MathDB

3

variable point on the line BC

ABC

, let

X

be a variable point on the line

BC

such that

C

lies between

B

and

X

and the incircles of the triangles

ABX

and

ACX

intersect at two distinct points

P

and

Q.

Prove that the line

PQ

passes through a point independent of

X

.

triangle with vertices at three of the given points

Let

p

be an odd prime and

n

a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length

p^{n}

. Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by

p^{n+1}

.
Proposed by Alexander Ivanov, Bulgaria

Most challenging inequality on the ISL 2004

Let

{a_1,a_2,\dots,a_n}

be positive real numbers,

{n>1}

. Denote by

g_n

their geometric mean, and by

A_1,A_2,\dots,A_n

the sequence of arithmetic means defined by

A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n.

Let

G_n

be the geometric mean of

A_1,A_2,\dots,A_n

. Prove the inequality n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 and establish the cases of equality.
Proposed by Finbarr Holland, Ireland

8

2

Cyclic quadrilateral geometry in the style of V. Thebault

Given a cyclic quadrilateral

ABCD

, let

M

be the midpoint of the side

CD

, and let

N

be a point on the circumcircle of triangle

ABM

. Assume that the point

N

is different from the point

M

and satisfies

\frac{AN}{BN}=\frac{AM}{BM}

. Prove that the points

E

,

F

,

N

are collinear, where

E=AC\cap BD

and

F=BC\cap DA

.
Proposed by Dusan Dukic, Serbia and Montenegro

g(G)^3 <= c * f(G)^4

For a finite graph

G

, let

f(G)

be the number of triangles and

g(G)

the number of tetrahedra formed by edges of

G

. Find the least constant

c

such that

g(G)^3\le c\cdot f(G)^4

for every graph

G

.
Proposed by Marcin Kuczma, Poland

1

2

problem involving the tau function, Australia TST 1

Let

\tau(n)

denote the number of positive divisors of the positive integer

n

. Prove that there exist infinitely many positive integers

a

such that the equation

\tau(an)=n

does not have a positive integer solution

n

.

10001 students at an university

There are

10001

students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of

k

societies. Suppose that the following conditions hold:
i.) Each pair of students are in exactly one club.
ii.) For each student and each society, the student is in exactly one club of the society.
iii.) Each club has an odd number of students. In addition, a club with

{2m+1}

students (

m

is a positive integer) is in exactly

m

societies.
Find all possible values of

k

.
Proposed by Guihua Gong, Puerto Rico

4

2

m=4k^2-5

Let

k

be a fixed integer greater than 1, and let

{m=4k^2-5}

. Show that there exist positive integers

a

and

b

such that the sequence

(x_n)

defined by x_0=a, x_1=b, x_{n+2}=x_{n+1}+x_n \text{for} n=0,1,2,\dots, has all of its terms relatively prime to

m

.
Proposed by Jaroslaw Wroblewski, Poland

sum of its entries not exceeding C in absolute value

Consider a matrix of size

n\times n

whose entries are real numbers of absolute value not exceeding

1

. The sum of all entries of the matrix is

0

. Let

n

be an even positive integer. Determine the least number

C

such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding

C

in absolute value.
Proposed by Marcin Kuczma, Poland

5

3

Colombia TST [regular n-gon and angles summing up to 180°]

Let

A_1A_2A_3\ldots A_n

be a regular

n

-gon. Let

B_1

and

B_{n-1}

be the midpoints of its sides

A_1A_2

and

A_{n-1}A_n

. Also, for every

i\in\left\{2,3,4,\ldots ,n-2\right\}

. Let

S

be the point of intersection of the lines

A_1A_{i+1}

and

A_nA_i

, and let

B_i

be the point of intersection of the angle bisector bisector of the angle

\measuredangle A_iSA_{i+1}

with the segment

A_iA_{i+1}

.
Prove that

\sum_{i=1}^{n-1} \measuredangle A_1B_iA_n=180^{\circ}

.
Proposed by Dusan Dukic, Serbia and Montenegro

ab+bc+ca = 1

If

a

,

b

,

c

are three positive real numbers such that

ab+bc+ca = 1

, prove that

\sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}.

Game

A

and

B

play a game, given an integer

N

,

A

writes down

1

first, then every player sees the last number written and if it is

n

then in his turn he writes

n+1

or

2n

, but his number cannot be bigger than

N

. The player who writes

N

wins. For which values of

N

does

B

win?
Proposed by A. Slinko & S. Marshall, New Zealand

3

4

2

4

6

4

2004 IMO Shortlist

Subcontests

variable point on the line BC

triangle with vertices at three of the given points

Most challenging inequality on the ISL 2004

Cyclic quadrilateral geometry in the style of V. Thebault

g(G)^3 <= c * f(G)^4

problem involving the tau function, Australia TST 1

10001 students at an university

m=4k^2-5

sum of its entries not exceeding C in absolute value

Colombia TST [regular n-gon and angles summing up to 180°]

ab+bc+ca = 1

Game

2004 IMO Shortlist

Subcontests

variable point on the line BC

triangle with vertices at three of the given points

Most challenging inequality on the ISL 2004

Cyclic quadrilateral geometry in the style of V. Thebault

g(G)^3 &lt;= c * f(G)^4

problem involving the tau function, Australia TST 1

10001 students at an university

m=4k^2-5

sum of its entries not exceeding C in absolute value

Colombia TST [regular n-gon and angles summing up to 180&deg;]

ab+bc+ca = 1

Game

g(G)^3 <= c * f(G)^4

Colombia TST [regular n-gon and angles summing up to 180°]