MathDB

3

Part of 2018 Czech-Polish-Slovak Junior Match

Problems(2)

teacher gave each of her 37 students 36 pencils in different colors

Source: : Czech-Polish-Slovak Junior Match 2018, individual p3 CPSJ

2/1/2020
The teacher gave each of her 3737 students 3636 pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed.
Coloringcombinatorics
| ax^2 + bx + c | <= 1 => |cx^2 + bx + a | <= r

Source: Czech-Polish-Slovak Junior Match 2018, Team p3 CPSJ

3/7/2020
Calculate all real numbers rr with the following properties: If real numbers a,b,ca, b, c satisfy the inequality∣ax2+bx+c∣≤1 | ax^2 + bx + c | \le 1 for each x∈[−1,1]x \in [ - 1, 1], then they also satisfy the inequality ∣cx2+bx+a∣≤r| cx^2 + bx + a | \le r for each x∈[−1,1] x \in [- 1, 1].
trinomialalgebrainequalities