MathDB

10.4

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(4)

16x^3=(11x^2+x-1)\sqrt{x^2 - x+1} - VI Soros Olympiad 1999-00 Round 1 10.4

Source:

5/21/2024
Solve the equation 16x3=(11x2+x−1)x2−x+1.16x^3 = (11x^2 + x -1)\sqrt{x^2 - x + 1}.
algebra
r^2+r_a^2+r_b^2+ r_c^2 >= 2S

Source: VI Soros Olympiad 1990-00 R1 10.4 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
Prove that the inequality r2+ra2+rb2+rc2≥2S r^2+r_a^2+r_b^2+ r_c^2 \ge 2S holds for an arbitrary triangle, where rr is the radius of the circle inscribed in the triangle, rar_a, rbr_b, rcr_c are the radii of its three excribed circles, SS is the area of the triangle.
geometrygeometric inequalityGeometric Inequalitiesexradiusinequalities
congruent triangles if r, R, and area are equal >

Source: VI Soros Olympiad 1990-00 R2 10.4 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
Can we say that two triangles are congruent if the radii of the inscribed circles, the radii of the circumscribed circles, and the areas of these triangles are equal?
geometrycongruent triangles
AS // PQ wanted, starting with 2 intersecting circles

Source: VI Soros Olympiad 1990-00 R3 10.4 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
The circles ω1\omega_1 and ω2\omega_2 intersect at two points AA and BB. On the circle ω2\omega_2, point CC is taken in such a way that CACA is tangent to the circle ω1\omega_1. Through point AA, a straight line is drawn that intersects the circles ω1\omega_1, and ω2\omega_2 at points MM and NN, respectively , different from point AA. Point PP is the midpoint of the segment ACAC, QQ is the midpoint of MNMN, and SS is the intersection point of the line BQBQ with the circle ω1\omega_1, different from point BB. Prove that the lines ASAS and PQPQ are parallel.
geometryparallel