**A. 644.** Let \(\displaystyle f(x,y)\) be a polynomial with two variables and integer coefficients such that \(\displaystyle f\) is constant neither in \(\displaystyle x\)- nor in \(\displaystyle y\)-direction. Prove that \(\displaystyle \max_{a,b\in[-2,2]}\big|f(a,b)\big|\ge4\).

Based on the idea of *Tamás Erdélyi,* College Station, Texas

(5 points)

**A. 646.** Ginger and Rocky play the following game. First Ginger hides two bones in the corners of a rectangular garden. She may dig 45 cm deep altogether, that is, she may either hide the two bones in two different corners, where the sum of their depths may be at most 45 cm, or she may hide them in the same place, both bones at a maximum depth of 45 cm. She levels the ground carefully so that it is impossible to see where she has dug. Then Rocky may dig holes with a total depth of 1 m. Rocky's goal is to maximize the probability of finding both bones, while Ginger's goal is to maximize the probability of keeping at least one for herself.

\(\displaystyle (a)\) Show that if Ginger plays well she can achieve a probability of more than \(\displaystyle 1/2\) for at least one bone remaining hidden, independently of Rocky's search strategy.

\(\displaystyle (b)\) What are the chances of Ginger's success if both dogs play optimally?

Proposed by: *Endre Csóka,* Warwick

(5 points)

**C. 1300.** The lengths of the sides of a convex quadrilateral, in this order, are \(\displaystyle \sqrt{a}\), \(\displaystyle \sqrt{a+3}\), \(\displaystyle \sqrt{a+2}\) and \(\displaystyle \sqrt{2a+5}\). The length of each diagonal is \(\displaystyle \sqrt{2a+5}\). Determine the greatest angle of the quadrilateral.

(5 points)

This problem is for grade 11 - 12 students only.