MathDB

Problem 2

Part of 2001 Federal Math Competition of S&M

Problems(3)

square on a sphere

Source: Serbia & Montenegro 2001 2nd Grade P2

6/2/2021
Vertices of a square ABCDABCD of side 254\frac{25}4 lie on a sphere. Parallel lines passing through points A,B,CA,B,C and DD intersect the sphere at points A1,B1,C1A_1,B_1,C_1 and D1D_1, respectively. Given that AA1=2AA_1=2, BB1=10BB_1=10, CC1=6CC_1=6, determine the length of the segment DD1DD_1.
geometry3D geometrysphere
5 segments, triangles formed, one must be acute

Source: Serbia & Montenegro 2001 1st Grade P2

6/1/2021
Given are 55 segments, such that from any three of them one can form a triangle. Prove that from some three of them one can form an acute-angled triangle.
geometrycombinatoricscombinatorial geometry
inequality, sum of (x_n^2)/(n^3)

Source: Serbia & Montenegro 2001 3,4th Grade P2

6/2/2021
Let x1,x2,…,x2001x_1,x_2,\ldots,x_{2001} be positive numbers such that xi2≥x12+x2223+x3233+…+xi−12(i−1)3for 2≤i≤2001.x_i^2\ge x_1^2+\frac{x_2^2}{2^3}+\frac{x_3^2}{3^3}+\ldots+\frac{x_{i-1}^2}{(i-1)^3}\enspace\text{for }2\le i\le2001.Prove that ∑i=22001xix1+x2+…+xi−1>1.999\sum_{i=2}^{2001}\frac{x_i}{x_1+x_2+\ldots+x_{i-1}}>1.999.
inequalitiesSummation