MathDB

3

Part of 2016 Dutch IMO TST

Problems(3)

gcm and lcm

Source: Netherlands Team Selection Test 2016 Day 1-Problem 3

9/22/2016
Find all positive integers kk for which the equation: lcm(m,n)−gcd(m,n)=k(m−n) \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n) has no solution in integers positive (m,n)(m,n) with m≠nm\neq n.
number theoryleast common multiple
isosceles AB=AC, BF=BE, angle bisector, BD=EF iff AF=EC 2016 Dutch IMO TST2 P3

Source:

8/4/2019
Let △ABC\vartriangle ABC be an isosceles triangle with ∣AB∣=∣AC∣|AB| = |AC|. Let D,ED, E and FF be points on line segments BC,CABC, CA and ABAB, respectively, such that ∣BF∣=∣BE∣|BF| = |BE| and such that EDED is the internal angle bisector of ∠BEC\angle BEC. Prove that ∣BD∣=∣EF∣|BD|= |EF| if and only if ∣AF∣=∣EC∣|AF| = |EC|.
geometryequal segmentsisoscelesangle bisector
numbers satisfying s(n) < s(2n), are as many as ...s(n) > s(2n) [sum of digits\

Source: Dutch IMO TST 2016 day 3 p3

8/30/2019
Let kk be a positive integer, and let s(n)s(n) denote the sum of the digits of nn. Show that among the positive integers with kk digits, there are as many numbers nn satisfying s(n)<s(2n)s(n) < s(2n) as there are numbers nn satisfying s(n)>s(2n)s(n) > s(2n).
number theorysum of digitsDigits