1

Part of 1998 China Team Selection Test

Problems(2)

Binomial coefficients forming arithmetic progression

Source: China TST 1998, problem 1

k \in \mathbb{N}

such that
a.) For any

n \in \mathbb{N}

, there does not exist

j \in \mathbb{Z}

which satisfies the conditions

0 \leq j \leq n - k + 1

\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots, \left( \begin{array}{c} n\\ j + k - 1\end{array} \right)

forms an arithmetic progression.
b.) There exists

n \in \mathbb{N}

such that there exists

j

which satisfies

0 \leq j \leq n - k + 2

\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots , \left( \begin{array}{c} n\\ j + k - 2\end{array} \right)

forms an arithmetic progression.
Find all

n

which satisfies part b.)

arithmetic sequencecombinatorics unsolvedcombinatoricsarithmeticArithmetic ProgressionChinaTST

Orthocenter, circumcenter and incenter

Source: China TST 1998, problem 4

In acute-angled

\bigtriangleup ABC

H

is the orthocenter,

O

is the circumcenter and

I

is the incenter. Given that

\angle C > \angle B > \angle A

I

\bigtriangleup BOH

geometrycircumcircleincentertrigonometrygeometry unsolved