MathDB

1

Part of 2017 Dutch IMO TST

Problems(3)

disks in combinatorics

Source: Netherlands TST for IMO 2017 day 1,problem 1

2/1/2018
Let nn be a positive integer. Suppose that we have disks of radii 1,2,...,n.1, 2, . . . , n. Of each size there are two disks: a transparent one and an opaque one. In every disk there is a small hole in the centre, with which we can stack the disks using a vertical stick. We want to make stacks of disks that satisfy the following conditions: i)i) Of each size exactly one disk lies in the stack. ii)ii) If we look at the stack from directly above, we can see the edges of all of the nn disks in the stack. (So if there is an opaque disk in the stack,no smaller disks may lie beneath it.) Determine the number of distinct stacks of disks satisfying these conditions. (Two stacks are distinct if they do not use the same set of disks, or, if they do use the same set of disks and the orders in which the disks occur are different.)
combinatorics
geometry problem

Source: Netherlands TST for IMO 2017 day 3 problem 1

2/1/2018
A circle ω\omega with diameter AKAK is given. The point MM lies in the interior of the circle, but not on AKAK. The line AMAM intersects ω\omega in AA and QQ. The tangent to ω\omega at QQ intersects the line through MM perpendicular to AKAK, at PP. The point LL lies on ω\omega, and is such that PLPL is tangent to ω\omega and L≠QL\neq Q. Show that K,LK, L, and MM are collinear.
geometry
Trivial NT in TST

Source: Netherlands TST for IMO 2017,day 2 problem 1

2/1/2018
Let a,b,ca, b,c be distinct positive integers, and suppose that p=ab+bc+cap = ab+bc+ca is a prime number. (a)(a) Show that a2,b,c2a^2,b^,c^2 give distinct remainders after division by pp. (b) Show that a3,b3,c3a^3,b^3,c^3 give distinct remainders after division by pp.
number theory