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Part of 2024 Iran MO (3rd Round)

Problems(5)

Iran 3rd Round Combo

Source: Iran 2024 3rd Round Test 1 P2

8/29/2024
Two intelligent people playing a game on the 1403×14031403 \times 1403 table with 140321403^2 cells. The first one in each turn chooses a cell that didn't select before and draws a vertical line segment from the top to the bottom of the cell. The second person in each turn chooses a cell that didn't select before and draws a horizontal line segment from the left to the right of the cell. After 140321403^2 steps the game will be over. The first person gets points equal to the longest verticals line segment and analogously the second person gets point equal to the longest horizonal line segment. At the end the person who gets the more point will win the game. What will be the result of the game?
tablecombinatoricscellsgame strategyGame Theory
A complex FE from Iran

Source: Iran 2024 3rd round algebra exam P2

8/29/2024
A surjective function g:C→Cg: \mathbb{C} \to \mathbb C is given. Find all functions f:C→Cf: \mathbb{C} \to \mathbb C such that for all x,y∈Cx,y\in \mathbb C we have ∣f(x)+g(y)∣=∣f(y)+g(x)∣. |f(x)+g(y)| = | f(y) + g(x)|.
algebrafunctional equation
Tabular Triangle Combo

Source: Iran 2024 3rd Round Combinatorics Test P2

8/29/2024
Consider the main diagonal and the cells above it in an n×n n \times n grid. These cells form what we call a tabular triangle of length n n . We want to place a real number in each cell of a tabular triangle of length n n such that for each cell, the sum of the numbers in the cells in the same row and the same column (including itself) is zero. For example, the sum of the cells marked with a circle is zero. It is known that the number in the topmost and leftmost cell is 1.1. Find all possible ways to fill the remaining cells.
combinatoricstableTabular Trianglecells
Config geo with line perpendicular to the A-median

Source: Iran MO 3rd Round 2024 Geometry Exam P2

8/27/2024
Let MM be the midpoint of the side BCBC of the △ABC\triangle ABC. The perpendicular at AA to AMAM meets (ABC)(ABC) at KK. The altitudes BE,CFBE,CF of the triangle ABCABC meet AKAK at P,QP, Q. Show that the radical axis of the circumcircles of the triangles PKE,QKFPKE, QKF is perpendicular to BCBC.
geometry
Set of prime divisors are same!

Source: Iran 2024 3rd round number theory exam p2

8/27/2024
For all positive integers nn Prove that one can find pairwise coprime integers a,b,c>na,b,c>n such that the set of prime divisors of the numbers a+b+ca+b+c and ab+bc+acab+bc+ac coincides.
number theory