MathDB

Problem 2

Part of 2003 Federal Math Competition of S&M

Problems(3)

not rational

Source:

9/13/2012
Let ABCD be a square inscribed in a circle k and P be an arbitrary point of that circle. Prove that at least one of the numbers PA, PB, PC and PD is not rational.
geometrycircumcirclegeometry unsolved
maximum number of lattice unit squares intersecting segment

Source: Serbia 2003 1st Grade P2

5/12/2021
Given a segment ABAB of length 20032003 in a coordinate plane, determine the maximal number of unit squares with vertices in the lattice points whose intersection with the given segment is non-empty.
geometrycombinatorial geometrycombinatorics
A functional inequality

Source:

10/11/2016
Let f:[0,1]→ R f : [0, 1] \to\ R be a function such that :- 1.)1.) f(x)≥0f(x) \ge 0 for all xx in [0,1][0,1] . 2.)2.) f(1)=1f(1) = 1 . 3.)3.) If x1,x2x_1 , x_2 are in [0,1][0,1] such that x1+x2≤1x_1 + x_2 \le 1 , then f(x1)+f(x2)≤f(x1+x2)f(x_1) + f(x_2) \le f(x_1 + x_2) . Show that f(x)≤2xf(x) \le 2x for all xx in [0,1] [0,1] .
functional equationFunctional inequalityinequalities