MathDB

2

Part of 2010 Polish MO Finals

Problems(2)

Polish MO Final 2010, 2nd problem (equation in rationals)

Source:

11/7/2010
Positive rational number aa and bb satisfy the equality a3+4a2b=4a2+b4.a^3 + 4a^2b = 4a^2 + b^4. Prove that the number a−1\sqrt{a}-1 is a square of a rational number.
functionalgebrapolynomialnumber theory proposednumber theory
Polish MO Final 2010, 5th problem (remainder of the product)

Source:

11/7/2010
Prime number p>3p>3 is congruent to 22 modulo 33. Let ak=k2+k+1a_k = k^2 + k +1 for k=1,2,…,p−1k=1, 2, \ldots, p-1. Prove that product a1a2…ap−1a_1a_2\ldots a_{p-1} is congruent to 33 modulo pp.
modular arithmeticquadraticsOnline Math Opentrigonometrynumber theorynumber theory proposed