MathDB

5

Part of 2019 China Team Selection Test

Problems(4)

Weird FE on Rationals

Source: China TST Test 1 Day 2 Q5

3/11/2019
Determine all functions f:Q→Qf: \mathbb{Q} \to \mathbb{Q} such that f(2xy+12)+f(x−y)=4f(x)f(y)+12f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2} for all x,y∈Qx,y \in \mathbb{Q}.
functional equationfunctionChina TST
Length condition and prove angles sum to 90

Source: China TST Test 2 Day 2 Q5

3/11/2019
Let MM be the midpoint of BCBC of triangle ABCABC. The circle with diameter BCBC, ω\omega, meets AB,ACAB,AC at D,ED,E respectively. PP lies inside △ABC\triangle ABC such that ∠PBA=∠PAC,∠PCA=∠PAB\angle PBA=\angle PAC, \angle PCA=\angle PAB, and 2PM⋅DE=BC22PM\cdot DE=BC^2. Point XX lies outside ω\omega such that XM∥APXM\parallel AP, and XBXC=ABAC\frac{XB}{XC}=\frac{AB}{AC}. Prove that ∠BXC+∠BAC=90∘\angle BXC +\angle BAC=90^{\circ}.
geometryChina TST
Weird Geometry

Source: 2019 China TST Test 3 P5

3/29/2019
In ΔABC\Delta ABC, AD⊥BCAD \perp BC at DD. E,FE,F lie on line ABAB, such that BD=BE=BFBD=BE=BF. Let I,JI,J be the incenter and AA-excenter. Prove that there exist two points P,QP,Q on the circumcircle of ΔABC\Delta ABC , such that PB=QCPB=QC, and ΔPEI∼ΔQFJ\Delta PEI \sim \Delta QFJ .
geometryincentercircumcircle
Inequality about power sum

Source: 2019 China TST Test 4 P5

4/5/2019
Find all integer nn such that the following property holds: for any positive real numbers a,b,c,x,y,za,b,c,x,y,z, with max(a,b,c,x,y,z)=amax(a,b,c,x,y,z)=a , a+b+c=x+y+za+b+c=x+y+z and abc=xyzabc=xyz, the inequality an+bn+cn≥xn+yn+zna^n+b^n+c^n \ge x^n+y^n+z^n holds.
inequalitiesChina TSTChina