The prompt production of charmonium $chi_{c}$ and $J/psi$ states is studied in proton-proton collisions at a centre-of-mass energy of $sqrt{s}=7$ TeV at the Large Hadron Collider. The $chi_{c}$ and $J/psi$ mesons are identified through their decays $chi_{c}rightarrow J/psi gamma$ and $J/psirightarrow mu^+mu^-$ using 36 pb$^{-1}$ of data collected by the LHCb detector in 2010. The ratio of the prompt production cross-sections for $chi_{c}$ and $J/psi$, $sigma (chi_{c}rightarrow J/psi gamma)/ sigma (J/psi)$, is determined as a function of the $J/psi$ transverse momentum in the range $2 < p_{mathrm T}^{J/psi} < 15$ GeV/$c$. The results are in excellent agreement with next-to-leading order non-relativistic expectations and show a significant discrepancy compared with the colour singlet model prediction at leading order, especially in the low $p_{mathrm T}^{J/psi}$ region.
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