<li>Define a good word as a sequence that only consits only of the letters A, B , C-some of these letters may not appear in sequence- and in which A is never immedialely followed by A, How many seven-letter good words are there?</li>
<li>Find the eighth term of a sequence 1440 , 1716 , 1848…, whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.</li>
<li>Let N be the greatest integer multiple of 8 , no two of whose digits are the same. What is the reminder when N is divided by 1000?</li>
</ol>
<p>in #1 when you ‘may not always appear in sequence’ do you mean they may not appear at all or they do not have to follow alphabetical order?</p>
<p>they may not appear at all</p>
<p>the answer is 120.</p>
<p>to get the largest integer multiple of 8 such that no two digits are the same implies that N has 10 digits (0 - 9).
so N = 1234567890 x 8 = 9876543120.
Dividing N by 1000, we have 9876543.12
so the remainder is 120.</p>
<h1>1 is a perms and combs question. still working on it. my math is crappy some.</h1>
<p>Jrock, I don’t think your solution to #3 is correct…</p>
<p>If N is the greatest integer multiple of 8 has non-repeating digits from 0-9, then [edit: N/8] would have to begin with the greatest digit from 0-9:
N = 9876543210 x 8 = 79012345680</p>
<p>Then divided by 1000 is 79012345.68
So the remainder is 680</p>
<p>I don’t know if this is correct… it just might be a little more correct. ;)</p>
<p>answer = 348
the two A.P.'s are:
8,11,14,17,20,… and
180, 156, 132, 108, 84, etc
8 x 180 = 1440
11 x 156 = 1716
14 x 132 = 1848</p>
<p>your N is greater than mine. so your answer ‘is more correct’.
that is actually the correct answer.</p>
<p>but sprezz’s N has two 0’s…</p>
<p>then I was right initially.</p>
<p>ooooooooh damn. lol
you were right jrock</p>
<p>So where’s penn253, who brought us all into this boat?</p>
<p>Probably waiting for you guys to solve the first problem.</p>
<p>Problems 1, 2, and 3 are problems #2, 3, and 8, respectively, of 2003 AIME2. The answers are here:</p>
<p><a href=“http://www.kalva.demon.co.uk/aime/soln/answ03b.html[/url]”>http://www.kalva.demon.co.uk/aime/soln/answ03b.html</a></p>
<p>Oops! I meant 3, 8, and 2.</p>
<p>Acutally we solved them all with explanations on the Columbia board two days ago, and I was just messing with you guys. That thing has some cool problems on it though.</p>
<p>thx tsuna. The solution to the #2 question was really cool. I was trying to establish a relationship between the given figures, but couldn’t get one.</p>
<p>i got my answers by trial and error.</p>
<p>this AIME stuff is quite challenging! </p>
<p>Any other Math Freaks on the forum? you’re invited.</p>