MathDB

1

Part of 2018 China Team Selection Test

Problems(4)

Existence of reals satisfying cyclic relation

Source: 2018 China TST Day 1 Q1

1/2/2018
Let p,qp,q be positive reals with sum 1. Show that for any nn-tuple of reals (y1,y2,...,yn)(y_1,y_2,...,y_n), there exists an nn-tuple of reals (x1,x2,...,xn)(x_1,x_2,...,x_n) satisfying p⋅max⁡{xi,xi+1}+q⋅min⁡{xi,xi+1}=yip\cdot \max\{x_i,x_{i+1}\} + q\cdot \min\{x_i,x_{i+1}\} = y_i for all i=1,2,...,2017i=1,2,...,2017, where x2018=x1x_{2018}=x_1.
algebra
Sad Geometry

Source: 2018 China TST Day 3 Q 1

1/21/2018
Given a triangle ABCABC. DD is a moving point on the edge BCBC. Point EE and Point FF are on the edge ABAB and ACAC, respectively, such that BE=CDBE=CD and CF=BDCF=BD. The circumcircle of △BDE\triangle BDE and △CDF\triangle CDF intersects at another point PP other than DD. Prove that there exists a fixed point QQ, such that the length of QPQP is constant.
geometry
Internal common tangents and angle condition

Source: China TST 3 2018 Day 1 Q1

3/27/2018
Let ω1,ω2\omega_1,\omega_2 be two non-intersecting circles, with circumcenters O1,O2O_1,O_2 respectively, and radii r1,r2r_1,r_2 respectively where r1<r2r_1 < r_2. Let AB,XYAB,XY be the two internal common tangents of ω1,ω2\omega_1,\omega_2, where A,XA,X lie on ω1\omega_1, B,YB,Y lie on ω2\omega_2. The circle with diameter ABAB meets ω1,ω2\omega_1,\omega_2 at PP and QQ respectively. If ∠AO1P+∠BO2Q=180∘,\angle AO_1P+\angle BO_2Q=180^{\circ}, find the value of PXQY\frac{PX}{QY} (in terms of r1,r2r_1,r_2).
geometry
A Rational Root Problem on Polymonial Sequence

Source: 2018 China TST 4 Day 1 Problem 1

3/27/2018
Define the polymonial sequence {fn(x)}n≥1\left \{ f_n\left ( x \right ) \right \}_{n\ge 1} with f1(x)=1f_1\left ( x \right )=1, f2n(x)=xfn(x),  f2n+1(x)=fn(x)+fn+1(x),  n≥1.f_{2n}\left ( x \right )=xf_n\left ( x \right ), \; f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), \; n\ge 1. Look for all the rational number aa which is a root of certain fn(x).f_n\left ( x \right ).
polynomialPolynomialsalgebraRational Root TheoremSequence