MathDB

2016 Israel National Olympiad

Part of Israel National Olympiad

Subcontests

(7)
7
1
6
1

Israel 2016 Q6 - Ratio in a regular 12-gon

Points A1,A2,A3,...,A12A_1,A_2,A_3,...,A_{12} are the vertices of a regular polygon in that order. The 12 diagonals A1A6,A2A7,A3A8,...,A11A4,A12A5A_1A_6,A_2A_7,A_3A_8,...,A_{11}A_4,A_{12}A_5 are marked, as in the picture below. Let XX be some point in the plane. From XX, we draw perpendicular lines to all 12 marked diagonals. Let B1,B2,B3,...,B12B_1,B_2,B_3,...,B_{12} be the feet of the perpendiculars, so that B1B_1 lies on A1A6A_1A_6, B2B_2 lies on A2A7A_2A_7 and so on.
Evaluate the ratio XA1+XA2+⋯+XA12B1B6+B2B7+⋯+B12B5\frac{XA_1+XA_2+\dots+XA_{12}}{B_1B_6+B_2B_7+\dots+B_{12}B_5}. https://i.imgur.com/DUuwFth.png
5
1
4
1
3
1

Israel 2016 Q3 - Summing digits repeatedly

Denote by S(n)S(n) the sum of digits of nn. Given a positive integer NN, we consider the following process: We take the sum of digits S(N)S(N), then take its sum of digits S(S(N))S(S(N)), then its sum of digits S(S(S(N)))S(S(S(N)))... We continue this until we are left with a one-digit number.
We call the number of times we had to activate S(⋅)S(\cdot) the depth of NN.
For example, the depth of 49 is 2, since S(49)=13→S(13)=4S(49)=13\rightarrow S(13)=4, and the depth of 45 is 1, since S(45)=9S(45)=9.
[*] Prove that every positive integer NN has a finite depth, that is, at some point of the process we get a one-digit number. [*] Define x(n)x(n) to be the minimal positive integer with depth nn. Find the residue of x(5776)mod  6x(5776)\mod 6. [*] Find the residue of x(5776)−x(5708)mod  2016x(5776)-x(5708)\mod 2016.
2
1
1
1