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9.7

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(2)

different distance in 2D (IV Soros Olympiad 1997-98 Correspondence 9.7)

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)=∣x1−x2∣+∣y1−y2∣r(A, B) = | 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 (you can take A(1,3)A(1, 3), B(3,7)B(3, 7)). 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 geometryalgebrageometry
3 solutions with different % (IV Soros Olympiad 1997-98 R3 9.7)

Source:

6/1/2024
There are three solutions with different percentages of alcohol. If you mix them in a ratio of 1:2:31:2:3, you get a 20%20\% solution. If you mix them in a ratio of 5:4:3,5: 4: 3, you will get a solution with 50%50\% alcohol content. What percentage of alcohol will the solution contain if equal amounts of the original solutions are mixed?
algebraPercentage