<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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<p>In regard to number 1, Sam has 20 coins and 21 coins (fake), if one has an even weight, the other must have an odd weight (fake coins are 1 gram less). Thus, if fake coins are odd, then there is an odd weight of coins plus an even weight of coins to make an odd weight of coins at the start. If the fake coins are even, then the weight of fake coins is even and the weight of real coins is even also, 20*odd number= odd number; thus, the total weight is even. Now, with regard to the first case, if a fake coin is taken away, he will be left with an even number, odd-odd is even, and he will have an odd number if a real coin is taken away. Now, with regard to the second case if a fake coin is taken away, the weight left is even, and if a real coin is taken away, he will have an odd weight. Thus, if the weight of all the coins is even, a fake coin was taken, and if the weight of all the coins is odd, a real coin was taken.</p>
<p>I've got a question, that is close to number 1 above, from a USSR Olympiad. Show that the total number of handshakes made in the world at any given time is an even number. More Questions to follow if you all can decipher the first one.</p>
<p>In regard to number 3, all you have to do is set up the inequality so that the total number of two candies is never greater than 100; thus, candy 1(x) +candy 2(y), x + candy3 (z), y+z is less than 100. Thus, since we are dealing with integers, pieces of candy, we know that if x is 50, then y is 49, and further, if x is 50, z is 49; therefore, we obtain a maximum of 50+49+49=148 pieces of candy.</p>
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<li> infinity?</li>
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<p>do u mean like doing a pigeon-hole, bipartite graph thingy for the handshakes?
To prove it I mean. It is pretty easy to logically deduce out that for any one person doing the handshake, the receipient also shook hands, so of course it is always even.</p>