MathDB

2

Part of 2007 Moldova Team Selection Test

Problems(4)

p|m_km_{\sigma(k)}-m_lm_{\sigma(l)}

Source: Moldova 2007 IMO-BMO TST I problem 2

3/5/2007
Consider pp a prime number and pp consecutive positive integers m1,m2,…,mpm_{1}, m_{2}, \ldots, m_{p}. Choose a permutation σ\sigma of 1,2,…,p1, 2, \ldots, p. Show that there exist two different numbers k,l∈{1,2,…,p}k,l \in \{1,2, \ldots, p\} such that mkmσ(k)−mlmσ(l)m_{k}m_{\sigma(k)}-m_{l}m_{\sigma(l)} is divisible by pp.
modular arithmeticalgebrapolynomialnumber theorynumber theory proposed
sum of AI less than 3R

Source: Moldova 2007 IMO-BMO TST II problem 2

3/23/2007
If II is the incenter of a triangle ABCABC and RR is the radius of its circumcircle then AI+BI+CI≤3RAI+BI+CI\leq 3R
geometryincentercircumcircletrigonometryinequalitiestrig identitiesLaw of Sines
A famous Cauchy theorem for polynomials

Source: Moldova 2007 IMO-BMO TST IV problem 2

3/25/2007
If b1,b2,…,bnb_{1}, b_{2}, \ldots, b_{n} are non-negative reals not all zero, then prove that the polynomial xn−b1xn−1−b2xn−2−…−bn=0x^{n}-b_{1}x^{n-1}-b_{2}x^{n-2}-\ldots-b_{n}=0 has only one positive root pp, which is simple. Moreover prove that any root of the polynomial does not exceed pp in absolute value.
algebrapolynomialcalculusderivativeinductionalgebra proposed
Polynomial of prime is prime

Source: Moldova 2007 IMO-BMO TST III problem 2

3/24/2007
Find all polynomials f∈Z[X]f\in \mathbb{Z}[X] such that if pp is prime then f(p)f(p) is also prime.
algebrapolynomialsearchnumber theoryDiophantine equationnumber theory proposed