3

Part of 1995 China Team Selection Test

Problems(2)

21 people take a test with 15 true or false questions

Source: China TST 1995, problem 3

21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on?

vectorcombinatorics unsolvedcombinatorics

split in black and white intervals for quadratic polynomials

Source: China TST 1995, problem 6

Prove that the interval

\lbrack 0,1 \rbrack

can be split into black and white intervals for any quadratic polynomial

P(x)

, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval

\lbrack a,b \rbrack

P(b) - P(a)

.) Does the same result hold with a degree 3 or degree 5 polynomial?

quadraticsalgebrapolynomialalgebra unsolved