MathDB

3

Part of 1992 China Team Selection Test

Problems(2)

p | (x^2_0 - x_0 + 3)

Source: China TST 1992, problem 3

6/27/2005
For any prime pp, prove that there exists integer x0x_0 such that p∣(x02−x0+3)p | (x^2_0 - x_0 + 3) ⇔\Leftrightarrow there exists integer y0y_0 such that p∣(y02−y0+25).p | (y^2_0 - y_0 + 25).
quadraticsmodular arithmeticnumber theory unsolvednumber theory
China TST 1992 inequality

Source: China TST 1992, problem 6

6/27/2005
For any n,T≥2,n,T∈Nn,T \geq 2, n, T \in \mathbb{N}, find all a∈Na \in \mathbb{N} such that ∀ai>0,i=1,2,…,n\forall a_i > 0, i = 1, 2, \ldots, n, we have ∑k=1na⋅k+a24Sk<T2⋅∑k=1n1ak,\sum^n_{k=1} \frac{a \cdot k + \frac{a^2}{4}}{S_k} < T^2 \cdot \sum^n_{k=1} \frac{1}{a_k}, where Sk=∑i=1kai.S_k = \sum^k_{i=1} a_i.
inequalitiesinequalities unsolved