MathDB

1

Part of 2005 China Team Selection Test

Problems(7)

concurrent or parallel

Source: China TST 2005-1

6/14/2005
Convex quadrilateral ABCDABCD is cyclic in circle (O)(O), PP is the intersection of the diagonals ACAC and BDBD. Circle (O1)(O_{1}) passes through PP and BB, circle (O2)(O_{2}) passes through PP and AA, Circles (O1)(O_{1}) and (O2)(O_{2}) intersect at PP and QQ. (O1)(O_{1}), (O2)(O_{2}) intersect (O)(O) at another points EE, FF (besides BB, AA), respectively. Prove that PQPQ, CECE, DFDF are concurrent or parallel.
geometry solvedgeometry
permutation

Source: China TST 2005-4

6/14/2005
Let a1a_{1}, a2a_{2}, …, a6a_{6}; b1b_{1}, b2b_{2}, …, b6b_{6} and c1c_{1}, c2c_{2}, …, c6c_{6} are all permutations of 11, 22, …, 66, respectively. Find the minimum value of ∑i=16aibici\sum_{i=1}^{6}a_{i}b_{i}c_{i}.
inequalitiesrearrangement inequalitycombinatorics unsolvedcombinatorics
concurrency related to projections

Source: China TST 2005 Quiz 1.1 (missing from contest collections)

11/21/2022
Point PP lies inside triangle ABCABC. Let the projections of PP onto sides BCBC,CACA,ABAB be DD, EE, FF respectively. Let the projections from AA to the lines BPBP and CPCP be MM and NN respectively. Prove that MEME, NFNF and BCBC are concurrent.
geometryconcurrencyconcurrentprojective geometry
Division of a set

Source: China TST 2005

6/27/2006
Let kk be a positive integer. Prove that one can partition the set {0,1,2,3,⋯ ,2k+1−1}\{ 0,1,2,3, \cdots ,2^{k+1}-1 \} into two disdinct subsets {x1,x2,⋯ ,x2k}\{ x_1,x_2, \cdots, x_{2k} \} and {y1,y2,⋯ ,y2k}\{ y_1, y_2, \cdots, y_{2k} \} such that ∑i=12kxim=∑i=12kyim\sum_{i=1}^{2^k} x_i^m =\sum_{i=1}^{2^k} y_i^m for all m∈{1,2,⋯ ,k}m \in \{ 1,2, \cdots, k \}.
inductioncombinatorics
Equal angle

Source: China TST 2005

6/27/2006
Triangle ABCABC is inscribed in circle ω\omega. Circle γ\gamma is tangent to ABAB and ACAC at points PP and QQ respectively. Also circle γ\gamma is tangent to circle ω\omega at point SS. Let the intesection of ASAS and PQPQ be TT. Prove that ∠BTP=∠CTQ\angle{BTP}=\angle{CTQ}.
trigonometrygeometryincentergeometric transformationhomothetypower of a pointgeometry unsolved
Sum of denominators

Source: China TST 2005

6/27/2006
Prove that for any nn (n≥2n \geq 2) pairwise distinct fractions in the interval (0,1)(0,1), the sum of their denominators is no less than 13n32\frac{1}{3} n^{\frac{3}{2}}.
inequalitiesalgebra
Inequality with guass function

Source: China TST 2005

6/27/2006
Find all positive integers mm and nn such that the inequality: [(m+n)α]+[(m+n)β]≥[mα]+[nβ]+[n(α+β)] [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] is true for any real numbers α\alpha and β\beta. Here [x][x] denote the largest integer no larger than real number xx.
inequalitiesfunctionfloor functioninequalities unsolved