MathDB

Problem 4

Part of 2000 Croatia National Olympiad

Problems(4)

paying bills with coins of 1,2,5,10,20,50 size

Source: Croatia 2000 1st Grade P4

5/8/2021
We are given coins of 1,2,5,10,20,501,2,5,10,20,50 lipas and of 11 kuna (Croatian currency: 11 kuna = 100100 lipas). Prove that if a bill of MM lipas can be paid by NN coins, then a bill of NN kunas can be paid by M coins.
algebra
triangle in square with area <1/20 when there are 2000 points in a square 20x20

Source: 2008 Indonesia TST stage 2 test 1 p1

12/14/2020
Let ABCDABCD be a square with side 2020 and T1,T2,...,T2000T_1, T_2, ..., T_{2000} are points in ABCDABCD such that no 33 points in the set S={A,B,C,D,T1,T2,...,T2000}S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\} are collinear. Prove that there exists a triangle with vertices in SS, such that the area is less than 1/101/10.
geometrycombinatorial geometrycombinatoricsarea of a triangle
log equality with floors

Source: Croatia 2000 3rd Grade P4

5/9/2021
If n≥2n\ge2 is an integer, prove the equality ⌊log⁡2n⌋+⌊log⁡3n⌋+…+⌊log⁡nn⌋=⌊n⌋+⌊n3⌋+…+⌊nn⌋.\lfloor\log_2n\rfloor+\lfloor\log_3n\rfloor+\ldots+\lfloor\log_nn\rfloor=\left\lfloor\sqrt n\right\rfloor+\left\lfloor\sqrt[3]n\right\rfloor+\ldots+\left\lfloor\sqrt[n]n\right\rfloor.
logarithmsalgebrafloor function
floor sum of squarefree numbers

Source: Croatia 2000 4th Grade P4

5/9/2021
Let SS be the set of all squarefree numbers and nn be a natural number. Prove that ∑k∈S⌊nk⌋=n.\sum_{k\in S}\left\lfloor\sqrt{\frac nk}\right\rfloor=n.
number theoryfloor function