MathDB

2016 Dutch Mathematical Olympiad

Part of Dutch Mathematical Olympiad

Subcontests

(6)
4 seniors
1

midpoint wanted, right angles and angle bisectors given

In the acute triangle ABCABC, the midpoint of side BCBC is called MM. Point XX lies on the angle bisector of ∠AMB\angle AMB such that ∠BXM=90o\angle BXM = 90^o. Point YY lies on the angle bisector of ∠AMC\angle AMC such that ∠CYM=90o\angle CYM = 90^o. Line segments AMAM and XYXY intersect in point ZZ. Prove that ZZ is the midpoint of XYXY .
[asy] import geometry;
unitsize (1.2 cm);
pair A, B, C, M, X, Y, Z;
A = (0,0); B = (4,1.5); C = (0.5,3); M = (B + C)/2; X = extension(M, incenter(A,B,M), B, B + rotate(90)*(incenter(A,B,M) - M)); Y = extension(M, incenter(A,C,M), C, C + rotate(90)*(incenter(A,C,M) - M)); Z = extension(A,M,X,Y);
draw(A--B--C--cycle); draw(A--M); draw(M--interp(M,X,2)); draw(M--interp(M,Y,2)); draw(B--X, dotted); draw(C--Y, dotted); draw(X--Y);
dot("AA", A, SW); dot("BB", B, E); dot("CC", C, N); dot("MM", M, NE); dot("XX", X, NW); dot("YY", Y, NE); dot("ZZ", Z, S); [/asy]
4 juniors
1

orthocenters inside a quadrilateral, parallelogram wanted

In a quadrilateral ABCDABCD the intersection of the diagonals is called PP. Point XX is the orthocentre of triangle PABPAB. (The orthocentre of a triangle is the point where the three altitudes of the triangle intersect.) Point YY is the orthocentre of triangle PCDPCD. Suppose that XX lies inside triangle PABPAB and YY lies inside triangle PCDPCD. Moreover, suppose that PP is the midpoint of line segment XYXY . Prove that ABCDABCD is a parallelogram. [asy] import geometry;
unitsize (1.5 cm);
pair A, B, C, D, P, X, Y;
A = (0,0); B = (2,-0.5); C = (3.5,2.2); D = A + C - B; P = (A + C)/2; X = orthocentercenter(A,B,P); Y = orthocentercenter(C,D,P);
draw(A--B--C--D--cycle); draw(A--C); draw(B--D); draw(A--extension(A,X,B,P), dotted); draw(B--extension(B,X,A,P), dotted); draw(P--extension(P,X,A,B), dotted); draw(C--extension(C,Y,D,P), dotted); draw(D--extension(D,Y,C,P), dotted); draw(P--extension(P,Y,C,D), dotted);
dot("AA", A, W); dot("BB", B, S); dot("CC", C, E); dot("DD", D, N); dot("PP", P, E); dot("XX", X, NW); dot("YY", Y, SE); [/asy]
3
1
2
1

smallest sum product value of a sequence of 2n numbers

For an integer n≥1n \ge 1 we consider sequences of 2n2n numbers, each equal to 0,−10, -1 or 11. The sum product value of such a sequence is calculated by first multiplying each pair of numbers from the sequence, and then adding all the results together. For example, if we take n=2n = 2 and the sequence 0,1,1,−10,1, 1, -1, then we find the products 0⋅1,0⋅1,0⋅−1,1⋅1,1⋅−1,1⋅−10\cdot 1, 0\cdot 1, 0\cdot -1, 1\cdot 1, 1\cdot -1, 1\cdot -1. Adding these six results gives the sum product value of this sequence: 0+0+0+1+(−1)+(−1)=−10+0+0+1+(-1)+(-1) = -1. The sum product value of this sequence is therefore smaller than the sum product value of the sequence 0,0,0,00, 0, 0, 0, which equals 00. Determine for each integer n≥1n \ge 1 the smallest sum product value that such a sequence of 2n2n numbers could have.
Attention: you are required to prove that a smaller sum product value is impossible.
1
1

encircling 999 numbers with red, green, blue chalk

(a) On a long pavement, a sequence of 999999 integers is written in chalk. The numbers need not be in increasing order and need not be distinct. Merlijn encircles 500500 of the numbers with red chalk. From left to right, the numbers circled in red are precisely the numbers 1,2,3,...,499,5001, 2, 3, ...,499, 500. Next, Jeroen encircles 500500 of the numbers with blue chalk. From left to right, the numbers circled in blue are precisely the numbers 500,499,498,...,2,1500, 499, 498, ...,2,1.
Prove that the middle number in the sequence of 999999 numbers is circled both in red and in blue.
(b) Merlijn and Jeroen cross the street and find another sequence of 999999 integers on the pavement. Again Merlijn circles 500500 of the numbers with red chalk. Again the numbers circled in red are precisely the numbers 1,2,3,...,499,5001, 2, 3, ...,499, 500 from left to right. Now Jeroen circles 500500 of the numbers, not necessarily the same as Merlijn, with green chalk. The numbers circled in green are also precisely the numbers 1,2,3,...,499,5001, 2, 3, ...,499, 500 from left to right.
Prove: there is a number that is circled both in red and in green that is not the middle number of the sequence of 999999 numbers.
5
1