MathDB

1

a_{n+1}=(a_{n}^{2}+a_{n-1}^{2})/ a_{n-2} sequence of integer terms

\left( a_{n}\right) _{n\in\mathbb{N}}

by

a_{1}=a_{2}=a_{3}=1

and

a_{n+1}=\dfrac{a_{n}^{2}+a_{n-1}^{2}}{a_{n-2}}

for every integer

n\geq3

. Show that all elements

a_{i}

of this sequence are integers.
(L. J. Mordell and apparently Dana Scott, see also http://oeis.org/A064098)

12

1

Per and Kari each have n pieces of paper, they write numbers 1 to 2n

Per and Kari each have

n

pieces of paper. They both write down the numbers from

1

to

2n

in an arbitrary order, one number on each side. Afterwards, they place the pieces of paper on a table showing one side. Prove that they can always place them so that all the numbers from

1

to

2n

are visible at once.

8

1

f(1) = 1 , f(1)+ f(2)+...+ f(n) = n^2f(n), f(1995) =?

Let a function

f

satisfy

f(1) = 1

and

f(1)+ f(2)+...+ f(n) = n^2f(n)

for all

n \in N

. Determine

f(1995)

.

3

1

function

Find all functions

f: \mathbb{R} \rightarrow \mathbb{R}

such that for all real numbers

x,y

: x f(x)\minus{}yf(y)\equal{}(x\minus{}y)f(x\plus{}y).

5

1

Coprime to infinetely many lines of Pascal's triangle

Find all positive integers

n

such that there are

\infty

many lines of Pascal's triangle that have entries coprime to

n

only. In other words: such that there are

\infty

many

k

with the property that the numbers

\binom{k}{0},\binom{k}{1},\binom{k}{2},...,\binom{k}{k}

are all coprime to

n

.

11

1

Sorry for this one: (a^2+2b^2)/(b+c) + ... >= 3/2 (a+b+c)

Guess I should stop proposing problems at 2:00 AM, as this can lead to ones like this here: Let

a

,

b

,

c

be three positive reals. Prove the inequality

\frac{a^2+2b^2}{b+c}+\frac{b^2+2c^2}{c+a}+\frac{c^2+2a^2}{a+b}\geq\frac32\left(a+b+c\right)

.

9

1

Projections Y and Y' are symmetric with respect to midparall

Let

ABC

be a triangle, and

C^{\prime}

and

A^{\prime}

the midpoints of its sides

AB

and

BC

. Consider two lines

g

and

g^{\prime}

which both pass through the vertex

A

and are symmetric to each other with respect to the angle bisector of the angle

CAB

. Further, let

Y

and

Y^{\prime}

be the orthogonal projections of the point

B

on these lines

g

and

g^{\prime}

. Show that the points

Y

and

Y^{\prime}

are symmetric to each other with respect to the line

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

.

6

1

Reflections of incircle tangency points with respect to exte

The incircle of a triangle

ABC

touches its sides

BC

,

CA

,

AB

at the points

X

,

Y

,

Z

, respectively. Let

X^{\prime}

,

Y^{\prime}

,

Z^{\prime}

be the reflections of these points

X

,

Y

,

Z

in the external angle bisectors of the angles

CAB

,

ABC

,

BCA

, respectively. Show that

Y^{\prime}Z^{\prime}\parallel BC

,

Z^{\prime}X^{\prime}\parallel CA

and

X^{\prime}Y^{\prime}\parallel AB

.

2

1

Abc divides (a+b+c)^(n^2+n+1)

Let

a

,

b

,

c

and

n

be positive integers such that

a^n

is divisible by

b

, such that

b^n

is divisible by

c

, and such that

c^n

is divisible by

a

. Prove that \left(a \plus{} b \plus{} c\right)^{n^2 \plus{} n \plus{} 1} is divisible by

abc

. An even broader generalization, though not part of the QEDMO problem and not quite number theory either: If

u

and

n

are positive integers, and

a_1

,

a_2

, ...,

a_u

are integers such that

a_i^n

is divisible by a_{i \plus{} 1} for every

i

such that

1\leq i\leq u

(we set a_{u \plus{} 1} \equal{} a_1 here), then show that \left(a_1 \plus{} a_2 \plus{} ... \plus{} a_u\right)^{n^{u \minus{} 1} \plus{} n^{u \minus{} 2} \plus{} ... \plus{} n \plus{} 1} is divisible by

a_1a_2...a_u

.

1

|EAC| * |EBD| = |EAB| * |ECD| + |EBC| * |EDA|

Peter is a pentacrat and spends his time drawing pentagrams. With the abbreviation

\left|XYZ\right|

for the area of an arbitrary triangle

XYZ

, he notes that any convex pentagon

ABCDE

satisfies the equality

\left|EAC\right|\cdot\left|EBD\right|=\left|EAB\right|\cdot\left|ECD\right|+\left|EBC\right|\cdot\left|EDA\right|

. Guess what you are supposed to do and do it.

4

1

not hard

Among the points corresponding to number

1,2,...,2n

on the real line,

n

are colored in blue and

n

in red. Let

a_1,a_2,...,a_n

be the blue points and

b_1,b_2,...,b_n

be the red points. Prove that the sum

\mid a_1-b_1\mid+...+\mid a_n-b_n\mid

does not depend on coloring , and compute its value. :roll:

7

1

Not easy

Given a table with

2^n * n

1*1 squares (

2^n

rows and n column). In any square we put a number in {1, -1} such that no two rows are the same. Then we change numbers in some squares by 0. Prove that in new table we can choose some rows such that sum of all numbers in these rows equal to 0.

2006 QEDMO 3rd

Subcontests

a_{n+1}=(a_{n}^{2}+a_{n-1}^{2})/ a_{n-2} sequence of integer terms

Per and Kari each have n pieces of paper, they write numbers 1 to 2n

f(1) = 1 , f(1)+ f(2)+...+ f(n) = n^2f(n), f(1995) =?

function

Coprime to infinetely many lines of Pascal's triangle

Sorry for this one: (a^2+2b^2)/(b+c) + ... >= 3/2 (a+b+c)

Projections Y and Y' are symmetric with respect to midparall

Reflections of incircle tangency points with respect to exte

Abc divides (a+b+c)^(n^2+n+1)

|EAC| * |EBD| = |EAB| * |ECD| + |EBC| * |EDA|

not hard

Not easy

2006 QEDMO 3rd

Subcontests

a_{n+1}=(a_{n}^{2}+a_{n-1}^{2})/ a_{n-2} sequence of integer terms

Per and Kari each have n pieces of paper, they write numbers 1 to 2n

f(1) = 1 , f(1)+ f(2)+...+ f(n) = n^2f(n), f(1995) =?

function

Coprime to infinetely many lines of Pascal's triangle

Sorry for this one: (a^2+2b^2)/(b+c) + ... &gt;= 3/2 (a+b+c)

Projections Y and Y' are symmetric with respect to midparall

Reflections of incircle tangency points with respect to exte

Abc divides (a+b+c)^(n^2+n+1)

|EAC| * |EBD| = |EAB| * |ECD| + |EBC| * |EDA|

not hard

Not easy

Sorry for this one: (a^2+2b^2)/(b+c) + ... >= 3/2 (a+b+c)