MathDB

2

Part of 1996 Vietnam Team Selection Test

Problems(2)

2^f(n) divides sum of binomials

Source: Vietnam TST 1996 for the 37th IMO, problem 2

6/26/2005
For each positive integer nn, let f(n)f(n) be the maximal natural number such that: 2f(n)2^{f(n)} divides ∑i=0⌊n−12⌋(n2⋅i+1)3i\sum^{\left\lfloor \frac{n - 1}{2}\right\rfloor}_{i=0} \binom{n}{2 \cdot i + 1} 3^i. Find all nn such that f(n)=1996.f(n) = 1996. [hide="old version"]For each positive integer nn, let f(n)f(n) be the maximal natural number such that: 2f(n)2^{f(n)} divides ∑i=1n+1/2(2⋅i+1n)\sum^{n + 1/2}_{i=1} \binom{2 \cdot i + 1}{n}. Find all nn such that f(n)=1996.f(n) = 1996.
floor functionnumber theory unsolvednumber theory
Can the number of people be 65?

Source: Vietnam TST 1996 for the 37th IMO, problem 5

6/26/2005
There are some people in a meeting; each doesn't know at least 56 others, and for any pair, there exist a third one who knows both of them. Can the number of people be 65?
parameterizationlinear algebramatrixcombinatorics unsolvedcombinatorics