MathDB

2

Part of 2019 Iran MO (3rd Round)

Problems(8)

Iran function (third round)

Source: Iran MO 3rd round midterm exam

7/28/2019
Find all function f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} such that for any three real number a,b,ca,b,c , if a+f(b)+f(f(c))=0 a + f(b) + f(f(c)) = 0 : f(a)3+bf(b)2+c2f(c)=3abc f(a)^3 + bf(b)^2 + c^2f(c) = 3abc .
Proposed by Amirhossein Zolfaghari
functional equationIran
Avoiding all subsequences of length $k$

Source: Iranian third round midterm Combinatorics exam problem 2

8/27/2019
Let n,kn,k be positive integers so that n≥kn \ge k.Find the maximum number of binary sequances of length nn so that fixing any arbitary kk bits they do not produce all binary sequances of length kk.For exmple if k=1k=1 we can only have one sequance otherwise they will differ in at least one bit which means that bit produces all binary sequances of length 11.
combinatorics
Iran geometry

Source: Iran MO 3rd round 2019 mid-terms - Geometry P2

8/2/2019
Consider an acute-angled triangle ABCABC with AB=ACAB=AC and ∠A>60∘\angle A>60^\circ. Let OO be the circumcenter of ABCABC. Point PP lies on circumcircle of BOCBOC such that BP∥ACBP\parallel AC and point KK lies on segment APAP such that BK=BCBK=BC. Prove that CKCK bisects the arc BCBC of circumcircle of BOCBOC.
geometrycircumcircle
Set of prime divisors

Source: Iran MO 3rd round 2019 mid-terms - Number theory P2

8/1/2019
Prove that for any positive integers m>nm>n, there is infinitely many positive integers a,ba,b such that set of prime divisors of am+bna^m+b^n is equal to set of prime divisors of a2019+b1398a^{2019}+b^{1398}.
number theory
Nice polynomial

Source: Iranian third round 2019 Finals Algebra exam problem 2

8/18/2019
P(x)P(x) is a monoic polynomial with integer coefficients so that there exists monoic integer coefficients polynomials p1(x),p2(x),…,pn(x)p_1(x),p_2(x),\dots ,p_n(x) so that for any natural number xx there exist an index jj and a natural number yy so that pj(y)=P(x)p_j(y)=P(x) and also deg(pj)≥deg(P)deg(p_j) \ge deg(P) for all jj.Show that there exist an index ii and an integer kk so that P(x)=pi(x+k)P(x)=p_i(x+k).
algebrapolynomial
Maximum and minimum number of intersection

Source: Iranian third round 2019 finals Combinatorics exam problem 2

8/27/2019
Let TT be a triangulation of a 100100-gon.We construct P(T)P(T) by copying the same 100100-gon and drawing a diagonal if it was not drawn in TT an there is a quadrilateral with this diagonal and two other vertices so that all the sides and diagonals(Except the one we are going to draw) are present in TT.Let f(T)f(T) be the number of intersections of diagonals in P(T)P(T).Find the minimum and maximum of f(T)f(T).
combinatorics
Iran geometry

Source: Iran MO 3rd round 2019 finals - Geometry P2

8/14/2019
In acute-angled triangle ABCABC altitudes BE,CFBE,CF meet at HH. A perpendicular line is drawn from HH to EFEF and intersects the arc BCBC of circumcircle of ABCABC (that doesn’t contain AA) at KK. If AK,BCAK,BC meet at PP, prove that PK=PHPK=PH.
geometrycircumcircle
Classical number theory

Source: Iranian third round 2019 Finals Number theory exam problem 2

8/15/2019
Call a polynomial P(x)=anxn+an−1xn−1+…a1x+a0P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots a_1x+a_0 with integer coefficients primitive if and only if gcd⁡(an,an−1,…a1,a0)=1\gcd(a_n,a_{n-1},\dots a_1,a_0) =1.
a)Let P(x)P(x) be a primitive polynomial with degree less than 13981398 and SS be a set of primes greater than 13981398.Prove that there is a positive integer nn so that P(n)P(n) is not divisible by any prime in SS.
b)Prove that there exist a primitive polynomial P(x)P(x) with degree less than 13981398 so that for any set SS of primes less than 13981398 the polynomial P(x)P(x) is always divisible by product of elements of SS.
number theory