MathDB

2021 Iran Team Selection Test

Part of Iran Team Selection Test

Subcontests

(6)
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Who can guess the polynomials ?

Prove there exist two relatively prime polynomials P(x),Q(x)P(x),Q(x) having integer coefficients and a real number u>0u>0 such that if for positive integers a,b,c,da,b,c,d we have: ∣ac−1∣2021≤u∣d∣∣c∣1010|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}} ∣(ac)2020−bd∣≤u∣d∣∣c∣1010| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}} Then we have : bP(ac)=dQ(ac)bP(\frac{a}{c})=dQ(\frac{a}{c})
(Two polynomials are relatively prime if they don't have a common root)
Proposed by Navid Safaii and Alireza Haghi

multiplicative functions on a Tst

There exist 44 positive integers a,b,c,da,b,c,d such that abcd≠1abcd \neq 1 and each pair of them have a GCD of 11. Two functions f,g:N→{0,1}f,g : \mathbb{N} \rightarrow \{0,1\} are multiplicative functions such that for each positive integer nn we have : f(an+b)=g(cn+d)f(an+b)=g(cn+d) Prove that at least one of the followings hold. i)i) for each positive integer nn we have f(an+b)=g(cn+d)=0f(an+b)=g(cn+d)=0 ii)ii) There exists a positive integer kk such that for all nn where (n,k)=1(n,k)=1 we have g(n)=f(n)=1g(n)=f(n)=1
(Function ff is multiplicative if for any natural numbers a,ba,b we have f(ab)=f(a)f(b)f(ab)=f(a)f(b))
Proposed by Navid Safaii
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2

Problem involving sets, as always

Prove that we can color every subset with nn element of a set with 3n3n elements with 88 colors . In such a way that there are no 33 subsets A,B,CA,B,C with the same color where : ∣A∩B∣≤1,∣A∩C∣≤1,∣B∩C∣≤1|A \cap B| \le 1,|A \cap C| \le 1,|B \cap C| \le 1
Proposed by Morteza Saghafian and Amir Jafari

Meh geo, as always

Point DD is chosen on the Euler line of triangle ABCABC and it is inside of the triangle. Points E,FE,F are were the line BD,CDBD,CD intersect with AC,ABAC,AB respectively. Point XX is on the line ADAD such that ∠EXF=180−∠A\angle EXF =180 - \angle A, also A,XA,X are on the same side of EFEF. If PP is the second intersection of circumcircles of CXF,BXECXF,BXE then prove the lines XP,EFXP,EF meet on the altitude of AA
Proposed by Alireza Danaie
5
2

Best Geo ever made

Point XX is chosen inside the non trapezoid quadrilateral ABCDABCD such that ∠AXD+∠BXC=180\angle AXD +\angle BXC=180. Suppose the angle bisector of ∠ABX\angle ABX meets the DD-altitude of triangle ADXADX in KK, and the angle bisector of ∠DCX\angle DCX meets the AA-altitude of triangle ADXADX in LL.We know BK⊥CXBK \perp CX and CL⊥BXCL \perp BX. If the circumcenter of ADXADX is on the line KLKL prove that KL⊥ADKL \perp AD.
Proposed by Alireza Dadgarnia

Nice numbers which are not really nice

Call a triple of numbers Nice if one of them is the average of the other two. Assume that we have 2k+12k+1 distinct real numbers with k2k^2 Nice triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios
Proposed by Morteza Saghafian
4
2

Easy problem to give us comfort

Find all functions f:N→Rf : \mathbb{N} \rightarrow \mathbb{R} such that for all triples a,b,ca,b,c of positive integers the following holds :
f(ac)+f(bc)−f(c)f(ab)≥1f(ac)+f(bc)-f(c)f(ab) \ge 1
Proposed by Mojtaba Zare

Random prime functions on polynomials

Assume Ω(n),ω(n)\Omega(n),\omega(n) be the biggest and smallest prime factors of nn respectively . Alireza and Amin decided to play a game. First Alireza chooses 14001400 polynomials with integer coefficients. Now Amin chooses 700700 of them, the set of polynomials of Alireza and Amin are B,AB,A respectively . Amin wins if for all nn we have : max⁡P∈A(Ω(P(n)))≥min⁡P∈B(ω(P(n)))\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n))) Who has the winning strategy.
Proposed by Alireza Haghi
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1
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Iran TST 2021 First Exam,P1

In acute scalene triangle ABCABC the external angle bisector of ∠BAC\angle BAC meet BCBC at point XX.Lines lbl_b and lcl_c which tangents of BB and CC with respect to (ABC)(ABC).The line pass through XX intersects lbl_b and lcl_c at points YY and ZZ respectively. Suppose (AYB)∩(AZC)=N(AYB)\cap(AZC)=N and lb∩lc=Dl_b\cap l_c=D. Show that NDND is angle bisector of ∠YNZ\angle YNZ. Proposed by Alireza Haghi

How original of us

Natural numbers are placed in an infinite grid. Such that the number in each cell is equal to the number of its adjacent cells having the same number. Find the most distinct numbers this infinite grid can have.
(Two cells of the grid are adjacent if they have a common vertex)