MathDB

Problem 4

Part of 2022 Kyiv City MO Round 2

Problems(3)

Game of stones

Source: Kyiv City MO 2022 Round 2, Problem 7.4

1/30/2022
Fedir and Mykhailo have three piles of stones: the first contains 100100 stones, the second 101101, the third 102102. They are playing a game, going in turns, Fedir makes the first move. In one move player can select any two piles of stones, let's say they have aa and bb stones left correspondently, and remove gcd(a,b)gcd(a, b) stones from each of them. The player after whose move some pile becomes empty for the first time wins. Who has a winning strategy?
As a reminder, gcd(a,b)gcd(a, b) denotes the greatest common divisor of a,ba, b.
(Proposed by Oleksii Masalitin)
combinatoricsgameGCD
NT: Smallest degree of polynomial

Source: Kyiv City MO 2022 Round 2, Problem 10.4

1/30/2022
Prime p>2p>2 and a polynomial QQ with integer coefficients are such that there are no integers 1≤i<j≤p−11 \le i < j \le p-1 for which (Q(j)−Q(i))(jQ(j)−iQ(i))(Q(j)-Q(i))(jQ(j)-iQ(i)) is divisible by pp. What is the smallest possible degree of QQ?
(Proposed by Anton Trygub)
number theorypolynomialalgebra
Crazy geometry on incircles

Source: Kyiv City MO 2022 Round 2, Problem 11.4

1/30/2022
Let ABCDABCD be the cyclic quadrilateral. Suppose that there exists some line ll parallel to BDBD which is tangent to the inscribed circles of triangles ABC,CDAABC, CDA. Show that ll passes through the incenter of BCDBCD or through the incenter of DABDAB.
(Proposed by Fedir Yudin)
geometry