MathDB

1

a,b greatest pos. integer such 2^a | (5^n - 3^n), 2^b <= n, prove a<=b+3

n \ge 2

be an integer. Let

a

be the greatest positive integer such that

2^a | 5^n - 3^n

. Let

b

be the greatest positive integer such that

2^b \le n

. Prove that

a \le b + 3

.

3

1

x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z| equation in R^3

Find all triples

(x, y, z)

of real numbers that satisfy

x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|

.

1

all colourings of all pos.ntegers either red or green under 3 conditions

All positive integers are coloured either red or green, such that the following conditions are satisfied: - There are equally many red as green integers. - The sum of three (not necessarily distinct) red integers is red. - The sum of three (not necessarily distinct) green integers is green. Find all colourings that satisfy these conditions.

5

1

|AB|= |CD| or |AB| +|CD| =|AD| in a trapezoid

A trapezoid

ABCD

is given with

BC // AD

. Assume that the bisectors of the angles

BAD

and

CDA

intersect on the perpendicular bisector of the line segment

BC

. Prove that

|AB|= |CD|

or

|AB| +|CD| =|AD|

.

2

1

CE is the bisector of <BCD, circumcircle related

In an acute triangle

ABC

the angle

\angle C

is greater than

\angle A

. Let

E

be such that

AE

is a diameter of the circumscribed circle

\Gamma

of \vartriangle ABC. Let

K

be the intersection of

AC

and the tangent line at

B

to

\Gamma

. Let

L

be the orthogonal projection of

K

on

AE

and let

D

be the intersection of

KL

and

AB

. Prove that

CE

is the bisector of

\angle BCD

.

2011 Dutch BxMO TST

Subcontests

a,b greatest pos. integer such 2^a | (5^n - 3^n), 2^b <= n, prove a<=b+3

x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z| equation in R^3

all colourings of all pos.ntegers either red or green under 3 conditions

|AB|= |CD| or |AB| +|CD| =|AD| in a trapezoid

CE is the bisector of <BCD, circumcircle related

2011 Dutch BxMO TST

Subcontests

a,b greatest pos. integer such 2^a | (5^n - 3^n), 2^b &lt;= n, prove a&lt;=b+3

x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z| equation in R^3

all colourings of all pos.ntegers either red or green under 3 conditions

|AB|= |CD| or |AB| +|CD| =|AD| in a trapezoid

CE is the bisector of &lt;BCD, circumcircle related

a,b greatest pos. integer such 2^a | (5^n - 3^n), 2^b <= n, prove a<=b+3

CE is the bisector of <BCD, circumcircle related