<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>
</ol>
<p>what competition?</p>
<p>to seee which IVIES will have correct answers first.</p>
<p>LMAO.</p>
<p>You can forget about Columbia because this board is practically dead compared to the other ivies'.</p>
<p>And sir, no offense, but you have quite some time on your hands to engage in something like this. ;)</p>
<p>1.) Yes, Real = x, fake = x - 1: 20x + 21x - 21 = 41x - 21. If he loses a real one, he will have 40x - 21 which is odd no matter what and wouldn't balance. If he loses a fake one, he will have 40x - 20 which is even no matter what and would balance so yes it is possible.
2.) If they are moving at the same speed, then Old Junky would win because it only has to go 2/3 of the distance in the time New Sporty has to go the whole distance. Your wording is quite awkward so you may have left out part of it.
3.) 148, because if he took all of any two kinds, the total has to be 99 or below.
4.) If I understand it it is 0 because "The distance between any two of them is greater than d."</p>
<p>I know I am probably late on these, but I have just now gotten on this whole weekend. Columbia is already the best, and we proved it last time anyways.</p>
<p>Thanks for clarifying Penn (he PMed me). If that is the case then New Sporty has twice the distance left. But, because Old Junky had to stop at 1/3 and wait for New Sporty to get to 2/3 that means that New Sporty is NOT going twice as fast or more as Old Junky, therefore, Old Junky will be the winner.</p>