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MathNet 2014 Question 86

Math Olympiad Vault · MathNet 2014

Question 86

Points $X$, $Y$ and $Z$ are marked on the sides $AD$, $AB$ and $BC$ of a parallelogram $ABCD$, respectively. It is known that $AX = CZ$.

a) Prove that at least one of the inequalities holds
$$
XY + YZ \geq AC \text{ or } XY + YZ \geq BD.
$$

b) Is it true that $XY + YZ \geq \frac{AC + BD}{2}$?