Problems

Four siblings received magic wands for Christmas. It turned out that any three magic wands can form a triangle in such a way that the areas of all four triangles are equal. Are all the magic wands necessarily the same length?

A goat and a cow would take $45$ days to eat a full cart of hay. It would take a cow and a sheep $60$ days, but a sheep and a goat would need $90$ days. How many days would it take for all three animals to eat all the hay?

Anna and Beth played rock paper scissors ten times. Rock beat scissors, scissors beat paper and paper beat rock. Anna used rock three times, scissors six times and paper once. Beth used rock twice, scissors four times and paper four times. None of the ten games was a tie. Who won more games?