MathDB

1

a=b+c+d-2\sqrt{bc+cd+db} if d=a+b+c+2\sqrt{ab+bc+ca}

a,

b,

c,

d

be four positive reals such that

d=a+b+c+2\sqrt{ab+bc+ca}.

Prove that

a=b+c+d-2\sqrt{bc+cd+db}.

Darij Grinberg

7

1

in 20 people, each person sends a letter to 10 of the others

In a group of

20

people, each person sends a letter to

10

of the others. Prove that there are two persons who send a letter to each other.

4

1

Seven permutations of Z/(nZ) imply that n is coprime to 6

Let

n

be a positive integer, and let

\left( a_{1},\ a_{2} ,\ ...,\ a_{n}\right)

,

\left( b_{1},\ b_{2},\ ...,\ b_{n}\right)

and

\left( c_{1},\ c_{2},\ ...,\ c_{n}\right)

be three sequences of integers such that for any two distinct numbers

i

and

j

from the set

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

, none of the seven integers a_{i}\minus{}a_{j}; \left( b_{i}\plus{}c_{i}\right) \minus{}\left( b_{j}\plus{}c_{j}\right); b_{i}\minus{}b_{j}; \left( c_{i}\plus{}a_{i}\right) \minus{}\left( c_{j}\plus{}a_{j}\right); c_{i}\minus{}c_{j}; \left( a_{i}\plus{}b_{i}\right) \minus{}\left( a_{j}\plus{}b_{j}\right); \left( a_{i}\plus{}b_{i}\plus{}c_{i}\right) \minus{}\left( a_{j}\plus{}b_{j}\plus{}c_{j}\right) is divisible by

n

. Prove that: a) The number

n

is odd. b) The number

n

is not divisible by

3

. [hide="Source of the problem"]Source of the problem: This question is a generalization of one direction of Theorem 2.1 in: Dean Alvis, Michael Kinyon, Birkhoff's Theorem for Panstochastic Matrices, American Mathematical Monthly, 1/2001 (Vol. 108), pp. 28-37. The original Theorem 2.1 is obtained if you require b_{i}\equal{}i and c_{i}\equal{}\minus{}i for all

i

, and add in a converse stating that such sequences

\left( a_{1},\ a_{2},\ ...,\ a_{n}\right)

,

\left( b_{1},\ b_{2},\ ...,\ b_{n}\right)

and

\left( c_{1} ,\ c_{2},\ ...,\ c_{n}\right)

indeed exist if

n

is odd and not divisible by

3

.

1

Easy number theory: 3 coprime pairs yield infinitely many

Let

a

,

b

and

k

be three positive integers. We define two sequences

\left( a_{n}\right)

and

\left( b_{n}\right)

by the starting values a_{1}\equal{}a and b_{1}\equal{}b and the recurrent equations a_{n\plus{}1}\equal{}ka_{n}\plus{}b_{n} and b_{n\plus{}1}\equal{}kb_{n}\plus{}a_{n} for each positive integer

n

. Prove that if

a_{1}\perp b_{1}

,

a_{2}\perp b_{2}

and

a_{3}\perp b_{3}

hold, then

a_{n}\perp b_{n}

holds for every positive integer

n

. Here, the abbreviation

x\perp y

stands for "the numbers

x

and

y

are coprime".

2

1

Two opposite angles of a quadrilateral equal

Let

ABCD

be a (not self-intersecting) quadrilateral satisfying \measuredangle DAB \equal{} \measuredangle BCD\neq 90^{\circ}. Let

X

and

Y

be the orthogonal projections of the point

D

on the lines

AB

and

BC

, and let

Z

and

W

be the orthogonal projections of the point

B

on the lines

CD

and

DA

. Establish the following facts: a) The quadrilateral

XYZW

is an isosceles trapezoid such that

XY\parallel ZW

. b) Let

M

be the midpoint of the segment

AC

. Then, the lines

XZ

and

YW

pass through the point

M

. c) Let

N

be the midpoint of the segment

BD

, and let

X^{\prime}

,

Y^{\prime}

,

Z^{\prime}

,

W^{\prime}

be the midpoints of the segments

AB

,

BC

,

CD

,

DA

. Then, the point

M

lies on the circumcircles of the triangles

W^{\prime}X^{\prime}N

and

Y^{\prime}Z^{\prime}N

. [hide="Notice"]Notice. This problem has been discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=172417 .

5

1

a = yz' - zy', b = zx' - xz', c = xy' - yx'

Let

a

,

b

,

c

be three integers. Prove that there exist six integers

x

,

y

,

z

,

x^{\prime}

,

y^{\prime}

,

z^{\prime}

such that a\equal{}yz^{\prime}\minus{}zy^{\prime};\ \ \ \ \ \ \ \ \ \ b\equal{}zx^{\prime}\minus{}xz^{\prime};\ \ \ \ \ \ \ \ \ \ c\equal{}xy^{\prime}\minus{}yx^{\prime}.

8

1

"Triple-ratios" are also projectively invariant

Let

A

,

B

,

C

,

A^{\prime}

,

B^{\prime}

,

C^{\prime}

,

X

,

Y

,

Z

,

X^{\prime}

,

Y^{\prime}

,

Z^{\prime}

and

P

be pairwise distinct points in space such that

A^{\prime} \in BC;\ B^{\prime}\in CA;\ C^{\prime}\in AB;\ X^{\prime}\in YZ;\ Y^{\prime}\in ZX;\ Z^{\prime}\in XY;

P \in AX;\ P\in BY;\ P\in CZ;\ P\in A^{\prime}X^{\prime};\ P\in B^{\prime}Y^{\prime};\ P\in C^{\prime}Z^{\prime}

. Prove that \frac {BA^{\prime}}{A^{\prime}C}\cdot\frac {CB^{\prime}}{B^{\prime}A}\cdot\frac {AC^{\prime}}{C^{\prime}B} \equal{} \frac {YX^{\prime}}{X^{\prime}Z}\cdot\frac {ZY^{\prime}}{Y^{\prime}X}\cdot\frac {XZ^{\prime}}{Z^{\prime}Y}.

6

1

functional equation

Find all functions

f: \mathbb{R}\to\mathbb{R}

that satisfy the equation: f\left(\left(f\left(x\right)\right)^2 \plus{} f\left(y\right)\right) \equal{} xf\left(x\right) \plus{} y for any two real numbers

x

and

y

.

2007 QEDMO 5th

Subcontests

a=b+c+d-2\sqrt{bc+cd+db} if d=a+b+c+2\sqrt{ab+bc+ca}

in 20 people, each person sends a letter to 10 of the others

Seven permutations of Z/(nZ) imply that n is coprime to 6

Easy number theory: 3 coprime pairs yield infinitely many

Two opposite angles of a quadrilateral equal

a = yz' - zy', b = zx' - xz', c = xy' - yx'

"Triple-ratios" are also projectively invariant

functional equation

2007 QEDMO 5th

Subcontests

a=b+c+d-2\sqrt{bc+cd+db} if d=a+b+c+2\sqrt{ab+bc+ca}

in 20 people, each person sends a letter to 10 of the others

Seven permutations of Z/(nZ) imply that n is coprime to 6

Easy number theory: 3 coprime pairs yield infinitely many

Two opposite angles of a quadrilateral equal

a = yz' - zy', b = zx' - xz', c = xy' - yx'

&quot;Triple-ratios&quot; are also projectively invariant

functional equation

"Triple-ratios" are also projectively invariant