What if you simply planted trees? They act as a carbon sink, sequestering carbon dioxide as biomass. One statistic says that one million trees will sequester 0.9 teragrams over their 40 year lifetime. That is:
0.9 Tg = 9×10¹¹ g 9×10¹¹ g ÷ 106 trees ÷ 4×101 a = 2.25×10⁴ g·tree-1·a-1
That is, one tree will fix 22.5 kg of carbon dioxide in one year. The Earth Challenge requires removal of 1 billion tonnes (1015 g) annually from the atmosphere. You would have to plant:
1015 g ÷ 2.25×10⁴ g·tree-1·a-1 × 1 a = 4.4×1010 tree
Ie. 44 billion trees. The same article states these would cost roughly ten cents apiece, or $4.4 billion—a bit much in comparison to the prize money, but not an astronomical figure. Also, estimating 500 trees per hectare, that's
4.4×1010 tree ÷ 5×10² tree·ha-1 = 8.8×10⁷ ha = 8.8×10⁵ km²
…slightly more than the 718,000 km² of Amazon rainforest that's been clear-cut since 1970, or 2.2% of the 3.952 billion hectares of global forest cover. Still, that $4.4 billion forest, almost the size of Ontario, would only offset about one twenty-fifth of estimated man-made emissions.