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6

Part of 1950 Miklós Schweitzer

Problems(2)

Miklos Schweitzer 1950_6

Source: first round of 1950

10/2/2008
Prove the following identity for determinants: |c_{ik} \plus{} a_i \plus{} b_k \plus{} 1|_{i,k \equal{} 1,...,n} \plus{} |c_{ik}|_{i,k \equal{} 1,...,n} \equal{} |c_{ik} \plus{} a_i \plus{} b_k|_{i,k \equal{} 1,...,n} \plus{} |c_{ik} \plus{} 1|_{i,k \equal{} 1,...,n}
linear algebralinear algebra unsolved
Miklos Schweitzer 1950_6

Source: second part of 1950

10/3/2008
Consider an arc of a planar curve; let the radius of curvature at any point of the arc be a differentiable function of the arc length and its derivative be everywhere different from zero; moreover, let the total curvature be less than π2 \frac{\pi}{2}. Let P1,P2,P3,P4,P5 P_1,P_2,P_3,P_4,P_5 and P6 P_6 be any points on this arc, subject to the only condition that the radius of curvature at Pk P_k is greater than at Pj P_j if j<k j<k. Prove that the radius of the circle passing through the points P1,P3 P_1,P_3 and P5 P_5 is less than the radius of the circle through P2,P4 P_2,P_4 and P6 P_6
functioncalculusderivativeadvanced fieldsadvanced fields unsolved