MathDB

3

Part of 1986 China Team Selection Test

Problems(2)

China TST 1986 construction of sequence

Source: China TST 1986, problem 3

5/16/2005
Given a positive integer AA written in decimal expansion: (an,an−1,…,a0)(a_{n},a_{n-1}, \ldots, a_{0}) and let f(A)f(A) denote ∑k=0n2n−k⋅ak\sum^{n}_{k=0} 2^{n-k}\cdot a_k. Define A1=f(A),A2=f(A1)A_1=f(A), A_2=f(A_1). Prove that:
I. There exists positive integer kk for which Ak+1=AkA_{k+1}=A_k. II. Find such AkA_k for 1986.19^{86}.
algebraNumber theoretic functionsChinaTSTdecimal representationDigits
China TST 1986 symmetric sum

Source: China TST 1986, problem 7

5/16/2005
Let xi,x_i, 1≤i≤n1 \leq i \leq n be real numbers with n≥3.n \geq 3. Let pp and qq be their symmetric sum of degree 11 and 22 respectively. Prove that: i) p2⋅n−1n−2q≥0p^2 \cdot \frac{n-1}{n}-2q \geq 0 ii) ∣xi−pn∣≤p2−2nqn−1⋅n−1n\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n} for every meaningful ii.
inequalitiesquadraticsinequalities unsolved