Math Competition

<li>Sam had 20 real coins and 21 fake coins. A fake coin weights 1 gram less than a real one.Sam lost 1 coin. He has a balance scales that shows the weight difference , but he can only use it only once. Is it possible to find out which coin he lost-real or fake?</li>
<li>Two cars- Old Junky and New Sporty-started to move from point A to point B at the same time. After driving ond third of the way Old Junky stopped and started moving only when reaching B New Sporty turned back and continued to drive towards A. Which car will come first- Old Junky to B or New Sporty to A? Assume that each car is moving with the constant speed.</li>
<li>Santa has three kinds of canides in his bga. He knows that if he takes out any 100 candies , he will definitely have all three kindas of candies. What is the maximum number of candies in his bag? </li>
<li>There are N points on the plane. The distance between any two of them is greater than d. Find the maximum number of pairs of points thatt are exactly at distance d.</li>
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