MathDB

2

Lamps on a 2014x2014 board

2014^2

squares of a

2014 \times 2014

-board a light bulb is put. Light bulbs can be either on or off. In the starting situation a number of the light bulbs is on. A move consists of choosing a row or column in which at least

1007

light bulbs are on and changing the state of all

2014

light bulbs in this row or column (from on to off or from off to on). Find the smallest non-negative integer

k

such that from each starting situation there is a finite sequence of moves to a situation in which at most

k

light bulbs are on.

Polynomial such that |P(10)-P(0)|<1000

Let

P(x)

be a polynomial of degree

n \le 10

with integral coefficients such that for every

k \in \{1, 2, \dots, 10\}

there is an integer

m

with

P(m) = k

. Furthermore, it is given that

|P(10) - P(0)| < 1000

. Prove that for every integer

k

there is an integer

m

such that

P(m) = k.

4

2

Prove that <EBF=<ECF

Let

\triangle ABC

be a triangle with

|AC|=2|AB|

and let

O

be its circumcenter. Let

D

be the intersection of the bisector of

\angle A

with

BC

. Let

E

be the orthogonal projection of

O

to

AD

and let

F\ne D

be the point on

AD

satisfying

|CD|=|CF|

. Prove that

\angle EBF=\angle ECF

.

Pairs of primes and perfect squares

Determine all pairs

(p, q)

of primes for which

p^{q+1}+q^{p+1}

is a perfect square.

3

2

Prove expression is rational

Let

a

,

b

and

c

be rational numbers for which

a+bc

,

b+ac

and

a+b

are all non-zero and for which we have

\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}.

Prove that

\sqrt{(c-3)(c+1)}

is rational.

Orthocenter of an acute triangle

Let

H

be the orthocentre of an acute triangle

ABC

. The line through

A

perpendicular to

AC

and the line through

B

perpendicular to

BC

intersect in

D

. The circle with centre

C

through

H

intersects the circumcircle of triangle

ABC

in the points

E

and

F

. Prove that

|DE| = |DF| = |AB|

.

2

Prove that AB=CE

Let

\triangle ABC

be a triangle. Let

M

be the midpoint of

BC

and let

D

be a point on the interior of side

AB

. The intersection of

AM

and

CD

is called

E

. Suppose that

|AD|=|DE|

. Prove that

|AB|=|CE|

.

Maximum number elements in A U B

The sets

A

and

B

are subsets of the positive integers. The sum of any two distinct elements of

A

is an element of

B

. The quotient of any two distinct elements of

B

(where we divide the largest by the smallest of the two) is an element of

A

. Determine the maximum number of elements in

A\cup B

.

1

2

Determine all paris (a,b)

Determine all pairs

(a,b)

of positive integers satisfying a^2+b\mid a^2b+a \text{and} b^2-a\mid ab^2+b.

Function with prime divisors

Let

f:\mathbb{Z}_{>0}\rightarrow\mathbb{R}

be a function such that for all

n > 1

there is a prime divisor

p

of

n

such that

f(n)=f\left(\frac{n}{p}\right)-f(p).

Furthermore, it is given that

f(2^{2014})+f(3^{2015})+f(5^{2016})=2013

. Determine

f(2014^2)+f(2015^3)+f(2016^5)

.

2014 Dutch IMO TST

Subcontests

Lamps on a 2014x2014 board

Polynomial such that |P(10)-P(0)|<1000

Prove that <EBF=<ECF

Pairs of primes and perfect squares

Prove expression is rational

Orthocenter of an acute triangle

Prove that AB=CE

Maximum number elements in A U B

Determine all paris (a,b)

Function with prime divisors

2014 Dutch IMO TST

Subcontests

Lamps on a 2014x2014 board

Polynomial such that |P(10)-P(0)|&lt;1000

Prove that &lt;EBF=&lt;ECF

Pairs of primes and perfect squares

Prove expression is rational

Orthocenter of an acute triangle

Prove that AB=CE

Maximum number elements in A U B

Determine all paris (a,b)

Function with prime divisors

Polynomial such that |P(10)-P(0)|<1000

Prove that <EBF=<ECF