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An analysis of the Millimetre Astronomy Legacy Team 90 GHz (MALT90) survey has produced a sample of 27 candidate dense molecular clumps with large collapse motions, as revealed by large "blue" asymmetrical line profiles of the optically thick HCO+ (1–0) line. New, more sensitive molecular line observations of this sample, conducted with the Mopra 22-m telescope, confirm the blue asymmetries in the HCO+(1–0) line profiles, with large, positive values of the asymmetry parameter A (HCO+) = 0.69±0.01, and decreasing values of A for lines with smaller optical depths, as expected for collapsing clumps. The hyperfine ratios for N2H+(1–0) are in their optically thin, LTE, values, but for HCN (1–0) they are not; the F = 1 → 1 hyperfine line shows abnormally weak intensities. A simple two-component model shows that self-absorption of the background F = 1 → 1 hyperfine line by the main F = 2 → 1 hyperfine line of a cold, foreground, redshifted cloud can reproduce the observed HCN (1–0) hyperfine intensities and match both the HCN (1–0) and HCO+(1–0) line profiles. All of these results are consistent with self-absorption of the optically thick lines on the red side of the profile, as expected for collapsing clumps. A simple two-cloud model suggests that this sample represents dense clumps with extreme collapse velocities, V_inf ∼ 2.4 km s−1, the highest known.