MathDB

5

Part of 2011 Iran MO (3rd Round)

Problems(5)

number of good sequences

Source: Iran 3rd round 2011-combinatorics exam-p5

9/4/2011
Suppose that nn is a natural number. we call the sequence (x1,y1,z1,t1),(x2,y2,z2,t2),.....,(xs,ys,zs,ts)(x_1,y_1,z_1,t_1),(x_2,y_2,z_2,t_2),.....,(x_s,y_s,z_s,t_s) of Z4\mathbb Z^4 good if it satisfies these three conditions:
i) x1=y1=z1=t1=0x_1=y_1=z_1=t_1=0.
ii) the sequences xi,yi,zi,tix_i,y_i,z_i,t_i be strictly increasing.
iii) xs+ys+zs+ts=nx_s+y_s+z_s+t_s=n. (note that ss may vary).
Find the number of good sequences.
proposed by Mohammad Ghiasi
functionalgebralinear equationcombinatorics proposedcombinatorics
distance between prime numbers in Z[i]

Source: Iran 3rd round 2011-number theory exam-p5

9/5/2011
Suppose that kk is a natural number. Prove that there exists a prime number in Z\mathbb Z_{} such that every other prime number in Z\mathbb Z_{} has a distance at least kk with it.
modular arithmeticnumber theory proposednumber theory
concurrent lines formed by incenter and excenters

Source: Iran 3rd round 2011-geometry exam-p5

9/6/2011
Given triangle ABCABC, DD is the foot of the external angle bisector of AA, II its incenter and IaI_a its AA-excenter. Perpendicular from II to DIaDI_a intersects the circumcircle of triangle in A′A'. Define B′B' and C′C' similarly. Prove that AA′,BB′AA',BB' and CC′CC' are concurrent.
proposed by Amirhossein Zabeti
geometryincentercircumcirclegeometric transformationangle bisectorgeometry proposed
f(x^n) is irreducible

Source: Iran 3rd round 2011-algebra exam-p5

9/7/2011
f(x)f(x) is a monic polynomial of degree 22 with integer coefficients such that f(x)f(x) doesn't have any real roots and also f(0)f(0) is a square-free integer (and is not 11 or −1-1). Prove that for every integer nn the polynomial f(xn)f(x^n) is irreducible over Z[x]\mathbb Z[x].
proposed by Mohammadmahdi Yazdi
algebrapolynomialtrigonometryalgebra proposed
golden prime numbers

Source: Iran 3rd round 2011-final exam-p5

9/12/2011
Suppose that α\alpha is a real number and a1<a2<.....a_1<a_2<..... is a strictly increasing sequence of natural numbers such that for each natural number nn we have an≤nαa_n\le n^{\alpha}. We call the prime number qq golden if there exists a natural number mm such that q∣amq|a_m. Suppose that q1<q2<q3<.....q_1<q_2<q_3<..... are all the golden prime numbers of the sequence {an}\{a_n\}.
a) Prove that if α=1.5\alpha=1.5, then qn≤1390nq_n\le 1390^n. Can you find a better bound for qnq_n?
b) Prove that if α=2.4\alpha=2.4, then qn≤13902nq_n\le 1390^{2n}. Can you find a better bound for qnq_n?
part a proposed by mahyar sefidgaran by an idea of this question that the nnth prime number is less than 22n−22^{2n-2} part b proposed by mostafa einollah zade
number theoryprime numbers