MathDB

10.4

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

length of projection, isosceles, circle (V Soros Olympiad 1998-99 Round 1 10.4)

Source:

5/25/2024
A straight line tangent to a circle circumscribed about an isosceles triangle ABCABC (AB=ACAB = AC) at point BB intersects straight line ACAC at point PP, EE is the midpoint of ABAB (fig.). What is the projection of DEDE onto ABAB if PA=aPA = a? https://cdn.artofproblemsolving.com/attachments/e/3/59c67e8f5eb3d399656d86613bc699c8baf1c1.png
geometry
x+\sqrt{x^2-9} = 2(x+3)/(x-3)^2 (V Soros Olympiad 1998-99 Round 2 10.4)

Source:

5/25/2024
Solve the equation x+x2−9=2(x+3)(x−3)2 x + \sqrt{x^2-9} = \frac{2(x+3)}{(x-3)^2}
algebra
min <MQA, MQ=QA, Q incenter (V Soros Olympiad 1998-99 Round 3 10.4)

Source:

5/26/2024
Let MM be the midpoint of side BCBC of triangle ABCABC, QQ the point of intersection of its angle bisectors. It is known that MQ=QAMQ=QA. Find the smallest possible value of angle ∠MQA\angle MQA.
geometryincenterangles