n \in \mathbb Z \}.

Show that every real number

x \geq 0

is an accumulation point of

S.

73

1

Subset G of the plane - IMO LongList 1992 TUR2

Let

\{A_n | n = 1, 2, \cdots \}

be a set of points in the plane such that for each

n

, the disk with center

A_n

and radius

2^n

contains no other point

A_j

. For any given positive real numbers

a < b

and

R

, show that there is a subset

G

of the plane satisfying:
(i) the area of

G

is greater than or equal to

R

;
(ii) for each point

P

in

G

,

a < \sum_{n=1}^{\infty} \frac{1}{|A_nP|} <b.

72

1

School and students - IMO LongList 1992 TUR1

In a school six different courses are taught: mathematics, physics, biology, music, history, geography. The students were required to rank these courses according to their preferences, where equal preferences were allowed. It turned out that:
* mathematics was ranked among the most preferred courses by all students;
* no student ranked music among the least preferred ones;
* all students preferred history to geography and physics to biology; and
* no two rankings were the same.
Find the greatest possible value for the number of students in this school.

71

1

There exist integers A1,A2 - IMO LongList 1992 THA2

Let

P_1(x, y)

and

P_2(x, y)

be two relatively prime polynomials with complex coefficients. Let

Q(x, y)

and

R(x, y)

be polynomials with complex coefficients and each of degree not exceeding

d

. Prove that there exist two integers

A_1, A_2

not simultaneously zero with

|A_i| \leq d + 1 \ (i = 1, 2)

and such that the polynomial

A_1P_1(x, y) + A_2P_2(x, y)

is coprime to

Q(x, y)

and

R(x, y).

70

1

Sequence of circles and inequality - ILL 1992 THA1

Let two circles

A

and

B

with unequal radii

r

and

R

, respectively, be tangent internally at the point

A_0

. If there exists a sequence of distinct circles

(C_n)

such that each circle is tangent to both

A

and

B

, and each circle

C_{n+1}

touches circle

C_{n}

at the point

A_n

, prove that

\sum_{n=1}^{\infty} |A_{n+1}A_n| < \frac{4 \pi Rr}{R+r}.

67

1

Geometric equations IMO LongList 1992 SPA2

In a triangle, a symmedian is a line through a vertex that is symmetric to the median with the respect to the internal bisector (all relative to the same vertex). In the triangle

ABC

, the median

m_a

meets

BC

at

A'

and the circumcircle again at

A_1

. The symmedian

s_a

meets

BC

at

M

and the circumcircle again at

A_2

. Given that the line

A_1A_2

contains the circumcenter

O

of the triangle, prove that:
(a)

\frac{AA'}{AM} = \frac{b^2+c^2}{2bc} ;

(b)

1+4b^2c^2 = a^2(b^2+c^2)

66

1

IMO LongList 1992 SPA1

A circle of radius

\rho

is tangent to the sides

AB

and

AC

of the triangle

ABC

, and its center

K

is at a distance

p

from

BC

.
(a) Prove that

a(p - \rho) = 2s(r - \rho)

, where

r

is the inradius and

2s

the perimeter of

ABC

. (b) Prove that if the circle intersect

BC

at

D

and

E

, then

DE=\frac{4\sqrt{rr_1(\rho-r)(r_1-\rho)}}{r_1-r}

where

r_1

is the exradius corresponding to the vertex

A.

65

1

Points in the space - IMO LongList 1992 SAF1

If

A, B, C

, and

D

are four distinct points in space, prove that there is a plane

P

on which the orthogonal projections of

A, B, C

, and

D

form a parallelogram (possibly degenerate).

64

1

The longest representation - IMO LongList 1992 ROM3

For any positive integer

n

consider all representations

n = a_1 + \cdots+ a_k

, where

a_1 > a_2 > \cdots > a_k > 0

are integers such that for all

i \in \{1, 2, \cdots , k - 1\}

, the number

a_i

is divisible by

a_{i+1}

. Find the longest such representation of the number

1992.

63

1

A very nice problem IMO LongList 1992 ROM2

Let

a

and

b

be integers. Prove that

\frac{2a^2-1}{b^2+2}

is not an integer.

