MathDB

2

Part of 2005 Taiwan TST Round 1

Problems(5)

area problem

Source: Taiwan 1st TST 2005, 3rd independent study, problem 2

8/12/2005
Let ABCDABCD be a convex quadrilateral. Is it possible to find a point PP such that the segments drawn between PP and the midpoints of the sides of ABCDABCD divide the quadrilateral into four sections of equal area? If PP exists, is it unique?
geometrygeometry proposed
Easy Algebra

Source: Taiwan 1st TST 2005, 1st independent study, problem 2

8/12/2005
Does there exist an positive integer nn, so that for any positive integer m<1002m<1002, there exists an integer kk so that m1002<kn<m+11003\displaystyle \frac{m}{1002} < \frac{k}{n} < \frac {m+1}{1003} holds? If nn does not exist, prove it; if nn exists, determine the minimum value of it. I know this problem was easy, but it still appeared on our TST, and so I posted it here.
algebra solvedalgebra
Recurrence...

Source: Taiwan 1st TST, 2nd independent study, problem 2

8/12/2005
The absolute value of every number in the sequence {an}\{a_n\} is smaller than 2005, and an+6=an+4+an+2−an.a_{n+6}=a_{n+4}+a_{n+2}-a_n. holds for all positive integers n. Prove that {an}\{a_n\} is periodic. Incredibly, this was probably the most difficult problem of our independent study problems in the 1st TST (excluding the final exam).
absolute valuealgebra solvedalgebra
angle problem

Source: Taiwan 1st TST 2005, final exam, first day, problem 2

8/12/2005
PP is a point in the interior of △ABC\triangle ABC, and ∠ABP=∠PCB=10∘\angle ABP = \angle PCB = 10^\circ. (a) If ∠PBC=10∘\angle PBC = 10^\circ and ∠ACP=20∘\angle ACP = 20^\circ, what is the value of ∠BAP\angle BAP? (b) If ∠PBC=20∘\angle PBC = 20^\circ and ∠ACP=10∘\angle ACP = 10^\circ, what is the value of ∠BAP\angle BAP?
trigonometrygeometryincentersymmetrycyclic quadrilateralgeometry proposed
tetrahedron

Source: Taiwan 1st TST 2005, final exam, second day, problem 5

8/12/2005
Show that for any tetrahedron, the condition that opposite edges have the same length is equivalent to the condition that the segment drawn between the midpoints of any two opposite edges is perpendicular to the two edges.
geometry3D geometrytetrahedrongeometry proposed