MathDB

2

Part of 2024 Iran MO (2nd Round)

Problems(2)

Iranian sequence

Source: Iran second round 2024 p2

4/18/2024
Find all sequences (an)n≥1(a_n)_{n\geq 1} of positive integers such that for all integers n≥3n\geq 3 we have
1a1a3+1a2a4+⋯+1an−2an=1−1a12+a22+⋯+an−12. \dfrac{1}{a_1 a_3} + \dfrac{1}{a_2a_4} + \cdots + \dfrac{1}{a_{n-2}a_n}= 1 - \dfrac{1}{a_1^2+a_2^2+\cdots +a_{n-1}^2}.
algebra and number theory
Recoloring Rows Alternatively

Source: Iran MO 2024 second round P5

4/19/2024
Sahand and Gholam play on a 1403×14031403\times 1403 table. Initially all the unit square cells are white. For each row and column there is a key for it (totally 2806 keys). Starting with Sahand players take turn alternatively pushing a button that has not been pushed yet, until all the buttons are pushed. When Sahand pushes a button all the cells of that row or column become black, regardless of the previous colors. When Gholam pushes a button all the cells of that row or column become red, regardless of the previous colors. Finally, Gholam's score equals the number of red squares minus the number of black squares and Sahand's score equals the number of black squares minus the number of red squares. Determine the minimum number of scores Gholam can guarantee without if both players play their best moves.
combinatoricsgames