MathDB

1

diophantine x^3 + 3x + 14 = 2 p^n , where p is prime

(x,n,p)

of positive integers

x

and

n

and primes

p

for which the following holds

x^3 + 3x + 14 = 2 p^n

2

1

triple (x, y, z) replaced by (y+z- x, z+x- y,x +y -z), a+b+c=2013, 10 steps

Consider a triple

(a, b, c)

of pairwise distinct positive integers satisfying

a + b + c = 2013

. A step consists of replacing the triple

(x, y, z)

by the triple

(y + z - x,z + x - y,x + y - z)

. Prove that, starting from the given triple

(a, b,c)

, after

10

steps we obtain a triple containing at least one negative number.

5

1

(AN)(NC) =(CD)(BN) in a cyclic ABCD with (AD)=(BD)

Let

ABCD

be a cyclic quadrilateral for which

|AD| =|BD|

. Let

M

be the intersection of

AC

and

BD

. Let

I

be the incentre of

\triangle BCM

. Let

N

be the second intersection pointof

AC

and the circumscribed circle of

\triangle BMI

. Prove that

|AN| \cdot |NC| = |CD | \cdot |BN|

.

1

equal areas of triangles in ABCD leads to parallel lines

In quadrilateral

ABCD

the sides

AB

and

CD

are parallel. Let

M

be the midpoint of diagonal

AC

. Suppose that triangles

ABM

and

ACD

have equal area. Prove that

DM // BC

.

4

1

Functional equation.

Determine all functions

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

satisfying

f(x+yf(x))=f(xf(y))-x+f(y+f(x))

2013 Dutch BxMO/EGMO TST

Subcontests

diophantine x^3 + 3x + 14 = 2 p^n , where p is prime

triple (x, y, z) replaced by (y+z- x, z+x- y,x +y -z), a+b+c=2013, 10 steps

(AN)(NC) =(CD)(BN) in a cyclic ABCD with (AD)=(BD)

equal areas of triangles in ABCD leads to parallel lines

Functional equation.