MathDB

2018 Bundeswettbewerb Mathematik

Part of Bundeswettbewerb Mathematik

Subcontests

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Perpendiculars, Parallels and six points on a circle

Let HH be the orthocenter of the acute triangle ABCABC. Let HaH_a be the foot of the perpendicular from AA to BCBC and let the line through HH parallel to BCBC intersect the circle with diameter AHaAH_a in the points PaP_a and QaQ_a. Similarly, we define the points Pb,QbP_b, Q_b and Pc,QcP_c,Q_c.
Show that the six points Pa,Qa,Pb,Qb,Pc,QcP_a,Q_a,P_b,Q_b,P_c,Q_c lie on a common circle.

Perpendicular lines, equal angles and a fixed point

Let TT be a point on a line segment ABAB such that TT is closer to BB than to AA. Show that for each point C≠TC \ne T on the line through TT perpendicular to ABAB there is exactly one point DD on the line segment ACAC with ∠CBD=∠BAC\angle CBD=\angle BAC. Moreover, show that the line through DD perpendicular to ACAC intersects the line ABAB in a point EE which is independent of the position of CC.
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An equation involving the floor function

Find all real numbers xx satisfying the equation ⌊20x+18⌋+⌊x+1820⌋=1.\left\lfloor \frac{20}{x+18}\right\rfloor+\left\lfloor \frac{x+18}{20}\right\rfloor=1.

Possible values for a sum under some functional equation

Consider all functions f:R→Rf:\mathbb{R} \to \mathbb{R} satisfying f(1−f(x))=xf(1-f(x))=x for all x∈Rx \in \mathbb{R}. a) By giving a concrete example, show that such a function exists. b) For each such function define the sum Sf=f(−2017)+f(−2016)+⋯+f(−1)+f(0)+f(1)+⋯+f(2017)+f(2018).S_f=f(-2017)+f(-2016)+\dots+f(-1)+f(0)+f(1)+\dots+f(2017)+f(2018). Determine all possible values of SfS_f.
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