MathDB

1

Number of sequences

n

be a positive integer. Find the number of sequences

a_1,a_2,...,a_k

of different numbers from

\{ 1,2,3,...,n\}

with the following property: for every number

a

of the sequence (except the first one) there exists a previous number

b

such that their difference is

1

(so

a-b= \pm 1

)

7 (C1)

1

Differences in a subset give all possible differences

Prove: From the set

\{1,2,...,n\}

, one can choose a subset with at most

2 \left\lfloor \sqrt n \right\rfloor +1

elements such that the set of the pairwise differences from this subset is

\{1,2,...,n-1\}

. (

\left\lfloor x \right\rfloor

means the greatest integer

\leq x

)

3 (C2)

1

Chain in a turnament

At a turnament between

n

persons, everyone playes exactly one time against everyone else, and at one game there is everytime a winner and a looser. Prove that one can arrange the participants in a chain

P_1 \to P_2 \to ... \to P_n

such that the

i

-th person has won against the

(i+1)

-th person.

10 (C3)

1

Two-element-subsets

Let

n\geq 3

be an integer. Let also

P_1,P_2,...,P_n

be different two-element-subsets of

M=\{1,2,...,n\}

, such that when for

i,j \in M , i\neq j

the sets

P_i,P_j

are not totally disjoint, then there is a

k \in M

with

P_k = \{ i,j\}

. Prove that every element of

M

occurse in exactly

2

of these subsets.

8 (Z2)

1

A²+ab+b²

Prove that if

n

can be written as

n=a^2+ab+b^2

, then also

7n

can be written that way.

4 (Z1)

1

X³+2y³+5z³=0

Solve the equation

x^3+2y^3+5z^3=0

in integers.

11 (Z3)

1

A+b+c divides a²+b²+c²

Let

a,b,c

be positive integers such that

a^2+b^2+c^2

is divisble by

a+b+c

. Prove that at least two of the numbers

a^3,b^3,c^3

leave the same remainder by division through

a+b+c

.

1 (Z4)

1

Sum of five cubes

Prove that every integer can be written as sum of

5

third powers of integers.

12 (U2)

1

Fear this inequality! [(b+c)^2 / (a^2+bc) + ... >= 6]

For any three positive real numbers

a

,

b

,

c

, prove the inequality

\frac{\left(b+c\right)^{2}}{a^{2}+bc}+\frac{\left(c+a\right)^{2}}{b^{2}+ca}+\frac{\left(a+b\right)^{2}}{c^{2}+ab}\geq 6.

Darij

6 (U1)

1

(a-b) (b-c) (c-d) (d-a) + (a-c)^2 (b-d)^2 >= 0

Prove that for any four real numbers

a

,

b

,

c

,

d

, the inequality

\left(a-b\right)\left(b-c\right)\left(c-d\right)\left(d-a\right)+\left(a-c\right)^2\left(b-d\right)^2\geq 0

holds. [hide="comment"]This is inequality (350) in: Mihai Onucu Drimbe, Inegalitati, idei si metode, Zalau: Gil, 2003.
Posted here only for the sake of completeness; in fact, it is more or less the same as http://www.mathlinks.ro/Forum/viewtopic.php?t=3152 . Darij

14 (G4)

1

The pentagram revisited

In the following, the abbreviation

g \cap h

will mean the point of intersection of two lines

g

and

h

. Let

ABCDE

be a convex pentagon. Let

A^{\prime}=BD\cap CE

,

B^{\prime}=CE\cap DA

,

C^{\prime}=DA\cap EB

,

D^{\prime}=EB\cap AC

and

E^{\prime}=AC\cap BD

. Furthermore, let

A^{\prime\prime}=AA^{\prime}\cap EB

,

B^{\prime\prime}=BB^{\prime}\cap AC

,

C^{\prime\prime}=CC^{\prime}\cap BD

,

D^{\prime\prime}=DD^{\prime}\cap CE

and

E^{\prime\prime}=EE^{\prime}\cap DA

. Prove that:

\frac{EA^{\prime\prime}}{A^{\prime\prime}B}\cdot\frac{AB^{\prime\prime}}{B^{\prime\prime}C}\cdot\frac{BC^{\prime\prime}}{C^{\prime\prime}D}\cdot\frac{CD^{\prime\prime}}{D^{\prime\prime}E}\cdot\frac{DE^{\prime\prime}}{E^{\prime\prime}A}=1.

Darij

9 (G3)

1

Standard triangle geometry: BQ bisects CA

Let

ABC

be a triangle with

AB\neq CB

. Let

C^{\prime}

be a point on the ray

[AB

such that

AC^{\prime}=CB

. Let

A^{\prime}

be a point on the ray

[CB

such that

CA^{\prime}=AB

. Let the circumcircles of triangles

ABA^{\prime}

and

CBC^{\prime}

intersect at a point

Q

(apart from

B

). Prove that the line

BQ

bisects the segment

CA

. Darij

2 (G2)

1

Prove that UV || CA

Let

ABC

be a triangle. Let

C^{\prime}

and

A^{\prime}

be the reflections of its vertices

C

and

A

, respectively, in the altitude of triangle

ABC

issuing from

B

. The perpendicular to the line

BA^{\prime}

through the point

C^{\prime}

intersects the line

BC

at

U

; the perpendicular to the line

BC^{\prime}

through the point

A^{\prime}

intersects the line

BA

at

V

. Prove that

UV \parallel CA

. Darij

5 (G1)

1

Feet of altitudes and circumcircle of a triangle ABC

Let

ABC

be a triangle, and let

C^{\prime}

and

A^{\prime}

be the feet of its altitudes issuing from the vertices

C

and

A

, respectively. Denote by

P

the midpoint of the segment

C^{\prime}A^{\prime}

. The circumcircles of triangles

AC^{\prime}P

and

CA^{\prime}P

have a common point apart from

P

; denote this common point by

Q

. Prove that: (a) The point

Q

lies on the circumcircle of the triangle

ABC

. (b) The line

PQ

passes through the point

B

. (c) We have

\frac{AQ}{CQ}=\frac{AB}{CB}

. Darij

2005 QEDMO 1st

Subcontests

Number of sequences

Differences in a subset give all possible differences

Chain in a turnament

Two-element-subsets

A²+ab+b²

X³+2y³+5z³=0

A+b+c divides a²+b²+c²

Sum of five cubes

Fear this inequality! [(b+c)^2 / (a^2+bc) + ... >= 6]

(a-b) (b-c) (c-d) (d-a) + (a-c)^2 (b-d)^2 >= 0

The pentagram revisited

Standard triangle geometry: BQ bisects CA

Prove that UV || CA

Feet of altitudes and circumcircle of a triangle ABC

2005 QEDMO 1st

Subcontests

Number of sequences

Differences in a subset give all possible differences

Chain in a turnament

Two-element-subsets

A²+ab+b²

X³+2y³+5z³=0

A+b+c divides a²+b²+c²

Sum of five cubes

Fear this inequality! [(b+c)^2 / (a^2+bc) + ... &gt;= 6]

(a-b) (b-c) (c-d) (d-a) + (a-c)^2 (b-d)^2 &gt;= 0

The pentagram revisited

Standard triangle geometry: BQ bisects CA

Prove that UV || CA

Feet of altitudes and circumcircle of a triangle ABC

Fear this inequality! [(b+c)^2 / (a^2+bc) + ... >= 6]

(a-b) (b-c) (c-d) (d-a) + (a-c)^2 (b-d)^2 >= 0