MathDB

1999 China Team Selection Test

Part of China Team Selection Test

Subcontests

(3)
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China TST 1999 any 3-element subset

Let S={1,2,…,15}S = \lbrace 1, 2, \ldots, 15 \rbrace. Let A1,A2,…,AnA_1, A_2, \ldots, A_n be nn subsets of SS which satisfy the following conditions: I. ∣Ai∣=7,i=1,2,…,n|A_i| = 7, i = 1, 2, \ldots, n; II. ∣Ai∩Aj∣≤3,1≤i<j≤n|A_i \cap A_j| \leq 3, 1 \leq i < j \leq n III. For any 3-element subset MM of SS, there exists AkA_k such that M⊂AkM \subset A_k. Find the smallest possible value of nn.

Permutation Minimum and Maximum

For every permutation τ \tau of 1,2,…,10 1, 2, \ldots, 10, \tau \equal{} (x_1, x_2, \ldots, x_{10}), define S(\tau) \equal{} \sum_{k \equal{} 1}^{10} |2x_k \minus{} 3x_{k \minus{} 1}|. Let x_{11} \equal{} x_1. Find I. The maximum and minimum values of S(τ) S(\tau). II. The number of τ \tau which lets S(τ) S(\tau) attain its maximum. III. The number of τ \tau which lets S(τ) S(\tau) attain its minimum.
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p = kq + r and a^2 | r

Find all prime numbers pp which satisfy the following condition: For any prime q<pq < p, if p=kq+r,0≤r<qp = kq + r, 0 \leq r < q, there does not exist an integer q>1q > 1 such that a2∣ra^{2} \mid r.

China TST 1999 sequence

For a fixed natural number m≥2m \geq 2, prove that a.) There exists integers x1,x2,…,x2mx_1, x_2, \ldots, x_{2m} such that xixm+i=xi+1xm+i−1+1,i=1,2,…,m(∗)x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*) b.) For any set of integers {x1,x2,…,x2m\lbrace x_1, x_2, \ldots, x_{2m} which fulfils (*), an integral sequence …,y−k,…,y−1,y0,y1,…,yk,…\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots can be constructed such that ykym+k=yk+1ym+k−1+1,k=0,±1,±2,…y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots such that yi=xi,i=1,2,…,2my_i = x_i, i = 1, 2, \ldots, 2m.
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