MathDB

3

Part of 2007 China Team Selection Test

Problems(7)

Points on the circle

Source: China TST 2007, Problem 3

12/29/2008
There are 63 63 points arbitrarily on the circle C \mathcal{C} with its diameter being 20 20. Let S S denote the number of triangles whose vertices are three of the 63 63 points and the length of its sides is no less than 9 9. Fine the maximum of S S.
algorithmcombinatorics unsolvedcombinatorics
A subset of (1, 2, ..., n)

Source: China TST 2007, Problem 6

12/29/2008
Let n n be a positive integer, let A A be a subset of {1,2,⋯ ,n} \{1, 2, \cdots, n\}, satisfying for any two numbers x,y∈A x, y\in A, the least common multiple of x x, y y not more than n n. Show that |A|\leq 1.9\sqrt {n} \plus{} 5.
least common multiplenumber theoryrelatively primenumber theory unsolved
Polynomial

Source: Chinese TST 2007 1st quiz P3

1/3/2009
Prove that for any positive integer n n, there exists only n n degree polynomial f(x), f(x), satisfying f(0) \equal{} 1 and (x \plus{} 1)[f(x)]^2 \minus{} 1 is an odd function.
algebrapolynomialfunctioninductionquadraticslimitintegration
There exist A, B, C such that lengths AB and AC are close

Source: Chinese TST 2007 2nd quiz P3

1/3/2009
Assume there are n≥3 n\ge3 points in the plane, Prove that there exist three points A,B,C A,B,C satisfying 1\le\frac{AB}{AC}\le\frac{n\plus{}1}{n\minus{}1}.
inequalitiescombinatorial geometryanalytic geometrygeometry
Power of 2

Source: Chinese TST 2007 4th quiz P3

1/3/2009
Let n n be positive integer, A,B⊆[0,n] A,B\subseteq[0,n] are sets of integers satisfying \mid A\mid \plus{} \mid B\mid\ge n \plus{} 2. Prove that there exist a∈A,b∈B a\in A, b\in B such that a \plus{} b is a power of 2. 2.
functioninductioncombinatorics proposedcombinatorics
Lovely inequality [Walther Janous's problem from 1989]

Source: Jack Garfunkel, P1490, CRUX 1989/9, p.270, Chinese TST 2007 5th quiz P3

11/7/2003
Find the smallest constant k k such that \frac {x}{\sqrt {x \plus{} y}} \plus{} \frac {y}{\sqrt {y \plus{} z}} \plus{} \frac {z}{\sqrt {z \plus{} x}}\leq k\sqrt {x \plus{} y \plus{} z} for all positive x x, y y, z z.
inequalities3-variable inequalitycyclic inequalitysquare root inequality
Table

Source: Chinese TST 2007 6th quiz P3

1/4/2009
Consider a 7×7 7\times 7 numbers table a_{ij} \equal{} (i^2 \plus{} j)(i \plus{} j^2), 1\le i,j\le 7. When we add arbitrarily each term of an arithmetical progression consisting of 7 7 integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.
invariantalgebrapolynomialcalculusfunctionarithmetic sequenceChina