62

1

Exist k_i for such c_i - IMO LongList 1992 ROM1

Let

c_1, \cdots, c_n \ (n \geq 2)

be real numbers such that

0 \leq \sum c_i \leq n

. Prove that there exist integers

k_1, \cdots , k_n

such that

\sum k_i=0

and

1-n \leq c_i + nk_i \leq n

for every

i = 1, \cdots , n.

61

1

Board and cards IMO LongList 1992 PRK3

There are a board with

2n \cdot 2n \ (= 4n^2)

squares and

4n^2-1

cards numbered with different natural numbers. These cards are put one by one on each of the squares. One square is empty. We can move a card to an empty square from one of the adjacent squares (two squares are adjacent if they have a common edge). Is it possible to exchange two cards on two adjacent squares of a column (or a row) in a finite number of movements?

59

1

Regular 7-gon IMO LongList 1992 PRK1

Let a regular

7

-gon

A_0A_1A_2A_3A_4A_5A_6

be inscribed in a circle. Prove that for any two points

P, Q

on the arc

A_0A_6

the following equality holds:

\sum_{i=0}^6 (-1)^{i} PA_i = \sum_{i=0}^6 (-1)^{i} QA_i .

57

1

Nice inequality with A,G,H - IMO LongList 1992

For positive numbers

a, b, c

define

A = \frac{(a + b + c)}{3}

,

G = \sqrt[3]{abc}

,

H = \frac{3}{(a^{-1} + b^{-1} + c^{-1})}.

Prove that

\left( \frac AG \right)^3 \geq \frac 14 + \frac 34 \cdot \frac AH.

56

1

IMO LongList 1992 - Graph problem

A directed graph (any two distinct vertices joined by at most one directed line) has the following property: If

x, u,

and

v

are three distinct vertices such that

x \to u

and

x \to v

, then

u \to w

and

v \to w

for some vertex

w

. Suppose that

x \to u \to y \to\cdots \to z

is a path of length

n

, that cannot be extended to the right (no arrow goes away from

z

). Prove that every path beginning at

x

arrives after

n

steps at

z.

54

1

ILL 92 - Prove that n>125

Suppose that

n > m \geq 1

are integers such that the string of digits

143

occurs somewhere in the decimal representation of the fraction

\frac{m}{n}

. Prove that

n > 125.

52

1

ILL 92 NET 2 - Lamps

Let

n

be an integer

> 1

. In a circular arrangement of

n

lamps

L_0, \cdots, L_{n-1}

, each one of which can be either ON or OFF, we start with the situation that all lamps are ON, and then carry out a sequence of steps,

Step_0, Step_1, \cdots

. If

L_{j-1}

(

j

is taken mod n) is ON, then

Step_j

changes the status of

L_j

(it goes from ON to OFF or from OFF to ON) but does not change the status of any of the other lamps. If

L_{j-1}

is OFF, then

Step_j

does not change anything at all. Show that:
(a) There is a positive integer

M(n)

such that after

M(n)

steps all lamps are ON again.
(b) If

n

has the form

2^k

, then all lamps are ON after

n^2 - 1

steps.
(c) If

n

has the form

2^k +1

, then all lamps are ON after

n^2 -n+1

steps.

50

1

ILL 92 nine-point circle

Let

N

be a point inside the triangle

ABC

. Through the midpoints of the segments

AN, BN

, and

CN

the lines parallel to the opposite sides of

\triangle ABC

are constructed. Let

AN, BN

, and

CN

be the intersection points of these lines. If

N

is the orthocenter of the triangle

ABC

, prove that the nine-point circles of

\triangle ABC

and

\triangle A_NB_NC_N

coincide.
[hide="Remark."]Remark. The statement of the original problem was that the nine-point circles of the triangles

A_NB_NC_N

and

A_MB_MC_M

coincide, where

N

and

M

are the orthocenter and the centroid of

ABC

. This statement is false.

