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1982 Bundeswettbewerb Mathematik

Part of Bundeswettbewerb Mathematik

Subcontests

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Bundeswettbewerb Mathematik 1982 Problem 1.2

In a convex quadrilateral ABCDABCD sides ABAB and DCDC are both divided into mm equal parts by points A,S1,S2,…,Sm−1,BA, S_1 , S_2 , \ldots , S_{m-1} ,B and D,T1,T2,…,Tm−1,C,D,T_1, T_2, \ldots , T_{m-1},C, respectively (in this order). Similarly, sides BCBC and ADAD are divided into nn equal parts by points B,U1,U2,…,Un−1,CB,U_1,U_2, \ldots, U_{n-1},C and A,V1,V2,…,Vn−1,DA,V_1,V_2, \ldots,V_{n-1}, D. Prove that for 1≤i≤m−11 \leq i \leq m-1 each of the segments SiTiS_i T_i is divided by the segments UjVjU_j V_j (1≤j≤n−11\leq j \leq n-1) into nn equal parts

Bundeswettbewerb Mathematik 1981 Problem 2.2

Decide whether every triangle ABCABC in space can be orthogonally projected onto a plane such that the projection is an equilateral triangle A′B′C′A'B'C'.
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point inside convex polygon

Suppose PP is a point inside a convex 2n2n-gon, such that PP does not lie on any diagonal. Show that PP lies inside an even number of triangles whose vertices are vertices of the polygon.

Bundeswettbewerb Mathematik 1982 Problem 2.3

Given that a1,a2,...,ana_1, a_2, . . . , a_n are nonnegative real numbers with a1+⋯+an=1a_1 + \cdots + a_n = 1, prove that the expression a11+a2+a3+⋯+an  +  a21+a1+a3+⋯+an  +  ⋯  + an1+a1+a2+⋯+an−1 \frac{a_1}{1+a_2 +a_3 +\cdots +a_n }\; +\; \frac{a_2}{1+a_1 +a_3 +\cdots +a_n }\; +\; \cdots \; +\, \frac{a_n }{1+a_1 +a_2+\cdots +a_{n-1} } attains its minimum, and determine this minimum.
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