MathDB

Problem 4

Part of 1999 Croatia National Olympiad

Problems(4)

operation on (3,4,12) to get (2,8,10)

Source: Croatia 1999 1st Grade P4

5/17/2021
A triple of numbers (a1,a2,a3)=(3,4,12)(a_1,a_2,a_3)=(3,4,12) is given. The following operation is performed a finite number of times: choose two numbers a,ba,b from the triple and replace them by 0.6x−0.8y0.6x-0.8y and 0.8x+0.6y0.8x+0.6y. Is it possible to obtain the (unordered) triple (2,8,10)(2,8,10)?
number theory
equation of # of wins/losses in tournament

Source: Croatia 1999 2nd Grade P4

5/17/2021
In a basketball competition, nn teams took part. Each pair of teams played exactly one match, and there were no draws. At the end of the competition the ii-th team had xix_i wins and yiy_i defeats (i=1,…,n)(i=1,\ldots,n). Prove that x12+x22+…+xn2=y12+y22+…+yn2x_1^2+x_2^2+\ldots+x_n^2=y_1^2+y_2^2+\ldots+y_n^2.
combinatoricsTournament
a+b=c+d mod 20 for some four integers out of 9

Source: Croatia 1999 3rd Grade P4

5/17/2021
Given nine positive integers, is it always possible to choose four different numbers a,b,c,da,b,c,d such that a+ba+b and c+dc+d are congruent modulo 2020?
number theorypigeonhole principle
limit of point sequence defined by recurrence

Source: Croatia 1999 4th Grade P4

5/17/2021
On the coordinate plane is given the square with vertices T1(1,0),T2(0,1),T3(−1,0),T4(0,−1)T_1(1,0),T_2(0,1),T_3(-1,0),T_4(0,-1). For every n∈Nn\in\mathbb N, point Tn+4T_{n+4} is defined as the midpoint of the segment TnTn+1T_nT_{n+1}. Determine the coordinates of the limit point of TnT_n as n→∞n\to\infty, if it exists.
geometrySequence