49

1

4k+2 real numbers

Given real numbers

x_i \ (i = 1, 2, \cdots, 4k + 2)

such that

\sum_{i=1}^{4k +2} (-1)^{i+1} x_ix_{i+1} = 4m \qquad ( \ x_1=x_{4k+3} \ )

prove that it is possible to choose numbers

x_{k_{1}}, \cdots, x_{k_{6}}

such that

\sum_{i=1}^{6} (-1)^{i} k_i k_{i+1} > m \qquad ( \ x_{k_{1}} = x_{k_{7}} \ )

48

1

Functional equation with α, β

Find all the functions

f : \mathbb R^+ \to \mathbb R

satisfying the identity

f(x)f(y)=y^{\alpha}f\left(\frac x2 \right) + x^{\beta} f\left(\frac y2 \right) \qquad \forall x,y \in \mathbb R^+

Where

\alpha,\beta

are given real numbers.

46

1

The sequence does not contain numbers of the form 2^n - 1

Prove that the sequence

5, 12, 19, 26, 33,\cdots

contains no term of the form

2^n -1.

45

1

Number of ways to express n as sum of distinct numbers

Let

n

be a positive integer. Prove that the number of ways to express

n

as a sum of distinct positive integers (up to order) and the number of ways to express

n

as a sum of odd positive integers (up to order) are the same.

43

1

Find all positive integers n with φ(n) | n and a condition

Find the number of positive integers

n

satisfying

\phi(n) | n

such that

\sum_{m=1}^{\infty} \left( \left[ \frac nm \right] - \left[\frac{n-1}{m} \right] \right) = 1992

What is the largest number among them? As usual,

\phi(n)

is the number of positive integers less than or equal to

n

and relatively prime to

n.

41

1

Define omega

Let

S

be a set of positive integers

n_1, n_2, \cdots, n_6

and let

n(f)

denote the number

n_1n_{f(1)} +n_2n_{f(2)} +\cdots+n_6n_{f(6)}

, where

f

is a permutation of

\{1, 2, . . . , 6\}

. Let

\Omega=\{n(f) | f \text{ is a permutation of } \{1, 2, . . . , 6\} \}

Give an example of positive integers

n_1, \cdots, n_6

such that

\Omega

contains as many elements as possible and determine the number of elements of

\Omega

.

40

1

Find maximal value of N

The colonizers of a spherical planet have decided to build

N

towns, each having area

1/1000

of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of

N

?

39

1

Find the minimum k

Let

n \geq 2

be an integer. Find the minimum

k

for which there exists a partition of

\{1, 2, . . . , k\}

into

n

subsets

X_1,X_2, \cdots , X_n

such that the following condition holds: for any

i, j, 1 \leq i < j \leq n

, there exist

x_i \in X_1, x_j \in X_2

such that

|x_i - x_j | = 1.

37

1

Prove that A+B+C < pi

Let the circles

C_1, C_2

, and

C_3

be orthogonal to the circle

C

and intersect each other inside

C

forming acute angles of measures

A, B

, and

C

. Show that

A + B +C < \pi.

34

1

Prove that p_1, q_1, r_1, p_2, q_2, r_2 exist

Let

a, b, c

be integers. Prove that there are integers

p_1, q_1, r_1, p_2, q_2, r_2

such that

a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1, c = p_1q_2 - p_2q_1.

32

1

Functional equation - f_n(x) = x

Let

S_n = \{1, 2,\cdots, n\}

and

f_n : S_n \to S_n

be defined inductively as follows:

f_1(1) = 1, f_n(2j) = j \ (j = 1, 2, \cdots , [n/2])

and
* if

n = 2k \ (k \geq 1)

, then

f_n(2j - 1) = f_k(j) + k \ (j = 1, 2, \cdots, k);

* if

n = 2k + 1 \ (k \geq 1)

, then

f_n(2k + 1) = k + f_{k+1}(1), f_n(2j - 1) = k + f_{k+1}(j + 1) \ (j = 1, 2,\cdots , k).

Prove that

f_n(x) = x

if and only if

x

is an integer of the form

\frac{(2n + 1)(2^d - 1)}{2^{d+1} - 1}

for some positive integer

d.

