By the time a party is over, $ 28$ handshakes have occurred. If everyone shook everyone else's hand once, how many people attended the party?
Problem
Source:
Tags: MATHCOUNTS, geometry, geometric transformation, blogs
AIME15
30.04.2009 05:26
If everyone shakes everyone else's hands, then each pair of people produces a handshake. Thus, $ \binom{n}{2}=28$, where $ n$ is the number of people in the party. Thus:
\begin{align*}
\binom{n}{2}&=28
\\ \frac{n(n-1)}{2}&=28
\\ n(n-1)&=56
\\ n&=\boxed{8}
\end{align*}
There are $ \boxed{8}$ people in the party.
grn_trtle
30.04.2009 06:24
This was on their TST? Their IMO TST?
AIME15
30.04.2009 06:25
This could be on a MATHCOUNTS School Sprint round, arguably in the first 5-10 problems.
tonypr
30.04.2009 06:30
Unfortunately yes, it's a bit sad but you have to understand Puerto Rico's Math Olympics is Relatively new, like 8 or 9 maybe 10 years old, from what I believe, this problem is on the test so that the 7th and 8th graders don't feel discouraged from the rest of the test that they might not be able to get, (note 7th-12th graders take the same test). While the Students in the higher levels fight for the best solution.
grn_trtle
30.04.2009 06:32
Hmm... even so, they're kind of banking on the best students to make it through. If a beginner manages a perfect score on the TST and makes the IMO team, he/she won't be able to keep up with anyone else at the actual IMO.
lukezli
30.04.2009 06:44
yeah, but still.... itd be cool to go to imo! i wish i lived in puerto rico.
metafor
01.05.2009 03:45
grn_trtle wrote: Hmm... even so, they're kind of banking on the best students to make it through. If a beginner manages a perfect score on the TST and makes the IMO team, he/she won't be able to keep up with anyone else at the actual IMO. Haven't you seen our scores? That's mostly what happens (we can't keep up with other teams). Mathwise, we're way below average. Still, I enjoyed owning that test.
Aristocrat
01.05.2009 03:48
I wonder if I can travel to Puerto Rico, make IMO, and come back to the States and write that in my college applications...
tonypr
01.05.2009 04:38
It'd be pretty hard, first off normally they only have enough money to send 3 students to the IMO, maybe that's changing this year but I'm not sure, and plus the first round is always in spanish(From what I have seen), all the subsequent tests have translations to english, but this test was easier than other years, I would post the other years, but right now i'm too tired. Metafor do you want to post them?(I know you have them in your blog)
metafor
01.05.2009 05:50
You will find last year's "TST" here.