MathDB

2

Part of 2019 Dutch IMO TST

Problems(3)

max (a^2/b,b^2/a) max (c^2/d,d^2/c) =(min (a + b, c + d))^4

Source: Dutch IMO TST2 2019 p2

1/11/2020
Determine all 44-tuples (a,b,c,d)(a,b, c, d) of positive real numbers satisfying a+b+c+d=1a + b +c + d = 1 and max⁡(a2b,b2a)⋅max⁡(c2d,d2c)=(min⁡(a+b,c+d))4\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4
algebrasystem of equationsmax and minmaxmin
exactly one of the equalities f(g(x)) = x and g(f(x)) = x holds

Source: Dutch IMO TST1 2019 p2

1/10/2020
Write SnS_n for the set {1,2,...,n}\{1, 2,..., n\}. Determine all positive integers nn for which there exist functions f:Sn→Snf : S_n \to S_n and g:Sn→Sng : S_n \to S_n such that for every xx exactly one of the equalities f(g(x))=xf(g(x)) = x and g(f(x))=xg(f(x)) = x holds.
functionalgebra
n^2 + n + 1 cannot be product of 2 integers with difference <2\sqrt{n}

Source: Dutch IMO TST3 2019 p2

1/11/2020
Let nn be a positive integer. Prove that n2+n+1n^2 + n + 1 cannot be written as the product of two positive integers of which the difference is smaller than 2n2\sqrt{n}.
Productnumber theory