MathDB

Problem 2

Part of 2002 Federal Math Competition of S&M

Problems(3)

chords in a circle, show area inequality

Source: S&M 2002 2nd Grade P2

5/14/2021
Points A0,A1,…,A2kA_0,A_1,\ldots,A_{2k}, in this order, divide a circumference into 2k+12k+1 equal arcs. Point A0A_0 is connected by chords to all the other points. These 2k2k chords divide the interior of the circle into 2k+12k+1 parts. These parts are alternately painted red and blue so that there are k+1k+1 red and kk blue parts. Show that the blue area is larger than the red area.
geometrygeometric inequalityareainequalities
cevian inequality

Source: S&M 2002 1st Grade P2

5/14/2021
Let OO be a point inside a triangle ABCABC and let the lines AO,BOAO,BO, and COCO meet sides BC,CABC,CA, and ABAB at points A1,B1A_1,B_1, and C1C_1, respectively. If AA1AA_1 is the longest among the segments AA1,BB1,CC1AA_1,BB_1,CC_1, prove that OA1+OB1+OC1≤AA1.OA_1+OB_1+OC_1\le AA_1.
Geometric Inequalitiesgeometric inequalityinequalitiesgeometry
triangle, sides √(fibonacci #s)

Source: S&M 2002 3&4th Grade P2

5/15/2021
The (Fibonacci) sequence fnf_n is defined by f1=f2=1f_1=f_2=1 and fn+2=fn+1+fnf_{n+2}=f_{n+1}+f_n for n≥1n\ge1. Prove that the area of the triangle with the sides f2n+1,f2n+2,\sqrt{f_{2n+1}},\sqrt{f_{2n+2}}, and f2n+3\sqrt{f_{2n+3}} is equal to 12\frac12.
geometryFibonacci