MathDB

9.3

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(3)

machines in the workshop (IV Soros Olympiad 1997-98 Correspondence 9.3)

Source:

6/1/2024
Several machines were working in the workshop. After reconstruction, the number of machines decreased, and the percentage by which the number of machines decreased turned out to be equal to the number of remaining machines. What was the smallest number of machines that could have been in the workshop before the reconstruction?
algebra
OM + ON >= R

Source: IV Soros Olympiad 1997-98 R3 9.3 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/1/2024
Through point OO - the center of a circle circumscribed around an acute triangle - a straight line is drawn, perpendicular to one of its sides and intersecting the other two sides of the triangle (or their extensions) at points MM and NN. Prove that OM+ON≥ROM+ON \ge R, where RR is the radius of the circumscribed circle around the triangle.
geometrygeometric inequality
angle, ratio 2:1, 1:1 by orhocenter (IV Soros Olympiad 1997-98 R2 9.3)

Source:

6/1/2024
What is angle BB of triangleABC ABC, if it is known that the altitudes drawn from AA and CC intersect inside the triangle and one of them is divided by of intersection point into equal parts, and the other one in the ratio of 2:12: 1, counting from the vertex?
ratiogeometryorthocenterangles