MathDB

2011 Dutch IMO TST

Part of Dutch IMO TST

Subcontests

(5)
2
2

tilings of a rectangular m x n board with 1 x 2 tiles

We consider tilings of a rectangular m \times n-board with 1\times2-tiles. The tiles can be placed either horizontally, or vertically, but they aren't allowed to overlap and to be placed partially outside of the board. All squares on theboard must be covered by a tile. (a) Prove that for every tiling of a 4 \times 2010-board with 1\times2-tiles there is a straight line cutting the board into two pieces such that every tile completely lies within one of the pieces. (b) Prove that there exists a tiling of a 5 \times  2010-board with 1\times 2-tiles such that there is no straight line cutting the board into two pieces such that every tile completely lies within one of the pieces.

xf(x + xy) = xf(x) + f(x^2)f(y)

Find all functions f:R→Rf : R\to R satisfying xf(x+xy)=xf(x)+f(x2)f(y)xf(x + xy) = xf(x) + f(x^2)f(y) for all x,y∈Rx, y \in R.
4
2
1
2
5
2

DF divides the line segment EG into two equal parts

Let ABCABC be a triangle with ∣AB∣>∣BC∣|AB|> |BC|. Let DD be the midpoint of ACAC. Let EE be the intersection of the angular bisector of ∠ABC\angle ABC and the line ACAC. Let FF be the point on BEBE such that CFCF is perpendicular to BEBE. Finally, let GG be the intersection of CFCF and BDBD. Prove that DFDF divides the line segment EGEG into two equal parts.

a red, b blue and c green points such that a+b+c = 10

Find all triples (a,b,c)(a, b, c) of positive integers with a+b+c=10a+b+c = 10 such that there are aa red, bb blue and cc green points (all different) in the plane satisfying the following properties: ∙\bullet for each red point and each blue point we consider the distance between these two points, the sum of these distances is 3737, ∙\bullet for each green point and each red point we consider the distance between these two points, the sum of these distances is 3030, ∙\bullet for each blue point and each green point we consider the distance between these two points, the sum of these distances is 11.
3
2

angle chasing candidate, intersecting circles given, equal angles wanted

The circles Γ1\Gamma_1 and Γ2\Gamma_2 intersect at DD and PP. The common tangent line of the two circles closest to point DD touches Γ1\Gamma_1 in A and Γ2\Gamma_2 in BB. The line ADAD intersects Γ2\Gamma_2 for the second time in CC. Let MM be the midpoint of line segment BCBC. Prove that ∠DPM=∠BDC\angle DPM = \angle BDC.

three circles given, tangent wanted

Let Γ1\Gamma_1 and Γ2\Gamma_2 be two intersecting circles with midpoints respectively O1O_1 and O2O_2, such that Γ2\Gamma_2 intersects the line segment O1O2O_1O_2 in a point AA. The intersection points of Γ1\Gamma_1 and Γ2\Gamma_2 are CC and DD. The line ADAD intersects Γ1\Gamma_1 a second time in SS. The line CSCS intersects O1O2O_1O_2 in FF. Let Γ3\Gamma_3 be the circumcircle of triangle ADAD. Let EE be the second intersection point of Γ1\Gamma_1 and Γ3\Gamma_3. Prove that O1EO_1E is tangent to Γ3\Gamma_3.