MathDB

Easy number theory 2017 Taiwan TST

Source: 2017 Taiwan TST Round 1

4/13/2018

For postive integers

k,n

, let

f_k(n)=\sum_{m\mid n,m>0}m^k

Find all pairs of positive integer

(a,b)

such that

f_a(n)\mid f_b(n)

for every positive integer

n

.

number theory

Concentric Circles

Source: USAMO 1998

10/9/2005

Let

{\cal C}_1

and

{\cal C}_2

be concentric circles, with

{\cal C}_2

in the interior of

{\cal C}_1

. From a point

A

on

{\cal C}_1

one draws the tangent

AB

to

{\cal C}_2

(

B\in {\cal C}_2

). Let

C

be the second point of intersection of

AB

and

{\cal C}_1

, and let

D

be the midpoint of

AB

. A line passing through

A

intersects

{\cal C}_2

at

E

and

F

in such a way that the perpendicular bisectors of

DE

and

CF

intersect at a point

M

on

AB

. Find, with proof, the ratio

AM/MC

.

geometrycircumcircleperpendicular bisectorUSAMOpower of a point

Polynomial Functional Equation

Source: 2017 Taiwan TST Round 1, Day 3, Problem 1. Proposed by CSJL

4/14/2017

Find all polynomials

P

with real coefficients which satisfy P(x)P(x+1)=P(x^2-x+3) \forall x \in \mathbb{R}

algebrapolynomialFunctional Equationsfunctional equation

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