27

1

Define Σ to be the set of all circles with two properties

Let

ABC

be an arbitrary scalene triangle. Define

\sum

to be the set of all circles

y

that have the following properties:
(i)

y

meets each side of

ABC

in two (possibly coincident) points;
(ii) if the points of intersection of

y

with the sides of the triangle are labeled by

P, Q, R, S, T , U

, with the points occurring on the sides in orders

\mathcal B(B,P,Q,C), \mathcal B(C, R, S,A), \mathcal B(A, T,U,B)

, then the following relations of parallelism hold:

TS \parallel BC; PU\parallel CA; RQ\parallel AB

. (In the limiting cases, some of the conditions of parallelism will hold vacuously; e.g., if

A

lies on the circle

y

, then

T

,

S

both coincide with

A

and the relation

TS \parallel BC

holds vacuously.)
(a) Under what circumstances is

\sum

nonempty?
(b) Assuming that Σ is nonempty, show how to construct the locus of centers of the circles in the set

\sum

.
(c) Given that the set

\sum

has just one element, deduce the size of the largest angle of

ABC.

(d) Show how to construct the circles in

\sum

that have, respectively, the largest and the smallest radii.

25

1

Partitions of the set N

(a) Show that the set

\mathbb N

of all positive integers can be partitioned into three disjoint subsets

A, B

, and

C

satisfying the following conditions:

A^2 = A, B^2 = C, C^2 = B,

AB = B, AC = C, BC = A,

where

HK

stands for

\{hk | h \in H, k \in K\}

for any two subsets

H, K

of

\mathbb N

, and

H^2

denotes

HH.

(b) Show that for every such partition of

\mathbb N

,

\min\{n \in N | n \in A \text{ and } n + 1 \in A\}

is less than or equal to

77.

24

1

Function defined on positive rationals

(a) Show that there exists exactly one function

f : \mathbb Q^+ \to \mathbb Q^+

satisfying the following conditions: (i) if

0 < q < \frac 12

, then

f(q)=1+f \left( \frac{q}{1-2q} \right);

(ii) if

1 < q \leq 2

, then

f(q) = 1+f(q + 1);

(iii)

f(q)f(1/q) = 1

for all

q \in \mathbb Q^+.

(b) Find the smallest rational number

q \in \mathbb Q^+

such that

f(q) = \frac{19}{92}.

21

1

Inequality with 1/x +1/y +1/z = 2

Prove that if

x,y,z >1

and

\frac 1x +\frac 1y +\frac 1z = 2

, then

\sqrt{x+y+z} \geq \sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.

20

1

Coloring the points in the plane

Let

X

and

Y

be two sets of points in the plane and

M

be a set of segments connecting points from

X

and

Y

. Let

k

be a natural number. Prove that the segments from

M

can be painted using

k

colors in such a way that for any point

x \in X \cup Y

and two colors

\alpha

and

\beta

(\alpha \neq \beta)

, the difference between the number of

\alpha

-colored segments and the number of

\beta

-colored segments originating in

X

is less than or equal to

1

.

19

1

Prove the formula for A(n)

Denote by

a_n

the greatest number that is not divisible by

3

and that divides

n

. Consider the sequence

s_0 = 0, s_n = a_1 +a_2+\cdots+a_n, n \in \mathbb N

. Denote by

A(n)

the number of all sums

s_k \ (0 \leq k \leq 3^n, k \in \mathbb N_0)

that are divisible by

3

. Prove the formula

A(n) = 3^{n-1} + 2 \cdot 3^{(n/2)-1} \cos \left(\frac{n\pi}{6}\right), \qquad n\in \mathbb N_0.

18

1

Express a_n in terms of Fibonacci numbers

Fibonacci numbers are defined as follows:

F_0 = F_1 = 1, F_{n+2} = F_{n+1}+F_n, n \geq 0

. Let

a_n

be the number of words that consist of

n

letters

0

or

1

and contain no two letters

1

at distance two from each other. Express

a_n

in terms of Fibonacci numbers.

15

1

78 lines in plane that have 1992 points of intersection

Prove that there exist

78

lines in the plane such that they have exactly

1992

points of intersection.

14

1

Nice Number Theory problem with |a_k|=1

Integers

a_1, a_2, . . . , a_n

satisfy

|a_k| = 1

and

\sum_{k=1}^{n} a_ka_{k+1}a_{k+2}a_{k+3} = 2,

where

a_{n+j} = a_j

. Prove that

n \neq 1992.

