forum.texasmath.org

Your connection to the competitive math/science community.

Skip to content

AIME I 2010 #10

American Mathematics Contest & American Invitational Mathematics Examination. See http://www.unl.edu/amc for more information
Note:USAMO/IMO-level questions need to go in the Olympiad section.
Post a reply

AIME I 2010 #10

Postby stupidityismygam » Tue Mar 23, 2010 1:44 am

Let N be the number of ways to write 2010 in the form 2010=a_3\cdot 10^3+a_2 \cdot 10^2+a_1 \cdot 10+a_0, where the a_i's are integers, and 0 \leq a_i \leq 99. An example of such a representation is 1 \cdot 10^3+3 \cdot 10^2 + 67 \cdot 10^1+40 \cdot 10^0. Find N.
my avatar is pretty awesome
tytia
User avatar
stupidityismygam
Ubiquitous Spammer
 
Posts: 1413
Joined: Tue Nov 21, 2006 7:02 pm
Top

Re: AIME I 2010 #10

Postby 88bobcat » Fri Jul 02, 2010 11:57 am

stupidityismygam wrote:Let N be the number of ways to write 2010 in the form 2010=a_3\cdot 10^3+a_2 \cdot 10^2+a_1 \cdot 10+a_0, where the a_i's are integers, and 0 \leq a_i \leq 99. An example of such a representation is 1 \cdot 10^3+3 \cdot 10^2 + 67 \cdot 10^1+40 \cdot 10^0. Find N.



Because, 0 \leq a_i \leq 99, a_2 and a_0 have a much greater range of possibilities. a_3 and a_1 are the coefficients that limit the total number N.

For a_3 = 0, a_1 has 100 different possibilities, 0 \leq a_1 \leq 99.

For a_3 = 1, a_1 has 100 different possibilities, 0 \leq a_1 \leq 99.

For a_3 = 2, a_1 has 2 two different possilities, a_1 \in \{0,1\}.


Hence, N = 100 + 100 + 2 = \boxed{202}.



\displaystyle\lim_{x\rightarrow0}\frac{\sin(x)}{x}+\lim_{y\rightarrow\infty}\ln\left(1+\frac{1}{y}\right)^{y}+\frac{\cos^4(\theta)-\sin^4(\theta)}{\cos(2\theta)}=\sum_{n=0}^{\infty}\left(\frac{2}{3}\right)^{n}

User avatar
88bobcat
Celebrated Spammer
 
Posts: 447
Joined: Tue May 26, 2009 9:51 pm
Location: Orange, TX
Top

Re: AIME I 2010 #10

Postby AuSmith » Mon Jul 05, 2010 6:05 pm

So, you're saying that a_2=0 always?
log∙ic (loj'ik), n. 1. The art of being wrong with confidence.
User avatar
AuSmith
Grizzled Old Veteran
 
Posts: 1047
Joined: Sun Nov 05, 2006 12:24 am
Location: College Station, TX
Top

Re: AIME I 2010 #10

Postby stupidityismygam » Tue Jul 06, 2010 3:12 am

AuSmith wrote:So, you're saying that a_2=0 always?


No. What he is saying is given an a_3 and an a_1, there is only one possibility of what a_2 and a_0 can be.
my avatar is pretty awesome
tytia
User avatar
stupidityismygam
Ubiquitous Spammer
 
Posts: 1413
Joined: Tue Nov 21, 2006 7:02 pm
Top
Post a reply

Return to AMC/AIME

Who is online

Users browsing this forum: No registered users and 0 guests

Powered by phpBB® Forum Software © phpBB Group
cron