MathDB

4

Part of 2008 Postal Coaching

Problems(6)

l < m but a_l > a_m or a_l - l is an odd number

Source: Indian Postal Coaching 2008 set 1 p4

5/25/2020
Let n∈Nn \in N and kk be such that 1≤k≤n1 \le k \le n. Find the number of ordered kk-tuples (a1,a2,...,ak)(a_1, a_2,...,a_k) of integers such the 1≤aj≤n1 \le a_j \le n, for 1≤j≤k1 \le j \le k and either there exist l,m∈{1,2,...,k}l,m \in \{1, 2,..., k\} such that l<ml < m but al>ama_l > a_m or there exists l∈{1,2,...,k}l \in \{1, 2,..., k\} such that al−la_l - l is an odd number.
combinatoricsoddinequalities
f(xf(y))= (1 - y)f(xy) + x^2y^2f(y)

Source: Indian Postal Coaching 2008 set 2 p4

5/25/2020
Find all functions f:R→Rf : R \to R such that f(xf(y))=(1−y)f(xy)+x2y2f(y)f(xf(y))= (1 - y)f(xy) + x^2y^2f(y) for all reals x,yx, y.
functionalfunctional equationalgebra
P(x) = x^4 - 8p^2/q x^3 + 4qx^2 - 3px + p^2 = 0 4 positive roots

Source: Indian Postal Coaching 2008 set 3 p4

5/25/2020
Find all real numbersp,qp, q for which the polynomial equation P(x)=x4−8p2qx3+4qx2−3px+p2=0P(x) = x^4 - \frac{8p^2}{q}x^3 + 4qx^2 - 3px + p^2 = 0 has four positive roots.
algebrapolynomialroots
n distinct naturals whose sum is a square and whose product is a cube

Source: Indian Postal Coaching 2008 set 5 p4

5/25/2020
Show that for each natural number nn, there exist nn distinct natural numbers whose sum is a square and whose product is a cube.
number theoryPerfect Squareperfect cube
checkers on a 8x8 chessboard

Source: Indian Postal Coaching 2008 set 4 p4

5/25/2020
An 8×88\times 8 square board is divided into 6464 unit squares. A ’skew-diagonal’ of the board is a set of 88 unit squares no two of which are in the same row or same column. Checkers are placed in some of the unit squares so that ’each skew-diagonal contains exactly two squares occupied by checkers’. Prove that there exist two rows or two columns which contain all the checkers.
combinatoricsChessboard
partition with same cardinality and same sum of elements of {1,2,3,...,n}

Source: Indian Postal Coaching 2008 set 6 p4

5/25/2020
Consider the set A={1,2,...,n}A = \{1, 2, ..., n\}, where n∈N,n≥6n \in N, n \ge 6. Show that AA is the union of three pairwise disjoint sets, with the same cardinality and the same sum of their elements, if and only if nn is a multiple of 33.
partitioncombinatoricsSubsets