MathDB

1996 All-Russian Olympiad Regional Round

Part of All-Russian Olympiad Regional Round

Subcontests

(28)
9.4
1
11.6
1
11.8
1
11.7
1
11.5
1
11.4
1
11.3
1
11.2
1
10.6
1

sum of areas on n-1 #s <1/2 [ABC] - All-Russian MO 1996 Regional (R4) 10.6

Given triangle A0B0C0A_0B_0C_0. On the segment A0B0A_0B_0 points A1A_1, A2A_2, ......, AnA_n, and on the segment B0C0B_0C_0 - points C1C_1, C2C_2, ......, CnCn so that all segments AiCi+1A_iC_{i+1} (i=0i = 0, 11, ......,n−1n-1) are parallel to each other and all segments CiAi+1 C_iA_{i+1} (i=0i = 0, 11, ......,n−1n-1) are too. Segments C0A1C_0A_1, A1C2A_1C_2, A2C1A_2C_1 and C1A0C_1A_0 bound a certain parallelogram, segments C1A2C_1A_2, A2C3A_2C_3, A3C2A_3C_2 and C2A1C_2A_1 too, etc. Prove that the sum of the areas of all n−1n -1 resulting parallelograms less than half the area of triangle A0B0C0A_0B_0C_0.
10.8
1
10.4
1
10.3
1
10.2
1
10.1
1
9.8
1
9.7
1
9.5
1
9.3
1
9.2
1
9.1
1
8.8
1
8.7
1
8.6
1
8.5
1
8.4
1
8.3
1
8.2
1
8.1
1