MathDB

3

Part of 2005 India IMO Training Camp

Problems(2)

Ascent and descent

Source: Indian IMOTC 2005 Day 3 problem 3

9/23/2005
A merida path of order 2n2n is a lattice path in the first quadrant of xyxy- plane joining (0,0)(0,0) to (2n,0)(2n,0) using three kinds of steps U=(1,1)U=(1,1), D=(1,−1)D= (1,-1) and L=(2,0)L= (2,0), i.e. UU joins x,y)x,y) to (x+1,y+1)(x+1,y+1) etc... An ascent in a merida path is a maximal string of consecutive steps of the form UU. If S(n,k)S(n,k) denotes the number of merdia paths of order 2n2n with exactly kk ascents, compute S(n,1)S(n,1) and S(n,n−1)S(n,n-1).
analytic geometrycombinatorics unsolvedcombinatorics
A trignometric eqn

Source: Indian IMOTC 2005 Day 5 Problem 3

9/23/2005
For real numbers a,b,c,da,b,c,d not all equal to 00 , define a real function f(x)=a+bcos⁡2x+csin⁡5x+dcos⁡8xf(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}. Suppose f(t)=4af(t) = 4a for some real tt. prove that there exist a real number ss s.t. f(s)<0f(s)<0
functiontrigonometryinequalitiesalgebra unsolvedalgebra