13

1

Perpendicular lines with circumcenters

Let

ABCD

be a convex quadrilateral such that

AC = 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.

12

1

Short and nice problem, Prove that ABC is acute-angled

Given a triangle

ABC

such that the circumcenter is in the interior of the incircle, prove that the triangle

ABC

is acute-angled.

9

1

Congruent quadrilaterals

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; DF,AH; AH,BJ; BJ,CL

are denoted by

P_1,Q_1,R_1, S_1

, respectively, and the intersection points of the pairs of straight lines

AI,BK; BK,CE;

CE,DG; DG,AI

are denoted by

P_2,Q_2,R_2, S_2

, respectively. Prove that

P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2.

8

1

Unique function

Given two positive real numbers

a

and

b

, suppose that a mapping

f : \mathbb R^+ \to \mathbb R^+

satisfies the functional equation

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

Prove that there exists a unique solution of this equation.

6

1

Probablity for a modular equation

Suppose that n numbers

x_1, x_2, . . . , x_n

are chosen randomly from the set

\{1, 2, 3, 4, 5\}

. Prove that the probability that

x_1^2+ x_2^2 +\cdots+ x_n^2 \equiv 0 \pmod 5

is at least

\frac 15.

4

1

Geometric inequality with angles

Let

p, q

, and

r

be the angles of a triangle, and let

a = \sin2p, b = \sin2q

, and

c = \sin2r

. If

s = \frac{(a + b + c)}2

, show that

s(s - a)(s - b)(s -c) \geq 0.

When does equality hold?

3

1

Nine-point circle

Let

ABC

be a triangle,

O

its circumcenter,

S

its centroid, and

H

its orthocenter. Denote by

A_1, B_1

, and

C_1

the centers of the circles circumscribed about the triangles

CHB, CHA

, and

AHB

, respectively. Prove that the triangle

ABC

is congruent to the triangle

A_1B_1C_1

and that the nine-point circle of

\triangle ABC

is also the nine-point circle of

\triangle A_1B_1C_1

.

68

1

Trigonometric equation

Show that the numbers

\tan \left(\frac{r \pi }{15}\right)

, where

r

is a positive integer less than

15

and relatively prime to

15

, satisfy

x^8 - 92x^6 + 134x^4 - 28x^2 + 1 = 0.

47

1

Evaluate the floor

Evaluate

\left \lfloor \ \prod_{n=1}^{1992} \frac{3n+2}{3n+1} \ \right \rfloor

36

1

Rational solutions a^2 + c^2 + 17(b^2 + d^2)=21, ab + cd = 2

Find all rational solutions of

a^2 + c^2 + 17(b^2 + d^2) = 21,

ab + cd = 2.

23

1

Interesting problem on Egyptian Numbers

An Egyptian number is a positive integer that can be expressed as a sum of positive integers, not necessarily distinct, such that the sum of their reciprocals is

1

. For example,

32 = 2 + 3 + 9 + 18

is Egyptian because

\frac 12 +\frac 13 +\frac 19 +\frac{1}{18}=1

. Prove that all integers greater than

23

are Egyptian.

16

1

Solve the equation for x,y,z ∈ Z

Find all triples

(x, y, z)

of integers such that

\frac{1}{x^2}+\frac{2}{y^2}+\frac{3}{z^2} =\frac 23

11

1

φ(n,m) a new function for every n

Let

\phi(n,m), m \neq 1

, be the number of positive integers less than or equal to

n

that are coprime with

m.

Clearly,

\phi(m,m) = \phi(m)

, where

\phi(m)

is Euler’s phi function. Find all integers

m

that satisfy the following inequality:

\frac{\phi(n,m)}{n} \geq \frac{\phi(m)}{m}

for every positive integer

n.

5

1

AI || OH iff BAC = 120 degrees

Let

I,H,O

be the incenter, centroid, and circumcenter of the nonisosceles triangle

ABC

. Prove that

AI \parallel HO

if and only if

\angle BAC =120^{\circ}

.

2

1

Prove that x,y exist

Let

m

be a positive integer and

x_0, y_0

integers such that

x_0, y_0

are relatively prime,

y_0

divides

x_0^2+m

, and

x_0

divides

y_0^2+m

. Prove that there exist positive integers

x

and

y

such that

x

and

y

are relatively prime,

y

divides

x^2 + m

,

x

divides

y^2 + m

, and

x + y \leq m+ 1.

