**2016 Putnam B1.** Let be the sequence such that and for , (as usual, the function is the natural logarithm.

Show that the infinite series converges and find its sum.

**2016 Putnam B1.** Let be the sequence such that and for , (as usual, the function is the natural logarithm.

Show that the infinite series converges and find its sum.

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Following the recent IMO 2016, I have been meaning to do some analysis on IMO results. Unfortunately I have not had time to do so…

In the meantime, I thought I’d share the data I’ve scraped so far so that others who have the time and interest might have a go at analysing the data. Data is available at my Github repo.

All data was scraped from imo-official.org; the scripts I used to scrape them are in the ETL folder of the same repo. The data generally looks clean except some minor issues for “Contestant” (i.e. names of contestants). For example, my name is written in one order for 2003 and 2005 but in a different way for 2004. I have no idea how widespread this issue is, although a cursory glance at contestants in my country suggest the issue is a minor one.

This year’s International Mathematical Olympiad (IMO) took place in Hong Kong from 6-16 July. The problems can be downloaded from this page or viewed at the Art of Problem Solving (AoPS) forum page for IMO 2016 (here).

* Congratulations to my country, Singapore’s team, for coming in 4th overall!* And for winning the country’s:

**Best**medal haul (4G 2S, previous best was 4G 1S 1B in 2011)**Highest**absolute score (196 out of a possible 252, previous best was 179 in 2011)**2nd best**overall team ranking (4th, best was 3rd in 2011)

Some factoids on the results:

- The Gold, Silver and Bronze medal score cutoffs were
**29, 22 and 16**respectively. Nothing too out of the ordinary. (Interestingly, IMO 2014 had the exact same cutoffs.) - The top 3 countries were the
**United States, South Korea and China**. (These 3 countries were the top 3 in 2015 as well, just a different order: USA, China, South Korea.)- Team USA won a best possible 6 Gold medals. (Next best was 4G 2S by South Korea, China and Singapore.)

- There were
**6**contestants with perfect scores: 3 from South Korea, 2 from USA and 1 from China.- 3 of them were first-time participants of the competition. 2 of them had won Gold in 2015, while the remaining contestant (Allen Liu, USA) had won Gold in 2014 and 2015.

**Question 3**was the most difficult question, with a mean score of 0.251, 91% of students scoring 0 points and only 10 contestants scoring a perfect 7 points. The easiest was**Question 1**with a mean score of 5.272. (The questions from easiest to hardest: 1, 4, 2, 5, 6, 3.)

And finally, some useful links:

- IMO 2016 official website (At the time of writing of this post, the results have not been posted on this website.)
- AoPS forum on Contests and Problem Sets (All the latest threads are related to IMO 2016.)
- imo-official.org page for IMO 2016 (Contains all the stats you could dream of for IMO 2016.)

**2013 AIME I 15.** Let be the number of ordered triples of integers satisfying the conditions

(a) ,

(b) there exist integers , , and , and prime where ,

(c) divides , , and , and

(d) each ordered triple and each ordered triple form arithmetic sequences.

Find .

**2013 AIME I 15.** Let be the number of ordered triples of integers satisfying the conditions

(a) ,

(b) there exist integers , , and , and prime where ,

(c) divides , , and , and

(d) each ordered triple and each ordered triple form arithmetic sequences.

Find .

**2013 AIME I 14.** For , let

and

so that . Then where and are relatively prime positive integers. Find .

**2013 AIME I 14.** For , let

and

so that . Then where and are relatively prime positive integers. Find .

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