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10.3

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(3)

distance in 2D (IV Soros Olympiad 1997-98 Correspondence 10.3)

Source:

6/1/2024
For any two points A(x1,y1)A (x_1 , y_1) and B(x2,y2)B (x_2, y_2), the distance r(A,B)r (A, B) between them is determined by the equality r(A,B)=max{∣x1−x2∣,∣y1−y2∣}r(A, B) = max\{| x_1- x_2 | , | y_1 - y_2 |\}. Prove that the triangle inequality r(A,C)+r(C,B)≥r(A,B)r(A, C) + r(C, B) \ge r(A, B). holds for the distance introduced in this way .
Let AA and BB be two points of the plane . Find the locus of points CC for which a) r(A,C)+r(C,B)=r(A,B)r(A, C) + r(C, B) = r(A, B) b) r(A,C)=r(C,B).r(A, C) = r(C, B).
analytic geometrygeometryalgebra
distance of feet of 2 altitudes = 1/2 R

Source: IV Soros Olympiad 1997-98 R2 10.3 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/1/2024
What can angle BB of triangle ABCABC be equal to if it is known that the distance between the feet of the altitudes drawn from vertices AA and CC is equal to half the radius of the circle circumscribed around this triangle?
geometrycircumradius
3 3digit numbers in artihm. progession (IV Soros Olympiad 1997-98 R3 10.3)

Source:

6/2/2024
Three different digits were used to create three different three-digit numbers forming an arithmetic progression. (In each number, all the digits are different.) What is the largest difference in this progression?
number theoryArithmetic ProgressionDigits