MathDB

1996 China Team Selection Test

Part of China Team Selection Test

Subcontests

(3)
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larger number is a multiple of the smaller number

Let M \equal{} \lbrace 2, 3, 4, \ldots\, 1000 \rbrace. Find the smallest n∈N n \in \mathbb{N} such that any n n-element subset of M M contains 3 pairwise disjoint 4-element subsets S,T,U S, T, U such that I. For any 2 elements in S S, the larger number is a multiple of the smaller number. The same applies for T T and U U. II. For any s∈S s \in S and t∈T t \in T, (s,t) \equal{} 1. III. For any s∈S s \in S and u∈U u \in U, (s,u)>1 (s,u) > 1.

|ka + lb + mc| > 1/h

Does there exist non-zero complex numbers a,b,ca, b, c and natural number hh such that if integers k,l,mk, l, m satisfy ∣k∣+∣l∣+∣m∣≥1996|k| + |l| + |m| \geq 1996, then ∣ka+lb+mc∣>1h|ka + lb + mc| > \frac {1}{h} is true?
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2

f(n+1) >= f(n) >= n/(n=1) * f(2n)

SS is the set of functions f:N→Rf:\mathbb{N} \to \mathbb{R} that satisfy the following conditions: I. f(1)=2f(1) = 2 II. f(n+1)≥f(n)≥nn+1f(2n)f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n) for n=1,2,…n = 1, 2, \ldots Find the smallest M∈NM \in \mathbb{N} such that for any f∈Sf \in S and any n∈N,f(n)<Mn \in \mathbb{N}, f(n) < M.

China TST 1996 sum inequalities

Let α1,α2,…,αn\alpha_1, \alpha_2, \dots, \alpha_n, and β1,β2,…,βn\beta_1, \beta_2, \ldots, \beta_n, where n≥4n \geq 4, be 2 sets of real numbers such that ∑i=1nαi2<1and∑i=1nβi2<1.\sum_{i=1}^{n} \alpha_i^2 < 1 \qquad \text{and} \qquad \sum_{i=1}^{n} \beta_i^2 < 1. Define \begin{align*} A^2 &= 1 - \sum_{i=1}^{n} \alpha_i^2,\\ B^2 &= 1 - \sum_{i=1}^{n} \beta_i^2,\\ W &= \frac{1}{2} (1 - \sum_{i=1}^{n} \alpha_i \beta_i)^2. \end{align*} Find all real numbers λ\lambda such that the polynomial xn+λ(xn−1+⋯+x3+Wx2+ABx+1)=0,x^n + \lambda (x^{n-1} + \cdots + x^3 + Wx^2 + ABx + 1) = 0, only has real roots.
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each country has 9 representatives

3 countries A,B,CA, B, C participate in a competition where each country has 9 representatives. The rules are as follows: every round of competition is between 1 competitor each from 2 countries. The winner plays in the next round, while the loser is knocked out. The remaining country will then send a representative to take on the winner of the previous round. The competition begins with AA and BB sending a competitor each. If all competitors from one country have been knocked out, the competition continues between the remaining 2 countries until another country is knocked out. The remaining team is the champion. I. At least how many games does the champion team win? II. If the champion team won 11 matches, at least how many matches were played?

China TST 1996 diameter circle cut

Let side BCBC of △ABC\bigtriangleup ABC be the diameter of a semicircle which cuts ABAB and ACAC at DD and EE respectively. FF and GG are the feet of the perpendiculars from DD and EE to BCBC respectively. DGDG and EFEF intersect at MM. Prove that AM⊥BCAM \perp BC.