http://codeforces.com/contest/369/problem/E

http://www.spoj.com/problems/INVCNT/

http://codeforces.com/gym/100741/problem/A

https://www.hackerrank.com/challenges/similarpair

https://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&page=show_problem&problem=3977

http://codeforces.com/problemset/problem/296/C

http://codeforces.com/contest/276/problem/C

https://codeforces.com/contest/1354/problem/D

Find problems under Fenwick section in this blog

https://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=24&page=show_problem&problem=3238

http://www.lightoj.com/volume_showproblem.php?problem=1112

http://www.lightoj.com/volume_showproblem.php?problem=1266

http://www.codechef.com/problems/SPREAD

http://www.spoj.com/problems/CTRICK/

http://www.spoj.com/problems/MATSUM/

http://www.spoj.com/problems/DQUERY/

http://www.spoj.com/problems/YODANESS/

http://www.spoj.com/problems/NKTEAM/

https://community.topcoder.com/stat?c=problem_statement&pm=6551&rd=9990

http://www.spoj.com/problems/ADABEHIVE/

https://www.hackerearth.com/practice/data-structures/advanced-data-structures/fenwick-binary-indexed-trees/practice-problems/algorithm/counting-in-byteland/

https://devskill.com/CodingProblems/ViewProblem/300

http://codeforces.com/contest/669/problem/E

http://codeforces.com/contest/777/problem/E

http://www.spoj.com/problems/TULIPNUM/

http://www.spoj.com/problems/SUMSUM/

http://www.spoj.com/problems/SGIFT/

http://www.spoj.com/problems/TPGA/

http://www.spoj.com/problems/ZIGZAG2/

http://www.spoj.com/problems/CRAYON/

http://www.spoj.com/problems/DCEPC705/

http://www.spoj.com/problems/DCEPC206/

http://www.spoj.com/problems/KOPC12G/

http://www.spoj.com/problems/TRIPINV/

http://codeforces.com/contest/597/problem/C

http://codeforces.com/contest/12/problem/D

http://codeforces.com/gym/101047/problem/J

http://codeforces.com/contest/707/problem/E

http://codeforces.com/contest/749/problem/E

http://codeforces.com/problemset/gymProblem/101055/D

http://codeforces.com/contest/849/problem/E

http://www.spoj.com/problems/ADACABAA/

https://codeforces.com/problemset/problem/704/A

http://matcomgrader.com/problem/9346/fundraising/

And about 70 problems from hackerearth here:

https://www.hackerearth.com/practice/data-structures/advanced-data-structures/fenwick-binary-indexed-trees/practice-problems/

Nice list.

A good problem with a Fenwick tree. https://codeforces.com/problemset/problem/459/D

Does one need to learn BIT when he knows segment tree ? Are there any problems which an be solved only with BIT and not segment tree ? I know BIT has shorter code and is slightly faster, but is there any other advantage ?

insightful blog comments :)