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1 Cor 9:24 Know ye not that they which run in a race run all, but one receiveth the prize? So run, that ye may obtain.
Rom 9:16 So then it is not a matter of willing or running, but of God's mercy.
So do we run in order to obtain things or do we just sit and wait for God's mercy to grant our desires?
The context of 1 Cor 9:24 clearly indicates that Paul is using running a race as a metaphor for the Christian's carrying out of God's work in his lifetime. In particular, Paul is speaking of his own missionary efforts of proclaiming the gospel in the immediately preceding verses.
Second, the context of Rom 9:16 speaks of justification , that is, becoming saved, with particular reference to the Jews.
Therefore, completely different things are being talked about here, and there is absolutely no case for contradiction.