MathDB

1991 Vietnam Team Selection Test

Part of Vietnam Team Selection Test

Subcontests

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fourth root from product

Let {x}\{x\} be a sequence of positive reals x1,x2,…,xnx_1, x_2, \ldots, x_n, defined by: x1=1,x2=9,x3=9,x4=1x_1 = 1, x_2 = 9, x_3=9, x_4=1. And for n≥1n \geq 1 we have: xn+4=xn⋅xn+1⋅xn+2⋅xn+34.x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}. Show that this sequence has a finite limit. Determine this limit.

5 non-empty disjoint sets

Let a set XX be given which consists of 2⋅n2 \cdot n distinct real numbers (n≥3n \geq 3). Consider a set KK consisting of some pairs (x,y)(x, y) of distinct numbers x,y∈Xx, y \in X, satisfying the two conditions: I. If (x,y)∈K(x, y) \in K then (y,x)∉K(y, x) \not \in K. II. Every number x∈Xx \in X belongs to at most 19 pairs of KK. Show that we can divide the set XX into 5 non-empty disjoint sets X1,X2,X3,X4,X5X_1, X_2, X_3, X_4, X_5 in such a way that for each i=1,2,3,4,5i = 1, 2, 3, 4, 5 the number of pairs (x,y)∈K(x, y) \in K where x,yx, y both belong to XiX_i is not greater than 3⋅n3 \cdot n.
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Greater or equal than n / (x+y)

For a positive integer n>2 n>2, let (a1,a2,…,an) \left(a_{1}, a_{2}, \ldots, a_{n}\right) be a sequence of n n positive reals which is either non-decreasing (this means, we have a1≤a2≤…≤an a_{1}\leq a_{2}\leq \ldots \leq a_{n}) or non-increasing (this means, we have a1≥a2≥…≥an a_{1}\geq a_{2}\geq \ldots \geq a_{n}), and which satisfies a1≠an a_{1}\neq a_{n}. Let x x and y y be positive reals satisfying xy≥a1−a2a1−an \frac{x}{y}\geq \frac{a_{1}-a_{2}}{a_{1}-a_{n}}. Show that: a1a2⋅x+a3⋅y+a2a3⋅x+a4⋅y+…+an−1an⋅x+a1⋅y+ana1⋅x+a2⋅y≥nx+y. \frac{a_{1}}{a_{2}\cdot x+a_{3}\cdot y}+\frac{a_{2}}{a_{3}\cdot x+a_{4}\cdot y}+\ldots+\frac{a_{n-1}}{a_{n}\cdot x+a_{1}\cdot y}+\frac{a_{n}}{a_{1}\cdot x+a_{2}\cdot y}\geq \frac{n}{x+y}.

canonical prime factorisation and iterated function

For every natural number nn we define f(n)f(n) by the following rule: f(1)=1f(1) = 1 and for n>1n>1 then f(n)=1+a1⋅p1+…+ak⋅pkf(n) = 1 + a_1 \cdot p_1 + \ldots + a_k \cdot p_k, where n=p1a1⋯pkakn = p_1^{a_1} \cdots p_k^{a_k} is the canonical prime factorisation of nn (p1,…,pkp_1, \ldots, p_k are distinct primes and a1,…,aka_1, \ldots, a_k are positive integers). For every positive integer ss, let fs(n)=f(f(…f(n))…)f_s(n) = f(f(\ldots f(n))\ldots), where on the right hand side there are exactly ss symbols ff. Show that for every given natural number aa, there is a natural number s0s_0 such that for all s>s0s > s_0, the sum fs(a)+fs−1(a)f_s(a) + f_{s-1}(a) does not depend on ss.
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Find the answer for n = 1991 and n = 2000

1.) In the plane let us consider a set SS consisting of n≥3n \geq 3 distinct points satisfying the following three conditions: I. The distance between any two points ∈S\in S is not greater than 1. II. For every point A∈SA \in S, there are exactly two “neighbor” points, i.e. two points X,Y∈SX, Y \in S for which AX=AY=1AX = AY = 1. III. For arbitrary two points A,B∈SA, B \in S, let A′,A′′A', A'' be the two neighbors of A,B′,B′′A, B', B'' the two neighbors of BB, then A′AA′′=B′BB′′A'AA'' = B'BB''. Is there such a set SS if n=1991n = 1991? If n=2000n = 2000 ? Explain your answer.

tetrahedron with side not greater than 1

Let TT be an arbitrary tetrahedron satisfying the following conditions: I. Each its side has length not greater than 1, II. Each of its faces is a right triangle. Let s(T)=SABC2+SBCD2+SCDA2+SDAB2s(T) = S^2_{ABC} + S^2_{BCD} + S^2_{CDA} + S^2_{DAB}. Find the maximal possible value of s(T)s(T).