MathDB

2

Part of 2012 Indonesia TST

Problems(8)

Incircle cuts

Source: 2012 Indonesia Round 2 TST 2 Problem 2

3/4/2012
Let ABCABC be a triangle, and its incenter touches the sides BC,CA,ABBC,CA,AB at D,E,FD,E,F respectively. Let ADAD intersects the incircle of ABCABC at MM distinct from DD. Let DFDF intersects the circumcircle of CDMCDM at NN distinct from DD. Let CNCN intersects ABAB at GG. Prove that EC=3GFEC = 3GF.
geometryincentercircumcirclegeometry unsolved
Scores of a math competition

Source: 2012 Indonesia Round 2 TST 1 Problem 2

2/26/2012
A TV station holds a math talent competition, where each participant will be scored by 8 people. The scores are F (failed), G (good), or E (exceptional). The competition is participated by three people, A, B, and C. In the competition, A and B get the same score from exactly 4 people. C states that he has differing scores with A from at least 4 people, and also differing scores with B from at least 4 people. Assuming C tells the truth, how many scoring schemes can occur?
combinatorics proposedcombinatorics
Coloring columns that still differentiates rows

Source: 2012 Indonesia Round 2 TST 3 Problem 2

3/18/2012
An m×nm \times n chessboard where m≤nm \le n has several black squares such that no two rows have the same pattern. Determine the largest integer kk such that we can always color kk columns red while still no two rows have the same pattern.
inductionRoss Mathematics Programcombinatorics proposedcombinatorics
Locus of point in the line connecting foot of tangents

Source: 2012 Indonesia Round 2 TST 4 Problem 2

3/18/2012
Let ω\omega be a circle with center OO, and let ll be a line not intersecting ω\omega. EE is a point on ll such that OEOE is perpendicular with ll. Let MM be an arbitrary point on MM different from EE. Let AA and BB be distinct points on the circle ω\omega such that MAMA and MBMB are tangents to ω\omega. Let CC and DD be the foot of perpendiculars from EE to MAMA and MBMB respectively. Let FF be the intersection of CDCD and OEOE. As MM moves, determine the locus of FF.
geometry proposedgeometry
Coloring the integers, again

Source: 2012 Indonesia Round 2.5 TST 2 Problem 2

5/21/2012
The positive integers are colored with black and white such that: - There exists a bijection from the black numbers to the white numbers, - The sum of three black numbers is a black number, and - The sum of three white numbers is a white number.
Find the number of possible colorings that satisfies the above conditions.
combinatorics proposedcombinatorics
There is a,b in S where b divides 2a

Source: 2012 Indonesia Round 2.5 TST 1 Problem 2

5/10/2012
Suppose SS is a subset of {1,2,3,…,2012}\{1,2,3,\ldots,2012\}. If SS has at least 10001000 elements, prove that SS contains two different elements a,ba,b, where bb divides 2a2a.
combinatorics unsolvedcombinatorics
Indices of intersecting sets form a set

Source: 2012 Indonesia Round 2.5 TST 3 Problem 2

5/21/2012
Let P1,P2,…,PnP_1, P_2, \ldots, P_n be distinct 22-element subsets of {1,2,…,n}\{1, 2, \ldots, n\}. Suppose that for every 1≤i<j≤n1 \le i < j \le n, if Pi∩Pj≠∅P_i \cap P_j \neq \emptyset, then there is some kk such that Pk={i,j}P_k = \{i, j\}. Prove that if a∈Pia \in P_i for some ii, then a∈Pja \in P_j for exactly one value of jj not equal to ii.
combinatorics unsolvedcombinatorics
No three numbers use all digits

Source: 2012 Indonesia Round 2.5 TST 4 Problem 2

5/31/2012
Let TT be the set of all 2-digit numbers whose digits are in {1,2,3,4,5,6}\{1,2,3,4,5,6\} and the tens digit is strictly smaller than the units digit. Suppose SS is a subset of TT such that it contains all six digits and no three numbers in SS use all six digits. If the cardinality of SS is nn, find all possible values of nn.
combinatorics proposedcombinatorics