Suppose that T∈N is given. Find all functions f:Z→C such that, for all m∈Z we have f(m+T)=f(m) and: ∀a,b,c∈Z:f(a)¯f(a+b)f(a+c)f(a+b+c)=1.Where ¯a is the complex conjugate of a.
2024 Iran MO (3rd Round)
Day 1
Two intelligent people playing a game on the 1403×1403 table with 14032 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 14032 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?
Consider an acute scalene triangle △ABC. The interior bisector of A intersects BC at E and the minor arc of \overarcBC in circumcircle of △ABC at M. Suppose that D is a point on the minor arc of \overarcBC such that ED=EM. P is a point on the line segment of AD such that ∠ABP=∠ACP≠0. O is the circumcenter of △ABC. Prove that OP⊥AM.
Day 2
For a given positive integer number n find all subsets {r0,r1,⋯,rn}⊂N such that nn+nn−1+⋯+1|nrn+⋯+nr0. Proposed by Shayan Tayefeh
Let ABCD be a parallelogram and let AX and AY be the altitudes from A to CB,CD, respectively. A line ℓ⊥XY bisects AX and meets AB,BC at K,L. Similarly, a line d⊥XY bisects AY and meets DA,DC at P,Q. Show that the circumcircles of △BKL and △DPQ are tangent to each other.
Sequence of positive integers {xk}k≥1 is given such that x1=1 and for all n≥1 we have x2n+1+P(n)=xnxn+2where P(x) is a polynomial with non-negative integer coefficients. Prove that P(x) is the constant polynomial. Proposed by Navid Safaei
Algebra
For positive real numbers a,b,c,d such that a2b+c+d+b2a+c+d+c2a+b+d=3d2a+b+cprove that 3a+3b+3c+3d≥16a+b+2d+16b+c+2d+16a+c+2d. Proposed by Mojtaba Zare
A surjective function g:C→C is given. Find all functions f:C→C such that for all x,y∈C we have |f(x)+g(y)|=|f(y)+g(x)|. Proposed by Mojtaba Zare, Amirabbas Mohammadi
An integer number n≥2 and real numbers x1<x2<⋯<xn are given. f:R→R is a function defined as f(x)=|(x−x2)(x−x3)⋯(x−xn)(x1−x2)(x1−x3)⋯(x1−xn)|+⋯+|(x−x1)(x−x2)⋯(x−xn−1)(xn−x1)(xn−x2)⋯(xn−xn−1)|.Prove that there exists i∈{1,2,⋯,n−1} such that for all x∈(xi,xi+1) one has f(x)<√n. Proposed by Navid Safaei
Combinatorics
n≥4 is an integer number. For any permutation x1,x2,⋯,xn of the numbers 1,2⋯,n we write the number x1+2x2+⋯+nxnon the board. Compute the number of total distinct numbers written on the board.
Consider the main diagonal and the cells above it in an n×n grid. These cells form what we call a tabular triangle of length n. We want to place a real number in each cell of a tabular triangle of length 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. Find all possible ways to fill the remaining cells.
m,n are given integer numbers such that m+n is an odd number. Edges of a complete bipartie graph Km,n are labeled by −1,1 such that the sum of all labels is 0. Prove that there exists a spanning tree such that the sum of the labels of its edges is equal to 0.
Geometry
Let ABCD be a cyclic quadrilateral with circumcircle Γ. Let M be the midpoint of the arc ABC. The circle with center M and radius MA meets AD,AB at X,Y. The point Z∈XY with Z≠Y satisfies BY=BZ. Show that ∠BZD=∠BCD.
Let M be the midpoint of the side BC of the △ABC. The perpendicular at A to AM meets (ABC) at K. The altitudes BE,CF of the triangle ABC meet AK at P,Q. Show that the radical axis of the circumcircles of the triangles PKE,QKF is perpendicular to BC.
Let ABC be a triangle with altitudes AD,BE,CF and orthocenter H. The perpendicular bisector of HD meets EF at P and N is the center of the nine-point circle. Let L be a point on the circumcircle of ABC such that ∠PLN=90∘ and A,L are in distinct sides of the line PN. Show that ANDL is cyclic.
Number Theory
Given a sequence x1,x2,x3,⋯ of positive integers, Ali proceed the following algorythm: In the i-th step he markes all rational numbers in the interval [0,1] which have denominator equal to xi. Then he write down the number ai equal to the length of the smallest interval in [0,1] which both two ends of that is a marked number. Find all sequences x1,x2,x3,⋯ with x5=5 and such that for all n∈N we have a1+a2+⋯+an=2−1xn. Proposed by Mojtaba Zare
For all positive integers n Prove that one can find pairwise coprime integers a,b,c>n such that the set of prime divisors of the numbers a+b+c and ab+bc+ac coincides. Proposed by Mohsen Jamali and Hesam Rajabzadeh
The prime number p and a positive integer k are given. Assume that P(x)∈Z[X] is a polynomial with coefficients in the set {0,1,⋯,p−1} with least degree which satisfies the following property: There exists a permutaion of numbers 1,2,⋯,p−1 around a circle such that for any k consecutive numbers a1,a2,⋯,ak one has p|P(a1)+P(a2)+⋯+P(ak).Prove that P(x) is of the form axd+b. Proposed by Yahya Motevassel