MathDB

Problem 6

Part of 1999 Ukraine National Mathematical Olympiad

Problems(4)

heaxgonal grid, number of ways to reach hexagon

Source: Ukraine 1999 Grade 8 P6

5/10/2021
Consider the figure consisting of 1919 hexagonal cells, as shown in the picture. At the cell AA, there is a piece that is allowed to move one cell up, up-right, or down-right. How many ways are there for the piece to reach the cell BB, not passing through the cell CC? https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZi9mL2Y4MWY5NGMwM2EyMGJhMTcxYThkOTc5YTY4NDQwMjVjNGFmMGQ5LnBuZw==&rn=VWtyYWluZSAxOTk5LnBuZw==
combinatorics
k^L/L^k=k!/L!

Source: Ukraine 1999 Grade 9 P6

5/10/2021
Find all pairs (k,l)(k,l) of positive integers such that kllk=k!l!\frac{k^l}{l^k}=\frac{k!}{l!}.
number theoryDiophantine equation
m^3-n^3=7mn+5

Source: Ukraine 1999 Grade 10 P6

5/11/2021
Solve the equation m3−n3=7mn+5m^3-n^3=7mn+5 in positive integers.
number theoryDiophantine equation
2012 function

Source:

11/14/2015
Find all functions f:R  ⟹  R f:R \implies R , such for all x,y,zx,y,z f(xy)+f(xz)≥f(x)f(yz)+1f(xy)+f(xz)\geq f(x)f(yz) + 1
function