MathDB

2

exists triangle of medians (15th -a QEDMO p8 19. - 22. 10. 2017)

ABC

be a triangle of area

1

with medians

s_a, s_b,s_c

. Show that there is a triangle whose sides are the same length as

s_a, s_b

, and

s_c

, and determine its area.

f (-42)=? if f (x^2 + yf (z)) = xf (x) + zf (y) , (2017)> 0

For a function

f: R\to R

,

f (2017)> 0

as well as

f (x^2 + yf (z)) = xf (x) + zf (y)

for all

x,y,z \in R

is known. What is the value of

f (-42)

?

9

2

n integers in a circle, a_1 + a_2 +...+ a_k <= 2k -1

Iskandar arranged

n \in N

integer numbers in a circle, the sum of which is

2n-1

. Crescentia now selects one of these numbers and name the given numbers in clockwise direction with

a_1,a_2,...., a_n

. Show that she can choose the starting number such that for all

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

the inequality

a_1 + a_2 +...+ a_k \le 2k -1

holds.

n is prime if 2^{n-1}-1 and not 2^h-1 is divisible by n, where n = ph + 1

Let

p

be a prime number and

h

be a natural number smaller than

p

. We set

n = ph + 1

. Prove that if

2^{n-1}-1

, but not

2^h-1

, is divisible by

n

, then

n

is a prime number.

11

2

[(2^1+3^1)...(2^{2048}+3^{2048})+2^{4096}]/3^{4096}

Calculate

\frac{(2^1+3^1)(2^2+3^2)(2^4+3^4)(2^8+3^8)...(2^{2048}+3^{2048})+2^{4096}}{3^{4096}}

f (xy) = f (x) f (y) , f (x) =x^{-1} for more than 3/4 of members of finite gro

Let

G

be a finite group and

f: G \to G

a map, such that

f (xy) = f (x) f (y)

for all

x, y \in G

and

f (x) = x^{-1}

for more than

\frac34

of all

x \in G

is fulfilled. Show that

f (x) =x^{-1}

even holds for all

x \in G

holds.

12

2

2 octahedra with sides 1-16 such each side has same probability with sum

Jorn wants to cheat at the role play: he intends to cheat the sides to re-label its two octahedra, so that each of the numbers from

1

to

16

has the same probability as the sum of the dice occurs. So that the game master does not notice this so easily, he only wants to use numbers from

0

to

8

, if necessary several times or not at all. Is this possible?

for which n is a^n + 1/a^n integer if (a +1/a )^2=11 ?

Let

a

be a real number such that

\left(a + \frac{1}{a}\right)^2=11

. For which

n\in N

is

a^n + \frac{1}{a^n}

an integer? Does this depend on the exact value of

a

?

10

2

q is divisor of 2^{q-1}- 1 if q = \frac{4^p-1}{3}

Let

p> 3

be a prime number and let

q = \frac{4^p-1}{3}

. Show that

q

is a composite integer as well is a divisor of

2^{q-1}- 1

.

inradius of P A_j A_{j+k} independent of P, if inradii of P A_iA_{i+1} are equal

Let

\ell

be a straight line and

P \notin \ell

be a point in the plane. On

\ell

are, in this arrangement, points

A_1, A_2,...

such that the radii of the incircles of all triangles

P A_iA_{i + 1}

are equal. Let

k \in N

. Show that the radius of the incircle of the triangle

P A_j A_{j + k}

does not depend on the choice of

j \in N

.

5

2

s^n (F) = F U {a + n|a \in F }

Let

F

be a finite subset of the integer numbers. We define a new subset

s(F)

in that

a\in Z

lies in

s (F)

if and only if exactly one of the numbers

a

and

a -1

in

F

. In the same way one gets from

s (F)

the set

s^2(F) = s (s (F))

and by

n

-fold application of

s

then iteratively further subsets

s^n (F)

. Prove there are infinitely many natural numbers

n

for which

s^n (F) = F\cup \{a + n|a \in F\}

.

f (x) = x^n + x^{n-1} +...+ x + 1= g (h (x))

For which natural numbers

n

can the polynomial

f (x) = x^n + x^{n-1} +...+ x + 1

as write

f (x) = g (h (x))

, where

g

and

h

should be real polynomials of degrees greater than

1

?

7

2

x +(3x-y)/(x^2 + y^2) = 3, y - (x + 3y/(x^2 + y^2) = 0

Find all real solutions

x, y

of the system of equations

\begin{cases} x + \dfrac{3x-y}{x^2 + y^2} = 3 \\ \\ y-\dfrac{x + 3y}{x^2 + y^2} = 0 \end{cases}

integer matrix BA^kB^{-1} , det (A) = 1, det (B) \ne 0

Let

A, B

be integer

n\times n

-matrices, where

det (A) = 1

and

det (B) \ne 0

. Show that there is a

k \in N

, for which

BA^kB^{-1}

is a matrix with integer entries.

6

1

x^3 + y^3 = 3xy

Find all integers

x,y

satisfy the

x^3 + y^3 = 3xy

.

