MathDB

1

Part of 2011 China Team Selection Test

Problems(6)

Nine point circle is tangent to incircle and three excircles

Source: China TST 2011 Day 1

3/28/2011
In △ABC\triangle ABC we have BC>CA>ABBC>CA>AB. The nine point circle is tangent to the incircle, AA-excircle, BB-excircle and CC-excircle at the points T,TA,TB,TCT,T_A,T_B,T_C respectively. Prove that the segments TTBTT_B and lines TATCT_AT_C intersect each other.
geometryrectangleanalytic geometrytrapezoidgeometric transformationhomothetytrigonometry
AP is perpendicular to PC

Source: China TST 2011 - Quiz 1 - D2 - P1

5/19/2011
Let one of the intersection points of two circles with centres O1,O2O_1,O_2 be PP. A common tangent touches the circles at A,BA,B respectively. Let the perpendicular from AA to the line BPBP meet O1O2O_1O_2 at CC. Prove that AP⊥PCAP\perp PC.
All functions f(x-f(y))=f(x+y^n)+f(f(y)+y^n) with fixed n

Source: China TST 2011 - Quiz 2 - D1 - P1

5/20/2011
Let n≥2n\geq 2 be a given integer. Find all functions f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} such that f(x−f(y))=f(x+yn)+f(f(y)+yn),∀x,y∈R.f(x-f(y))=f(x+y^n)+f(f(y)+y^n), \qquad \forall x,y \in \mathbb R.
functionalgebra unsolvedalgebra
ABC is similar to XYZ

Source: China TST 2011 - Quiz 2 - D2 - P1

5/20/2011
Let AA′,BB′,CC′AA',BB',CC' be three diameters of the circumcircle of an acute triangle ABCABC. Let PP be an arbitrary point in the interior of △ABC\triangle ABC, and let D,E,FD,E,F be the orthogonal projection of PP on BC,CA,ABBC,CA,AB, respectively. Let XX be the point such that DD is the midpoint of A′XA'X, let YY be the point such that EE is the midpoint of B′YB'Y, and similarly let ZZ be the point such that FF is the midpoint of C′ZC'Z. Prove that triangle XYZXYZ is similar to triangle ABCABC.
geometrycircumcirclegeometric transformationreflectionperpendicular bisector
The largest M for which there exists an arrangement

Source: China TST 2011 - Quiz 3 - D1 - P1

5/20/2011
Let n≥3n\geq 3 be an integer. Find the largest real number MM such that for any positive real numbers x1,x2,⋯ ,xnx_1,x_2,\cdots,x_n, there exists an arrangement y1,y2,⋯ ,yny_1,y_2,\cdots,y_n of real numbers satisfying ∑i=1nyi2yi+12−yi+1yi+2+yi+22≥M,\sum_{i=1}^n \frac{y_i^2}{y_{i+1}^2-y_{i+1}y_{i+2}+y_{i+2}^2}\geq M, where yn+1=y1,yn+2=y2y_{n+1}=y_1,y_{n+2}=y_2.
inequalities unsolvedinequalities
Prove that the triangle KJM is isosceles

Source: China TST 2011 - Quiz 3 - D2 - P1

5/20/2011
Let HH be the orthocenter of an acute trangle ABCABC with circumcircle Γ\Gamma. Let PP be a point on the arc BCBC (not containing AA) of Γ\Gamma, and let MM be a point on the arc CACA (not containing BB) of Γ\Gamma such that HH lies on the segment PMPM. Let KK be another point on Γ\Gamma such that KMKM is parallel to the Simson line of PP with respect to triangle ABCABC. Let QQ be another point on Γ\Gamma such that PQ∥BCPQ \parallel BC. Segments BCBC and KQKQ intersect at a point JJ. Prove that △KJM\triangle KJM is an isosceles triangle.
geometrycircumcircletrigonometrytrapezoidgeometric transformationreflectionparallelogram