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11.3

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(3)

sum a^4 + 2(sum ab^2 )^2 < 1 - VI Soros Olympiad 1999-00 Round 1 11.3

Source:

5/21/2024
The numbers a,ba, b and cc are such that a2+b2+c2=1a^2 + b^2 + c^2 = 1. Prove that a4+b4+c4+2(ab2+bc2+ca2)2≤1.a^4 + b^4 + c^4 + 2(ab^2 + bc^2 + ca^2)^2\le 1. At what a,ba, b and cc does inequality turn into equality?
algebrainequalities
2S <=OA OC+ OBxOB for tangential ABCD

Source: VI Soros Olympiad 1990-00 R2 11.3 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
A convex quadrilateral ABCDABCD has an inscribed circle touching its sides ABAB, BCBC, CDCD, DADA at the points MM,NN,PP,KK, respectively. Let OO be the center of the inscribed circle, the area of the quadrilateral MNPKMNPK is equal to 88. Prove the inequality 2S≤OA⋅OC+OB⋅OD.2S \le OA \cdot OC+ OB \cdot OD.
geometrygeometric inequalitytangential
3 spheres intersect alone one circle

Source: VI Soros Olympiad 1990-00 R3 11.3 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
Three spheres s1s_1, s2s_2, s3s_3 intersect along one circle ω\omega. Let AA be an arbitrary point lying on the circle ω\omega. Ray ABAB intersects spheres s1s_1, s2s_2, s3s_3 at points B1B_1, B2B_2, B3B_3, respectively, ray ACAC intersects spheres s1s_1, s2s_2, s3s_3 at points C1C_1, C2C_2, C3C_3, respectively (Bi≠AiB_i \ne A_i, Ci≠AiC_i \ne A_i, i=1,2,3i=1,2,3). It is known that B2B_2 is the midpoint of the segment B1B3B_1B_3. Prove that C2C_2 is the midpoint of the segment C1C3C_1C_3.
geometry3D geometrysphereSpheres