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

 

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This problem is taken from the UKMT Mathematical Challenges.