1

Prove that AC + CF = AB + BF

Points

D

and

E

are chosen on the sides

AB

and

AC

of the triangle

ABC

in such a way that if

F

is the intersection point of

BE

and

CD

, then

AE + EF = AD + DF

. Prove that

AC + CF = AB + BF.

30

1

Find all m that P_m is divisible by 33^33

Let

P_n = (19 + 92)(19^2 +92^2) \cdots(19^n +92^n)

for each positive integer

n

. Determine, with proof, the least positive integer

m

, if it exists, for which

P_m

is divisible by

33^{33}.

33

1

S has at most six elements

Let

a, b, c

be positive real numbers and

p, q, r

complex numbers. Let

S

be the set of all solutions

(x, y, z)

in

\mathbb C

of the system of simultaneous equations

ax + by + cz = p,

ax2 + by2 + cz2 = q,

ax3 + bx3 + cx3 = r.

Prove that

S

has at most six elements.

1992 IMO Longlists

Subcontests

fn(X) becomes more circular - IMO LongList 1992 CAN4

Polynomials - IMO LongList 1992 VIE1 (Last Problem of ILL)

orthocenter and circumcenter - IMO LongList 1992 USA4

Graph and degree of vertices - IMO LongList 1992 USA3

Sequence with Fibonacci numbers - IMO LongList 1992 USA1

Choosing numbers - IMO LongList 1992 TWN3

Decomposable numbers - IMO LonList 1992 TWN2

Positive integers in the sequence - IMO LongList 1992 TWN1

Every real number - IMO LongList 1992 TUR3

Subset G of the plane - IMO LongList 1992 TUR2

School and students - IMO LongList 1992 TUR1

There exist integers A1,A2 - IMO LongList 1992 THA2

Sequence of circles and inequality - ILL 1992 THA1

Geometric equations IMO LongList 1992 SPA2

IMO LongList 1992 SPA1

Points in the space - IMO LongList 1992 SAF1

The longest representation - IMO LongList 1992 ROM3

A very nice problem IMO LongList 1992 ROM2

Exist k_i for such c_i - IMO LongList 1992 ROM1

Board and cards IMO LongList 1992 PRK3

Regular 7-gon IMO LongList 1992 PRK1

Nice inequality with A,G,H - IMO LongList 1992

IMO LongList 1992 - Graph problem

ILL 92 - Prove that n&gt;125

ILL 92 NET 2 - Lamps

ILL 92 nine-point circle

4k+2 real numbers

Functional equation with α, β

The sequence does not contain numbers of the form 2^n - 1

Number of ways to express n as sum of distinct numbers

Find all positive integers n with φ(n) | n and a condition

Define omega

Find maximal value of N

Find the minimum k

Prove that A+B+C &lt; pi

Prove that p_1, q_1, r_1, p_2, q_2, r_2 exist

Functional equation - f_n(x) = x

Define &Sigma; to be the set of all circles with two properties

Partitions of the set N

Function defined on positive rationals

Inequality with 1/x +1/y +1/z = 2

Coloring the points in the plane

Prove the formula for A(n)

Express a_n in terms of Fibonacci numbers

78 lines in plane that have 1992 points of intersection

Nice Number Theory problem with |a_k|=1

Perpendicular lines with circumcenters

Short and nice problem, Prove that ABC is acute-angled

Congruent quadrilaterals

Unique function

Probablity for a modular equation

Geometric inequality with angles

Nine-point circle

Trigonometric equation

Evaluate the floor

Rational solutions a^2 + c^2 + 17(b^2 + d^2)=21, ab + cd = 2

Interesting problem on Egyptian Numbers

Solve the equation for x,y,z ∈ Z

φ(n,m) a new function for every n

AI || OH iff BAC = 120 degrees

Prove that x,y exist

Prove that AC + CF = AB + BF

Find all m that P_m is divisible by 33^33

S has at most six elements

ILL 92 - Prove that n>125

Prove that A+B+C < pi

Define Σ to be the set of all circles with two properties