MathDB

4

Part of 2017 Dutch IMO TST

Problems(3)

problem involving tangencyand midpoints

Source: Netherlands TST for IMO 2017 day 1 problem 4

2/1/2018
Let ABCABC be a triangle, let MM be the midpoint of ABAB, and let NN be the midpoint of CMCM. Let XX be a point satisfying both ∠XMC=∠MBC\angle XMC = \angle MBC and ∠XCM=∠MCB\angle XCM = \angle MCB such that XX and BB lie on opposite sides of CMCM. Let ω\omega be the circumcircle of triangle AMXAMX. (a)(a) Show that CMCM is tangent to ω\omega. (b)(b) Show that the lines NXNX and ACAC intersect on ω\omega
geometry
functional equation

Source: Netherlands TST for IMO 2017 day 2 problem 4

2/1/2018
Find all functions f:R→Rf : \mathbb{R} \rightarrow \mathbb{R} such that (y+1)f(x)+f(xf(y)+f(x+y))=y(y + 1)f(x) + f(xf(y) + f(x + y))= y for all x,y∈Rx, y \in \mathbb{R}.
functional equationalgebra
combinatorial geometry

Source: Netherlands TST for IMO 2017 day 3 problem 4

2/1/2018
Let n≥2n \geq 2 be an integer. Find the smallest positive integer mm for which the following holds: given nn points in the plane, no three on a line, there are mm lines such that no line passes through any of the given points, and for all points X≠YX \neq Y there is a line with respect to which XX and YY lie on opposite sides
combinatoricsgeometry