MathDB

1

Part of 2007 China Team Selection Test

Problems(8)

from China TST 2007

Source: Chinese TST 2007 2nd quiz P1

3/20/2007
u,v,w>0 u,v,w > 0,such that u \plus{} v \plus{} w \plus{} \sqrt {uvw} \equal{} 4 prove that \sqrt {\frac {uv}{w}} \plus{} \sqrt {\frac {vw}{u}} \plus{} \sqrt {\frac {wu}{v}}\geq u \plus{} v \plus{} w
inequalitiestrigonometryquadraticsfunction
Collinear

Source: Friday, 13th - My friend give me

4/13/2007
Points A A and B B lie on the circle with center O. O. Let point C C lies outside the circle; let CS CS and CT CT be tangents to the circle. M M be the midpoint of minor arc AB AB of (O). (O). MS, MT MS,\,MT intersect AB AB at points E, F E,\,F respectively. The lines passing through E, F E,\,F perpendicular to AB AB cut OS, OT OS,\,OT at X X and Y Y respectively. A line passed through C C intersect the circle (O) (O) at P, Q P,\,Q (P P lies on segment CQ CQ). Let R R be the intersection of MP MP and AB, AB, and let Z Z be the circumcentre of triangle PQR. PQR. Prove that: X, Y, Z X,\,Y,\,Z are collinear.
geometrygeometric transformationhomothetypower of a pointradical axisgeometry proposed
fairly easy, routine function, from Q+ to Q+

Source: China TST 2007 Q4

2/21/2008
Find all functions f: \mathbb{Q}^{\plus{}} \mapsto \mathbb{Q}^{\plus{}} such that: f(x) \plus{} f(y) \plus{} 2xy f(xy) \equal{} \frac {f(xy)}{f(x\plus{}y)}.
functioninductionalgebra proposedalgebra
Equiangular Polygon

Source: Chinese TST 2007 1st quiz P1

1/3/2009
When all vertex angles of a convex polygon are equal, call it equiangular. Prove that p>2 p > 2 is a prime number, if and only if the lengths of all sides of equiangular p p polygon are rational numbers, it is a regular p p polygon.
algebrapolynomialgeometry proposedgeometry
Concurrent

Source: Problem 11.3 - MOSP 2007, Chinese TST 2007 3rd quiz P1

6/19/2007
Let ABC ABC be a triangle. Circle ω \omega­ passes through points B B and C. C. Circle ω1 \omega_{1} is tangent internally to ω \omega­ and also to sides AB AB and AC AC at T, P, T,\, P, and Q, Q, respectively. Let M M be midpoint of arc BC ( BC\, (containing T) T) of ­ω. \omega. Prove that lines PQ, BC, PQ,\,BC, and MT MT are concurrent.
geometryincentergeometric transformationhomothety
An inequality

Source: Chinese TST 2007 4th quiz P1

1/3/2009
Let a1,a2,⋯ ,an a_{1},a_{2},\cdots,a_{n} be positive real numbers satisfying a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{n} \equal{} 1. Prove that \left(a_{1}a_{2} \plus{} a_{2}a_{3} \plus{} \cdots \plus{} a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 \plus{} a_{2}} \plus{} \frac {a_{2}}{a_{3}^2 \plus{} a_{3}} \plus{} \cdots \plus{} \frac {a_{n}}{a_{1}^2 \plus{} a_{1}}\right)\ge\frac {n}{n \plus{} 1}
inequalitiesinequalities proposed
Segment equal

Source: Chinese TST 2007 5th quiz P1

1/4/2009
Let convex quadrilateral ABCD ABCD be inscribed in a circle centers at O. O. The opposite sides BA,CD BA,CD meet at H H, the diagonals AC,BD AC,BD meet at G. G. Let O1,O2 O_{1},O_{2} be the circumcenters of triangles AGD,BGC. AGD,BGC. O1O2 O_{1}O_{2} intersects OG OG at N. N. The line HG HG cuts the circumcircles of triangles AGD,BGC AGD,BGC at P,Q P,Q, respectively. Denote by M M the midpoint of PQ. PQ. Prove that NO \equal{} NM.
geometrycircumcircleparallelogramperpendicular bisectorgeometry proposed
The pairs of positive integers

Source: Chinese TST 2007 6th quiz P1

1/4/2009
Find all the pairs of positive integers (a,b) (a,b) such that a^2 \plus{} b \minus{} 1 is a power of prime number ; a^2 \plus{} b \plus{} 1 can divide b^2 \minus{} a^3 \minus{} 1, but it can't divide (a \plus{} b \minus{} 1)^2.
modular arithmeticnumber theory proposednumber theory