MathDB

1

angle between 2 equilateral triangles on the same side of line is 60^o

A, B

, and

C

are on a line in this order. Points

D

and

E

lie on the same side of this line, in such a way that triangles

ABD

and

BCE

are equilateral. The segments

AE

and

CD

intersect in point

S

. Prove that

\angle ASD = 60^o

.
[asy] unitsize(1.5 cm);
pair A, B, C, D, E, S;
A = (0,0); B = (1,0); C = (2.5,0); D = dir(60); E = B + 1.5*dir(60); S = extension(C,D,A,E);
fill(A--B--D--cycle, gray(0.8)); fill(B--C--E--cycle, gray(0.8)); draw(interp(A,C,-0.1)--interp(A,C,1.1)); draw(A--D--B--E--C); draw(A--E); draw(C--D); draw(anglemark(D,S,A,5));
dot("

A

", A, dir(270)); dot("

B

", B, dir(270)); dot("

C

", C, dir(270)); dot("

D

", D, N); dot("

E

", E, N); dot("

S

", S, N); [/asy]

3 juniors

1

equal segments inside a quadr. with 2 parallel sides, angle bisectors related

In quadrilateral

ABCD

sides

BC

and

AD

are parallel. In each of the four vertices we draw an angular bisector. The angular bisectors of angles

A

and

B

intersect in point

P

, those of angles

B

and

C

intersect in point

Q

, those of angles

C

and

D

intersect in point

R

, and those of angles

D

and

A

intersect in point S. Suppose that

PS

is parallel to

QR

. Prove that

|AB| =|CD|

.
[asy] unitsize(1.2 cm);
pair A, B, C, D, P, Q, R, S;
A = (0,0); D = (3,0); B = (0.8,1.5); C = (3.2,1.5); S = extension(A, incenter(A,B,D), D, incenter(A,C,D)); Q = extension(B, incenter(A,B,C), C, C + incenter(A,B,D) - A); P = extension(A, S, B, Q); R = extension(D, S, C, Q);
draw(A--D--C--B--cycle); draw(B--Q--C); draw(A--S--D);
dot("

A

", A, SW); dot("

B

", B, NW); dot("

C

", C, NE); dot("

D

", D, SE); dot("

P

", P, dir(90)); dot("

Q

", Q, dir(270)); dot("

R

", R, dir(90)); dot("

S

", S, dir(90)); [/asy]
Attention: the figure is not drawn to scale.

5

1

|a - b| >= |c| , |b - c| >= |a| and |c - a| >= |b| => one is sum of other two

Given are (not necessarily positive) real numbers

a, b

, and

c

for which

|a - b| \ge |c| , |b - c| \ge |a|

and

|c - a| \ge |b|

. Prove that one of the numbers

a, b

, and

c

is the sum of the other two.

2

1

max no of dominoes in a certain way in a 1000x1000 board

On a

1000\times 1000

-board we put dominoes, in such a way that each domino covers exactly two squares on the board. Moreover, two dominoes are not allowed to be adjacent, but are allowed to touch in a vertex. Determine the maximum number of dominoes that we can put on the board in this way.
Attention: you have to really prove that a greater number of dominoes is impossible.

1

2015 groups of 5 distinct numbers, for any 2 groups exactly 4 no occur in both

We make groups of numbers. Each group consists of five distinct numbers. A number may occur in multiple groups. For any two groups, there are exactly four numbers that occur in both groups. (a) Determine whether it is possible to make

2015

groups. (b) If all groups together must contain exactly six distinct numbers, what is the greatest number of groups that you can make? (c) If all groups together must contain exactly seven distinct numbers, what is the greatest number of groups that you can make?

4

1

prime diophantine 7pq^2 + p = q^3 + 43p^3 + 1

Find all pairs of prime numbers

(p, q)

for which

7pq^2 + p = q^3 + 43p^3 + 1

2015 Dutch Mathematical Olympiad

Subcontests

angle between 2 equilateral triangles on the same side of line is 60^o

equal segments inside a quadr. with 2 parallel sides, angle bisectors related

|a - b| >= |c| , |b - c| >= |a| and |c - a| >= |b| => one is sum of other two

max no of dominoes in a certain way in a 1000x1000 board

2015 groups of 5 distinct numbers, for any 2 groups exactly 4 no occur in both

prime diophantine 7pq^2 + p = q^3 + 43p^3 + 1

2015 Dutch Mathematical Olympiad

Subcontests

angle between 2 equilateral triangles on the same side of line is 60^o

equal segments inside a quadr. with 2 parallel sides, angle bisectors related

|a - b| &gt;= |c| , |b - c| &gt;= |a| and |c - a| &gt;= |b| =&gt; one is sum of other two

max no of dominoes in a certain way in a 1000x1000 board

2015 groups of 5 distinct numbers, for any 2 groups exactly 4 no occur in both

prime diophantine 7pq^2 + p = q^3 + 43p^3 + 1

|a - b| >= |c| , |b - c| >= |a| and |c - a| >= |b| => one is sum of other two