MathDB

3

Part of 1998 Estonia National Olympiad

Problems(4)

a race with 5 boxes and length 1998 kms

Source: 1998 Estonia National Olympiad Final Round grade 9 p3

11/4/2020
On a closed track, clockwise, there are five boxes A,B,C,DA, B, C, D and EE, and the length of the track section between boxes AA and BB is 11 km, between BB and CC - 55 km, between CC and DD - 22 km, between DD and EE - 1010 km, and between EE and AA - 33 km. On the track, they drive in a clockwise direction, the race always begins and ends in the box. What box did you start from if the length of the race was exactly 19981998 km?
algebranumber theorycombinatorics
a hotel has 13 rooms

Source: 1998 Estonia National Olympiad Final Round grade 10 p3

11/4/2020
The hotel has 1313 rooms with rooms from 11 to 1313, located on one side of a straight corridor in ascending order of numbers. During the tourist season, which lasts from May 11st to October 11st, the hotel visitor has the opportunity to rent either one room for two days in a row, or two adjacent rooms together for one day. How much could a hotel owner earn in a season if it is known that on October 11, rooms 11 and 1313 were empty, and the payment for one room was one tugrik per day?
combinatorics
f (x) \ne 0 and f (x+2) = f (x-1) f (x+5) => f (x+18) = f (x)

Source: 1998 Estonia National Olympiad Final Round grade 11 p3

3/11/2020
A function ff satisfies the conditions f(x)≠0f (x) \ne 0 and f(x+2)=f(x−1)f(x+5)f (x+2) = f (x-1) f (x+5) for all real x. Show that f(x+18)=f(x)f (x+18) = f (x) for any real xx.
functionperiodicfunctional equationfunctionalalgebra
irrational ratio of triangle areas, perpendicular related

Source: 1998 Estonia National Olympiad Final Round grade 12 p3

3/11/2020
In a triangle ABCABC, the bisector of the largest angle ∠A\angle A meets BCBC at point DD. Let EE and FF be the feet of perpendiculars from DD to ACAC and ABAB, respectively. Let RR denote the ratio between the areas of triangles DEBDEB and DFCDFC. (a) Prove that, for every real number r>0r > 0, one can construct a triangle ABC for which RR is equal to rr. (b) Prove that if RR is irrational, then at least one side length of △ABC\vartriangle ABC is irrational. (c) Give an example of a triangle ABCABC with exactly two sides of irrational length, but with rational RR.
ratioperpendiculartriangle areaareasgeometryirrational