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Part of 2023 Iran MO (3rd Round)

Problems(5)

Finally a general problem in Iran

Source: Iran MO 3rd round 2023 , Day 1 P3

8/16/2023
For each kk , find the least nn in terms of kk st the following holds: There exists nn real numbers a1,a2,⋅⋅⋅,ana_1 , a_2 ,\cdot \cdot \cdot , a_n st for each ii : 0<ai+1−ai<ai−ai−10 < a_{i+1} - a_{i} < a_i - a_{i-1} And , there exists kk pairs (i,j)(i,j) st ai−aj=1a_i - a_j = 1.
combinatorics
The desert of IranNT2023

Source: Iran MO 3rd round 2023 , NT exam , P3

8/17/2023
Let KK be an odd number st S2(K)=2S_2{(K)} = 2 and let ab=Kab=K where a,ba,b are positive integers. Show that if a,b>1a,b>1 and l,m>2l,m >2 are positive integers st:S2(a)<lS_2{(a)} < l and S2(b)<mS_2{(b)} < m then : K≤2lm−6+1K \leq 2^{lm-6} +1 (S2(n)S_2{(n)} is the sum of digits of nn written in base 2)
number theory
Tiling game

Source: Iran MO 2023 3rd round , Combinatorics exam P3

8/19/2023
There's infinity of the following blocks on the table:1∗1,1∗2,1∗3,..,1∗n1*1 , 1*2 , 1*3 ,.., 1*n. We have a n∗nn*n table and Ali chooses some of these blocks so that the sum of their area is at least n2n^2. Then , Amir tries to cover the n∗nn*n table so that none of blocks go out of the table and they don't overlap and he wanna maximize the covered area in the n∗nn*n table with those blocks chosen by Ali. Let kk be the maximum coverable area independent of Ali's choice. Prove that: n2−⌈n24⌉≤k≤n2−⌊n28⌋n^2 - \lceil \frac{n^2}{4} \rceil \leq k \leq n^2 - \lfloor \frac{n^2}{8} \rfloor
*Note : the blocks can be placed only vertically or horizontally.
combinatoricsceiling function
Again , maybe a P1

Source: Iran MO 2023 3rd round , geometry exam P3

8/23/2023
In triangle △ABC\triangle ABC points M,NM,N lie on BCBC st : ∠BAM=∠MAN=∠NAC\angle BAM= \angle MAN= \angle NAC . Points P,QP,Q are on the angle bisector of BACBAC, on the same side of BCBC as A , st : 13∠BAC=12∠BPC=∠BQC\frac{1}{3} \angle BAC = \frac{1}{2} \angle BPC = \angle BQC Let E=AM∩CQE = AM \cap CQ and F=AN∩BQF = AN \cap BQ . Prove that the common tangents to (EPF),(EQF)(EPF), (EQF) and the circumcircle of △ABC\triangle ABC , are concurrent.
geometryangle bisectorcircumcircle
Polynomials being a power of sth

Source: Iran MO 2023 3rd round , Algebra exam P3

8/21/2023
For numbers a,b∈Ra,b \in \mathbb{R} we consider the sets: A={an∣n∈N},B={bn∣n∈N}A=\{a^n | n \in \mathbb{N}\} , B=\{b^n | n \in \mathbb{N}\} Find all a,b>1a,b > 1 for which there exists two real , non-constant polynomials P,QP,Q with positive leading coefficients st for each r∈Rr \in \mathbb{R}: P(r)∈A  ⟺  Q(r)∈B P(r) \in A \iff Q(r) \in B
algebrapolynomial