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Weighing the baby at the clinic was a problem. The baby would not keep still and caused the scales to wobble. So I held the baby and stood on the scales while the nurse read off $78\;\mathrm{kg}$. Then the nurse held the baby while I read off $69\;\mathrm{kg}$. Finally I held the nurse while the baby read off $137\;\mathrm{kg}$. What was the combined weight of all three?

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This problem is taken from the UKMT Mathematical Challenges.