MathDB

2

basics of reductio ad absurdum

5

problems, each candidate has solved

3

problems. Among these candidates, for any group of

5

candidates that we might choose, we see that there is a problem which all members of the group had solved it. Prove that there is a problem solved by all candidates.

two colors induction

Let be a number

n\ge 2,

a binary funcion

b:\mathbb{Z}\rightarrow\mathbb{Z}_2,

and

\frac{n^3+5n}{6}

consecutive integers. Show that among these consecutive integers there are

n

of them, namely,

b_1,b_2,\ldots ,b_n,

that have the properties:

\text{(i)} b\left( b_1\right) =b\left( b_2\right) =\cdots =b\left( b_n\right)

\text{(ii)} 1\le b_2-b_1\le b_3-b_2\le \cdots\le b_n-b_{n-1}

3

2

inscriptible quadrilateral method

Let

ABC

be a right angle in

A,

and

M

be the mid of

BC.

On the perpendicular of

AM

through

A

choose a point

D

so that

DM

meets

AB

at a point, namely

P.

Let

E

be the projection of

D

on

BC.

Show that

\angle BPM =\angle EAC.

pentagon geometry

The convex pentagon

ABCDE

has the following properties:

\text{(i)} AB=BC

\text{(ii)} \angle ABE+\angle CBD =\angle DBE

\text{(iii)} \angle AEB +\angle BDC=180^{\circ}

Prove that the orthocenter of

BDE

lies on

AC.

2

angles of 7 and 17 degrees; discrete inequalities

n

lines meet at a point. Each one of the

2n

disjoint angles formed around this point by these lines has either

7^{\circ}

or

17^{\circ} .

a) Find

n.

b) Prove that among these lines there are at least two perpendicular ones.

fermat-like modular number theory

Let be a natural number

a\ge 2.

Prove that for any choice of primes which has the property that none of them divides any of the numbers

N_n=1+a+a^2+a^3+\cdots +a^{2n} ,

with natural

n,

there is another prime not among this choice which doesn't divide any of the numbers

N_n.

1

2

quick perfect square modular arithmetic

a) Show that

n^2+2n+2007

is squarefree for any natural number

n.

b) Prove that for any natural number

k\ge 2

there is a nonnegative integer

m

such that

m^2+2m+2k

is a perfect square.

diophantine cubic product substitution

Solve

x^3-y^3=2xy+7

in integers.

2007 Alexandru Myller

Subcontests

basics of reductio ad absurdum

two colors induction

inscriptible quadrilateral method

pentagon geometry

angles of 7 and 17 degrees; discrete inequalities

fermat-like modular number theory

quick perfect square modular arithmetic

diophantine cubic product substitution