MathDB

2

Part of 2006 China Team Selection Test

Problems(8)

Circumcircle passes through orthocentre [Fuhrmann extended]

Source: China TST 2006

6/18/2006
Let ω\omega be the circumcircle of △ABC\triangle{ABC}. PP is an interior point of △ABC\triangle{ABC}. A1,B1,C1A_{1}, B_{1}, C_{1} are the intersections of AP,BP,CPAP, BP, CP respectively and A2,B2,C2A_{2}, B_{2}, C_{2} are the symmetrical points of A1,B1,C1A_{1}, B_{1}, C_{1} with respect to the midpoints of side BC,CA,ABBC, CA, AB. Show that the circumcircle of △A2B2C2\triangle{A_{2}B_{2}C_{2}} passes through the orthocentre of △ABC\triangle{ABC}.
geometrycircumcirclegeometric transformationreflectionparallelogramratioanalytic geometry
Find (a,n)

Source: China TST 2006 (1)

3/24/2006
Find all positive integer pairs (a,n)(a,n) such that (a+1)n−ann\frac{(a+1)^n-a^n}{n} is an integer.
number theoryChina TSTHi
Functional equation

Source: China TST 2006

6/18/2006
The function f(n)f(n) satisfies f(0)=0f(0)=0, f(n)=n−f(f(n−1))f(n)=n-f \left( f(n-1) \right), n=1,2,3⋯n=1,2,3 \cdots. Find all polynomials g(x)g(x) with real coefficient such that f(n)=[g(n)],n=0,1,2⋯ f(n)= [ g(n) ], \qquad n=0,1,2 \cdots Where [g(n)][ g(n) ] denote the greatest integer that does not exceed g(n)g(n).
functionalgebrapolynomialalgebra unsolved
N variables inequality

Source: 2006 china tst

5/19/2006
x1,x2,⋯ ,xnx_{1}, x_{2}, \cdots, x_{n} are positive numbers such that ∑i=1nxi=1\sum_{i=1}^{n}x_{i}= 1. Prove that (∑i=1nxi)(∑i=1n11+xi)≤n2n+1\left( \sum_{i=1}^{n}\sqrt{x_{i}}\right) \left( \sum_{i=1}^{n}\frac{1}{\sqrt{1+x_{i}}}\right) \leq \frac{n^{2}}{\sqrt{n+1}}
inequalitiesfunctionCauchy Inequalityinequalities proposed
Prime divisor

Source: China TST 2006

6/18/2006
Prove that for any given positive integer mm and nn, there is always a positive integer kk so that 2k−m2^k-m has at least nn different prime divisors.
modular arithmeticnumber theory unsolvednumber theory
Sum of reciprocals

Source: China TST 2006

6/18/2006
Given positive integer nn, find the biggest real number CC which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than 11 is less than CC, then we can divide the set of numbers into no more than nn groups so that the sum of reciprocals of every group is less than 11.
algebrareciprocal sumExtremal combinatoricsmaximization
Inequality with x+y+z=1

Source: China TST 2006

6/18/2006
Given three positive real numbers x x, y y, z z such that x \plus{} y \plus{} z \equal{} 1, prove that \frac {xy}{\sqrt {xy \plus{} yz}} \plus{} \frac {yz}{\sqrt {yz \plus{} zx}} \plus{} \frac {zx}{\sqrt {zx \plus{} xy}} \le \frac {\sqrt {2}}{2}.
inequalitiesfunction
Integer and set

Source: China TST 2006

6/18/2006
Given positive integers mm, aa, bb, (a,b)=1(a,b)=1. AA is a non-empty subset of the set of all positive integers, so that for every positive integer nn there is an∈Aan \in A and bn∈Abn \in A. For all AA that satisfy the above condition, find the minimum of the value of ∣A∩{1,2,⋯ ,m}∣\left| A \cap \{ 1,2, \cdots,m \} \right|
number theory unsolvednumber theory