MathDB

4

Part of 2020 Dutch IMO TST

Problems(3)

1/r is an integer when r min of (a/b - c/d) when gcd (a, b) = 1, c<=a, d <= b

Source: 2020 Dutch IMO TST 1.4

11/21/2020
Let a,b≥2a, b \ge 2 be positive integers with gcd(a,b)=1gcd (a, b) = 1. Let rr be the smallest positive value that ab−cd\frac{a}{b}- \frac{c}{d} can take, where cc and dd are positive integers satisfying c≤ac \le a and d≤bd \le b. Prove that 1r\frac{1}{r} is an integer.
number theoryInteger
perpendicular wanted, tangents on circumcircle related

Source: 2020 Dutch IMO TST 2.4

11/22/2020
Let ABCABC be an acute-angled triangle and let PP be the intersection of the tangents at BB and CC of the circumscribed circle of △ABC\vartriangle ABC. The line through AA perpendicular on ABAB and cuts the line perpendicular on ACAC through CC at XX. The line through AA perpendicular on ACAC cuts the line perpendicular on ABAB through BB at YY. Show that AP⊥XYAP \perp XY.
geometryperpendicular
2 player game of m tiles of kx1 on a nxn board k <= n <= 2k - 1

Source: 2020 Dutch IMO TST 3.4

11/22/2020
Given are two positive integers kk and nn with k≤n≤2k−1k \le n \le 2k - 1. Julian has a large stack of rectangular k×1k \times 1 tiles. Merlin calls a positive integer mm and receives mm tiles from Julian to place on an n×nn \times n board. Julian first writes on every tile whether it should be a horizontal or a vertical tile. Tiles may be used the board should not overlap or protrude. What is the largest number mm that Merlin can call if he wants to make sure that he has all tiles according to the rule of Julian can put on the plate?
combinatoricstilesTiling