MathDB

3

Part of 2013 China Team Selection Test

Problems(6)

Painting balls into 4 colours

Source: 2013 China TST Quiz 1 Day 1 P3

3/30/2013
There arenn balls numbered 1,2,⋯ ,n1,2,\cdots,n, respectively. They are painted with 44 colours, red, yellow, blue, and green, according to the following rules: First, randomly line them on a circle. Then let any three clockwise consecutive balls numbered i,j,ki, j, k, in order. 1) If i>j>ki>j>k, then the ball jj is painted in red; 2) If i<j<ki<j<k, then the ball jj is painted in yellow; 3) If i<j,k<ji<j, k<j, then the ball jj is painted in blue; 4) If i>j,k>ji>j, k>j, then the ball jj is painted in green. And now each permutation of the balls determine a painting method. We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods.
Find out the number of all distinct painting methods.
inductioncombinatorics proposedcombinatorics
Find all positive reals r such that a set S exists

Source: 14th March 2013

4/1/2013
Find all positive real numbers r<1r<1 such that there exists a set S\mathcal{S} with the given properties: i) For any real number tt, exactly one of t,t+rt, t+r and t+1t+1 belongs to S\mathcal{S}; ii) For any real number tt, exactly one of t,t−rt, t-r and t−1t-1 belongs to S\mathcal{S}.
inductionmodular arithmeticalgebra proposedalgebra
China Team Selection Test 2013 TST 2 Day 2 Q3

Source: Nanjing high School , Jiangsu 19 Mar 2013

3/19/2013
Let n>1n>1 be an integer and let a0,a1,…,ana_0,a_1,\ldots,a_n be non-negative real numbers. Definite S_k=\sum_{i\equal{}0}^k \binom{k}{i}a_i for k=0,1,…,nk=0,1,\ldots,n. Prove that\frac{1}{n} \sum_{k\equal{}0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k\equal{}0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.
inequalitiesChina TSTalgebra
China Team Selection Test 2013 TST 2 Day 1 Q3

Source:

4/1/2013
Let AA be a set consisting of 6 points in the plane. denoted n(A)n(A) as the number of the unit circles which meet at least three points of AA. Find the maximum of n(A)n(A)
geometryparallelogramcombinatorial geometrycombinatorics proposedcombinatorics
2013 China IMO Team Selection Test 3 Day 1 Q3

Source: 24 Mar 2013

4/1/2013
101101 people, sitting at a round table in any order, had 1,2,...,1011,2,... , 101 cards, respectively. A transfer is someone give one card to one of the two people adjacent to him. Find the smallest positive integer kk such that there always can through no more than k k times transfer, each person hold cards of the same number, regardless of the sitting order.
combinatorics proposedcombinatorics
2013 China IMO Team Selection Test 3 Day 2 Q3

Source: 25 Mar 2013

4/1/2013
A point (x,y)(x,y) is a lattice point if x,y∈Zx,y\in\Bbb Z. Let E={(x,y):x,y∈Z}E=\{(x,y):x,y\in\Bbb Z\}. In the coordinate plane, PP and QQ are both sets of points in and on the boundary of a convex polygon with vertices on lattice points. Let T=P∩QT=P\cap Q. Prove that if T≠∅T\ne\emptyset and T∩E=∅T\cap E=\emptyset, then TT is a non-degenerate convex quadrilateral region.
analytic geometrygeometrycombinatorics proposedcombinatorics