MathDB

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Part of 2011 Bundeswettbewerb Mathematik

Problems(2)

q^2 + r = 2011 when ab = q (a + b) + r , 0 \le r <a + b

Source: Germany Federal - Bundeswettbewerb Mathematik 2011, round 1, p4

4/10/2020
Let aa and bb be positive integers. As is known, the division of of a⋅ba \cdot b with a+ba + b determines integers qq and rr uniquely such that a⋅b=q(a+b)+ra \cdot b = q (a + b) + r and 0≤r<a+b0 \le r <a + b. Find all pairs (a,b)(a, b) for which q2+r=2011q^2 + r = 2011.
Diophantine equationdiophantineEuclidean algorithmnumber theory
mininal sum of distances from a point to vertices of a tetrahedron

Source: Germany Federal - Bundeswettbewerb Mathematik 2011, round 2, p4

4/14/2020
Let ABCDABCD be a tetrahedron that is not degenerate and not necessarily regular, where sides ADAD and BCBC have the same length aa, sides BDBD and ACAC have the same length bb, side ABAB has length c1c_1 and the side CDCD has length c2c_2. There is a point PP for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities a,b,c1a, b, c_1 and c2c_2.
3D geometryminSumdistancestetrahedrongeometrygeometric inequality