Maths Olympiad Prep

Track / Stage 7 / 282 of 300 #1682 of 1964

Problem 1682

National Olympiad second round; IMO P1/P4
Geometry Difficulty 7.5 Find the answer

Convex hexagon ABCDEFABCDEF is drawn in the plane such that ACDFACDF and ABDEABDE are parallelograms with area 168. ACAC and BDBD intersect at GG. Given that the area of AGBAGB is 10 more than the area of CGBCGB, find the smallest possible area of hexagon ABCDEFABCDEF.

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Official solution

1. Given that ACDFACDF and ABDEABDE are parallelograms with area 168, we know that [ACDF]=[ABDE]=168[ACDF] = [ABDE] = 168.
2. Since ACDFACDF and ABDEABDE are parallelograms, AC∥DFAC \parallel DF and AB∥DEAB \parallel DE.
3. The diagonals ACAC and BDBD intersect at GG.
4. Let the area of △AGB\triangle AGB be 10+a10 + a and the area of △CGB\triangle CGB be aa.
5. Since [ACDF]=168[ACDF] = 168, we have [ACD]=[ACF]=84[ACD] = [ACF] = 84.
6. Similarly, since [ABDE]=168[ABDE] = 168, we have [ABD]=[ADE]=84[ABD] = [ADE] = 84.
7. Note that [ABD]=[ACD]=84[ABD] = [ACD] = 84, so AD∥BCAD \parallel BC.
8. This implies that ABCDABCD is a trapezoid.
9. Let [CGB]=a[CGB] = a. Then, [AGB]=10+a[AGB] = 10 + a and [CGD]=10+a[CGD] = 10 + a.
10. Since [AGD]=74−a[AGD] = 74 - a, we have △AGD∼△CGB\triangle AGD \sim \triangle CGB.
11. Using the similarity ratios and area ratios, we get:
a74−a=(aa+10)2 \frac{a}{74 - a} = \left(\frac{a}{a + 10}\right)^2
12. Solving the equation:
a74−a=(aa+10)2 \frac{a}{74 - a} = \left(\frac{a}{a + 10}\right)^2
a(74−a)=(a+10)2 a(74 - a) = (a + 10)^2
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
74a−a2=a2+20a+100 74a - a^2 = a^2 + 20a + 100
\[
74a - a

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.