MathDB

3

Part of 2007 India IMO Training Camp

Problems(3)

Find the number of "balanced" n-strings

Source: Indian IMO Training Camp 2007-ST 2 P3

3/5/2011
Given a finite string SS of symbols XX and OO, we denote Δ(s)\Delta(s) as the number ofX′X's in SS minus the number of O′O's (For example, Δ(XOOXOOX)=−1\Delta(XOOXOOX)=-1). We call a string SS balanced if every sub-string TT of (consecutive symbols) SS has the property −1≤Δ(T)≤2.-1\leq \Delta(T)\leq 2. (Thus XOOXOOXXOOXOOX is not balanced, since it contains the sub-string OOXOOOOXOO whose Δ\Delta value is −3.-3. Find, with proof, the number of balanced strings of length nn.
Putnamabsolute valuecombinatorics unsolvedcombinatorics
S={1,2,...,n}; T_{f}(j)=1,0; Determine sum(sum(T_f(j))

Source: Indian IMO Training Camp 2007-ST 1 P3

3/5/2011
Let X\mathbb X be the set of all bijective functions from the set S={1,2,⋯ ,n}S=\{1,2,\cdots, n\} to itself. For each f∈X,f\in \mathbb X, define Tf(j)={1,   if  f(12)(j)=j,0,   otherwiseT_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \ f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right. Determine ∑f∈X∑j=1nTf(j).\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j). (Here f(k)(x)=f(f(k−1)(x))f^{(k)}(x)=f(f^{(k-1)}(x)) for all k≥2.k\geq 2.)
functionprobabilityexpected valuecombinatorics unsolvedcombinatorics
Find all functional equations satisfying f(x+y)+f(x)f(y)=...

Source: Indian IMO Training Camp 2007-ST 4 P3

3/5/2011
Find all function(s) f:R→Rf:\mathbb R\to\mathbb R satisfying the equation f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy); For all x,y∈R.x,y\in\mathbb R.
functionratioalgebra proposedalgebra