MathDB

3

Part of 2006 China Team Selection Test

Problems(8)

Integer and inequality

Source: China TST 2006 (1)

3/24/2006
Given nn real numbers a1a_1, a2a_2 …\ldots ana_n. (n≥1n\geq 1). Prove that there exists real numbers b1b_1, b2b_2 …\ldots bnb_n satisfying: (a) For any 1≤i≤n1 \leq i \leq n, ai−bia_i - b_i is a positive integer. (b)∑1≤i<j≤n(bi−bj)2≤n2−112\sum_{1 \leq i < j \leq n} (b_i - b_j)^2 \leq \frac{n^2-1}{12}
inequalitiesfunctioninequalities unsolved
Least value of n

Source: China TST 2006

6/18/2006
Let aia_{i} and bib_{i} (i=1,2,⋯ ,ni=1,2, \cdots, n) be rational numbers such that for any real number xx there is: x2+x+4=∑i=1n(aix+b)2x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2} Find the least possible value of nn.
algebrapolynomialnumber theorygreatest common divisormodular arithmeticvectorlinear algebra
n-element set

Source: China TST 2006

6/18/2006
dd and nn are positive integers such that d∣nd \mid n. The n-number sets (x1,x2,⋯xn)(x_1, x_2, \cdots x_n) satisfy the following condition: (1) 0≤x1≤x2≤⋯≤xn≤n0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n (2) d∣(x1+x2+⋯xn)d \mid (x_1+x_2+ \cdots x_n) Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy xn=nx_n=n.
geometrygeometric transformationrotationfunctioncombinatorics unsolvedcombinatorics
Find polynomial

Source: China TST 2006

6/18/2006
Find all second degree polynomial d(x)=x2+ax+bd(x)=x^{2}+ax+b with integer coefficients, so that there exists an integer coefficient polynomial p(x)p(x) and a non-zero integer coefficient polynomial q(x)q(x) that satisfy: \left( p(x) \right)^{2}-d(x) \left( q(x) \right)^{2}=1,   \forall x \in \mathbb R.
algebrapolynomialalgebra unsolved
Coloured segments

Source: China TST 2006

6/18/2006
Given positive integers mm and nn so there is a chessboard with mnmn 1×11 \times 1 grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.
number theory unsolvednumber theory
Polynomial

Source: China TST 2006

6/18/2006
kk and nn are positive integers that are greater than 11. NN is the set of positive integers. A1,A2,⋯AkA_1, A_2, \cdots A_k are pairwise not-intersecting subsets of NN and A1∪A2∪⋯∪Ak=NA_1 \cup A_2 \cup \cdots \cup A_k = N. Prove that for some i∈{1,2,⋯ ,k}i \in \{ 1,2,\cdots,k \}, there exsits infinity many non-factorable n-th degree polynomials so that coefficients of one polynomial are pairwise distinct and all the coeficients are in AiA_i.
algebrapolynomialpigeonhole principlealgebra unsolved
Good and bad

Source: China TST 2006

6/18/2006
For a positive integer MM, if there exist integers aa, bb, cc and dd so that: M≤a<b≤c<d≤M+49,ad=bc M \leq a < b \leq c < d \leq M+49, \qquad ad=bc then we call MM a GOOD number, if not then MM is BAD. Please find the greatest GOOD number and the smallest BAD number.
number theory unsolvednumber theory
Cover

Source: China TST 2006

6/18/2006
△ABC\triangle{ABC} can cover a convex polygon MM.Prove that there exsit a triangle which is congruent to △ABC\triangle{ABC} such that it can also cover MM and has one side line paralel to or superpose one side line of MM.
geometry unsolvedgeometry