MathDB

2

Part of 2004 China Team Selection Test

Problems(7)

Primes don't exceed 2004

Source: China Team Selection Test 2004, Day 2, Problem 2

10/14/2005
Let p1,p2,…,p25p_1, p_2, \ldots, p_{25} are primes which don’t exceed 2004. Find the largest integer TT such that every positive integer ≤T\leq T can be expressed as sums of distinct divisors of (p1⋅p2⋅…⋅p25)2004.(p_1\cdot p_2 \cdot \ldots \cdot p_{25})^{2004}.
LaTeXnumber theory unsolvednumber theory
Chinese factorial equation

Source: China Team Selection Test 2004, Day 1, Problem 2

10/13/2005
Let u be a fixed positive integer. Prove that the equation n!=uα−uβn! = u^{\alpha} - u^{\beta} has a finite number of solutions (n,α,β).(n, \alpha, \beta).
floor functionlogarithmsnumber theory
Collinearity

Source: China TST 2004 Quiz

2/1/2009
Two equal-radii circles with centres O1 O_1 and O2 O_2 intersect each other at P P and Q Q, O O is the midpoint of the common chord PQ PQ. Two lines AB AB and CD CD are drawn through P P ( AB AB and CD CD are not coincide with PQ PQ ) such that A A and C C lie on circle O1 O_1 and B B and D D lie on circle O2 O_2. M M and N N are the mipoints of segments AD AD and BC BC respectively. Knowing that O1 O_1 and O2 O_2 are not in the common part of the two circles, and M M, N N are not coincide with O O. Prove that M M, N N, O O are collinear.
geometryincenterperimetergeometric transformationreflectiongeometry unsolved
Find k

Source: China TST 2004 Quiz

2/1/2009
Find the largest positive real k k, such that for any positive reals a,b,c,d a,b,c,d, there is always: (a\plus{}b\plus{}c) \left[ 3^4(a\plus{}b\plus{}c\plus{}d)^5 \plus{} 2^4(a\plus{}b\plus{}c\plus{}2d)^5 \right] \geq kabcd^3
inequalities unsolvedinequalities
Sets and Inequality

Source: China TST 2004 Quiz

2/1/2009
Let k k be a positive integer. Set A⊆Z A \subseteq \mathbb{Z} is called a k \minus{ set} if there exists x1,x2,⋯ ,xk∈Z x_1, x_2, \cdots, x_k \in \mathbb{Z} such that for any i≠j i \neq j, (x_i \plus{} A) \cap (x_j \plus{} A) \equal{} \emptyset, where x \plus{} A \equal{} \{ x \plus{} a \mid a \in A \}. Prove that if Ai A_i is k_i \minus{ set}( i \equal{} 1,2, \cdots, t), and A_1 \cup A_2 \cup \cdots \cup A_t \equal{} \mathbb{Z}, then \displaystyle \frac {1}{k_1} \plus{} \frac {1}{k_2} \plus{} \cdots \plus{} \frac {1}{k_t} \geq 1.
inequalitiescombinatorics unsolvedcombinatorics
Distance between Points

Source: China TST 2004 Quiz

2/1/2009
There are n≥5 n \geq 5 pairwise different points in the plane. For every point, there are just four points whose distance from which is 1 1. Find the maximum value of n n.
combinatorics unsolvedcombinatorics
Cyclic Quadrilateral Calculation

Source: China TST 2004 Quiz

2/1/2009
Convex quadrilateral ABCD ABCD is inscribed in a circle, \angle{A}\equal{}60^o, BC\equal{}CD\equal{}1, rays AB AB and DC DC intersect at point E E, rays BC BC and AD AD intersect each other at point F F. It is given that the perimeters of triangle BCE BCE and triangle CDF CDF are both integers. Find the perimeter of quadrilateral ABCD ABCD.
geometryperimeterinequalitiestrigonometrytrig identitiesLaw of Sinesgeometry unsolved