MathDB

1

Part of 2001 Romania Team Selection Test

Problems(4)

If the complex a,b,c satisfy equations they are real

Source: Romanian TST 2001

1/16/2011
Show that if a,b,ca,b,c are complex numbers that such that (a+b)(a+c)=b(b+c)(b+a)=c(c+a)(c+b)=a (a+b)(a+c)=b \qquad (b+c)(b+a)=c \qquad (c+a)(c+b)=a then a,b,ca,b,c are real numbers.
trigonometrycomplex numbersalgebra proposedalgebra
Nice polynomial equation

Source: Romanian TST 2001

1/16/2011
Find all polynomials with real coefficients PP such that P(x)P(2x2−1)=P(x2)P(2x−1) P(x)P(2x^2-1)=P(x^2)P(2x-1) for every x∈Rx\in\mathbb{R}.
algebrapolynomialnumber theoryrelatively primealgebra proposed
There exists c such that n^k divides f(c) for all k

Source: Romanian TST 2001

1/16/2011
Let nn be a positive integer and f(x)=amxm+…+a1X+a0f(x)=a_mx^m+\ldots + a_1X+a_0, with m≥2m\ge 2, a polynomial with integer coefficients such that: a) a2,a3…ama_2,a_3\ldots a_m are divisible by all prime factors of nn, b) a1a_1 and nn are relatively prime. Prove that for any positive integer kk, there exists a positive integer cc, such that f(c)f(c) is divisible by nkn^k.
algebrapolynomialnumber theory proposednumber theory
Find all pair fulfilling divisibility condition

Source: Romania 2001

11/10/2004
Find all pairs (m,n)\left(m,n\right) of positive integers, with m,n≥2m,n\geq2, such that an−1a^n-1 is divisible by mm for each a∈{1,2,3,…,n}a\in \left\{1,2,3,\ldots,n\right\}.
modular arithmeticnumber theory proposednumber theorypolynomial congruence