4

2

a^3 +1/a^3=? and a^5 +1/a^5 =? if (a+1/a)^2=11

Let

a

be a real number such that

\left(a + \frac{1}{a}\right)^2=11

. What possible values can

a^3 + \frac{1}{a^3}

and

a^5 + \frac{1}{a^5}

take?

image f ([a, b]) is interval of length b - a.

Find all functions

f: R \to R

for which the image

f ([a, b])

for all real

a \le b

is (not necessarily closed!) interval of length

b - a

.

3

1

ab, bc, ca, ab + bc + ca perfect squares if a^2+b^2+c^2=(a-b)^2+(b-c)^2+(c-a)^2

Let

a,b,c

natural numbers for which

a^2 + b^2 + c^2 = (a-b) ^2 + (b-c)^ 2 + (c-a) ^2

. Show that

ab, bc, ca

and

ab + bc + ca

are perfect squares .

2

4 colours in a 41x5 chessboard

Markers in the colors violet, cyan, octarine and gamma were placed on all fields of a

41\times 5

chessboard. Show that there are four squares of the same color that form the vertices of a rectangle whose edges are parallel to those of the board.

AXB = BXA if AXB + A + B = 0 for matrices

Let

A, B, X

be real

n\times n

matrices for which

AXB + A + B = 0

holds. Prove that

AXB = BXA

.

1

x^4-10y^4 + 3z^6 = 21

Find all integers

x, y, z

satisfy the

x^4-10y^4 + 3z^6 = 21

.

2017 QEDMO 15th

Subcontests

exists triangle of medians (15th -a QEDMO p8 19. - 22. 10. 2017)

f (-42)=? if f (x^2 + yf (z)) = xf (x) + zf (y) , (2017)> 0

n integers in a circle, a_1 + a_2 +...+ a_k <= 2k -1

n is prime if 2^{n-1}-1 and not 2^h-1 is divisible by n, where n = ph + 1

[(2^1+3^1)...(2^{2048}+3^{2048})+2^{4096}]/3^{4096}

f (xy) = f (x) f (y) , f (x) =x^{-1} for more than 3/4 of members of finite gro

2 octahedra with sides 1-16 such each side has same probability with sum

for which n is a^n + 1/a^n integer if (a +1/a )^2=11 ?

q is divisor of 2^{q-1}- 1 if q = \frac{4^p-1}{3}

inradius of P A_j A_{j+k} independent of P, if inradii of P A_iA_{i+1} are equal

s^n (F) = F U {a + n|a \in F }

f (x) = x^n + x^{n-1} +...+ x + 1= g (h (x))

x +(3x-y)/(x^2 + y^2) = 3, y - (x + 3y/(x^2 + y^2) = 0

integer matrix BA^kB^{-1} , det (A) = 1, det (B) \ne 0

x^3 + y^3 = 3xy

a^3 +1/a^3=? and a^5 +1/a^5 =? if (a+1/a)^2=11

image f ([a, b]) is interval of length b - a.

ab, bc, ca, ab + bc + ca perfect squares if a^2+b^2+c^2=(a-b)^2+(b-c)^2+(c-a)^2

4 colours in a 41x5 chessboard

AXB = BXA if AXB + A + B = 0 for matrices

x^4-10y^4 + 3z^6 = 21

2017 QEDMO 15th

Subcontests

exists triangle of medians (15th -a QEDMO p8 19. - 22. 10. 2017)

f (-42)=? if f (x^2 + yf (z)) = xf (x) + zf (y) , (2017)&gt; 0

n integers in a circle, a_1 + a_2 +...+ a_k &lt;= 2k -1

n is prime if 2^{n-1}-1 and not 2^h-1 is divisible by n, where n = ph + 1

[(2^1+3^1)...(2^{2048}+3^{2048})+2^{4096}]/3^{4096}

f (xy) = f (x) f (y) , f (x) =x^{-1} for more than 3/4 of members of finite gro

2 octahedra with sides 1-16 such each side has same probability with sum

for which n is a^n + 1/a^n integer if (a +1/a )^2=11 ?

q is divisor of 2^{q-1}- 1 if q = \frac{4^p-1}{3}

inradius of P A_j A_{j+k} independent of P, if inradii of P A_iA_{i+1} are equal

s^n (F) = F U {a + n|a \in F }

f (x) = x^n + x^{n-1} +...+ x + 1= g (h (x))

x +(3x-y)/(x^2 + y^2) = 3, y - (x + 3y/(x^2 + y^2) = 0

integer matrix BA^kB^{-1} , det (A) = 1, det (B) \ne 0

x^3 + y^3 = 3xy

a^3 +1/a^3=? and a^5 +1/a^5 =? if (a+1/a)^2=11

image f ([a, b]) is interval of length b - a.

ab, bc, ca, ab + bc + ca perfect squares if a^2+b^2+c^2=(a-b)^2+(b-c)^2+(c-a)^2

4 colours in a 41x5 chessboard

AXB = BXA if AXB + A + B = 0 for matrices

x^4-10y^4 + 3z^6 = 21

f (-42)=? if f (x^2 + yf (z)) = xf (x) + zf (y) , (2017)> 0

n integers in a circle, a_1 + a_2 +...+ a_k <= 2k -1