MathDB

1

3 right isosceles triangles on sides of a triangle, parallelogram wanted

ABC

, isosceles right-angled triangles

AUB, CVB

, and

AWC

are placed. These three triangles have their right angles at vertices

U, V

, and

W

, respectively. Triangle

AUB

lies completely inside triangle

ABC

and triangles

CVB

and

AWC

lie completely outside

ABC

. See the figure. Prove that quadrilateral

UVCW

is a parallelogram. [asy] import markers;
unitsize(1.5 cm);
pair A, B, C, U, V, W;
A = (0,0); B = (2,0); C = (1.7,2.5); U = (B + rotate(90,A)*(B))/2; V = (B + rotate(90,C)*(B))/2; W = (C + rotate(90,A)*(C))/2;
draw(A--B--C--cycle); draw(A--W, StickIntervalMarker(1,1,size=2mm)); draw(C--W, StickIntervalMarker(1,1,size=2mm)); draw(B--V, StickIntervalMarker(1,2,size=2mm)); draw(C--V, StickIntervalMarker(1,2,size=2mm)); draw(A--U, StickIntervalMarker(1,3,size=2mm)); draw(B--U, StickIntervalMarker(1,3,size=2mm)); draw(rightanglemark(A,U,B,5)); draw(rightanglemark(B,V,C,5)); draw(rightanglemark(A,W,C,5));
dot("

A

", A, S); dot("

B

", B, S); dot("

C

", C, N); dot("

U

", U, NE); dot("

V

", V, NE); dot("

W

", W, NW); [/asy]

2 juniors

1

isosceles wanted, parallelogram and equal segments given, for juniors

Let

ABCD

be a parallelogram with an acute angle at

A

. Let

G

be a point on the line

AB

, distinct from

B

, such that

|CG| = |CB|

. Let

H

be a point on the line

BC

, distinct from

B

, such that

|AB| =|AH|

. Prove that triangle

DGH

is isosceles.
[asy] unitsize(1.5 cm);
pair A, B, C, D, G, H;
A = (0,0); B = (2,0); D = (0.5,1.5); C = B + D - A; G = reflect(A,B)*(C) + C - B; H = reflect(B,C)*(H) + A - B;
draw(H--A--D--C--G); draw(interp(A,G,-0.1)--interp(A,G,1.1)); draw(interp(C,H,-0.1)--interp(C,H,1.1)); draw(D--G--H--cycle, dashed);
dot("

A

", A, SW); dot("

B

", B, SE); dot("

C

", C, E); dot("

D

", D, NW); dot("

G

", G, NE); dot("

H

", H, SE); [/asy]

5

1

divide a 1 by 1 square into rectangles with the same circumference

We consider the ways to divide a

1

by

1

square into rectangles (of which the sides are parallel to those of the square). All rectangles must have the same circumference, but not necessarily the same shape. a) Is it possible to divide the square into 20 rectangles, each having a circumference of

2:5

? b) Is it possible to divide the square into 30 rectangles, each having a circumference of

2

?

4

1

Leiden quadruple (p, a, b, c) with p + 2 = (a+b+c)/3

A quadruple

(p, a, b, c)

of positive integers is called a Leiden quadruple if -

p

is an odd prime number, -

a, b

, and

c

are distinct and -

ab + 1, bc + 1

and

ca + 1

are divisible by

p

. a) Prove that for every Leiden quadruple

(p, a, b, c)

we have

p + 2 \le \frac{a+b+c}{3}

. b) Determine all numbers

p

for which a Leiden quadruple

(p, a, b, c)

exists with

p + 2 = \frac{a+b+c}{3}

3

1

no of teams at a volleyball tournament can be 7, but cannot be 6

At a volleyball tournament, each team plays exactly once against each other team. Each game has a winning team, which gets

1

point. The losing team gets

0

points. Draws do not occur. In the nal ranking, only one team turns out to have the least number of points (so there is no shared last place). Moreover, each team, except for the team having the least number of points, lost exactly one game against a team that got less points in the final ranking. a) Prove that the number of teams cannot be equal to

6

. b) Show, by providing an example, that the number of teams could be equal to

7

.

1

diophantine abc = 2(a + b + c) with 0<a <=b <=c

Determine all triples

(a,b,c)

, where

a, b

, and

c

are positive integers that satisfy

a \le b \le c

and

abc = 2(a + b + c)

.

2014 Dutch Mathematical Olympiad

Subcontests

3 right isosceles triangles on sides of a triangle, parallelogram wanted

isosceles wanted, parallelogram and equal segments given, for juniors

divide a 1 by 1 square into rectangles with the same circumference

Leiden quadruple (p, a, b, c) with p + 2 = (a+b+c)/3

no of teams at a volleyball tournament can be 7, but cannot be 6

diophantine abc = 2(a + b + c) with 0<a <=b <=c

2014 Dutch Mathematical Olympiad

Subcontests

3 right isosceles triangles on sides of a triangle, parallelogram wanted

isosceles wanted, parallelogram and equal segments given, for juniors

divide a 1 by 1 square into rectangles with the same circumference

Leiden quadruple (p, a, b, c) with p + 2 = (a+b+c)/3

no of teams at a volleyball tournament can be 7, but cannot be 6

diophantine abc = 2(a + b + c) with 0&lt;a &lt;=b &lt;=c

diophantine abc = 2(a + b + c) with 0<a <=b <=c