3
Part of 2022 Saudi Arabia JBMO TST
Problems(2)
2000 consecutive integers on the board
Source: 2022 Saudi Arabia JBMO TST 1.3
11/3/2022
consecutive integers (not necessarily positive) are written on the board. A student takes several turns. On each turn, he partitions the integers into pairs, and substitutes each pair by the difference arid the sum of that pair (note that the difference does not need to be positive as the student may choose to subtract the greater number from the smaller one; in addition, all the operations are carried simultaneously). Prove that the student will never again write consecutive integers on the board.
combinatoricsconsecutive
incenter of PA_0Q lies on altitude of ABC
Source: 2022 Saudi Arabia JBMO TST 2.3
11/1/2022
Let and be the altitudes of acute-angled triangle , and is the midpoint of . Lines and meet the line passing through and parallel to at points and . Prove that the incenter of triangle lies on the altitude of triangle .
geometryincenter