MathDB

3

Part of 2014 Iran MO (3rd Round)

Problems(5)

polynomials

Source: Iran 3rd round 2014-Algebra exam-P3

8/31/2014
Let p,q∈R[x]p,q\in \mathbb{R}[x] such that p(z)q(z‾)p(z)q(\overline{z}) is always a real number for every complex number zz. Prove that p(x)=kq(x)p(x)=kq(x) for some constant k∈Rk \in \mathbb{R} or q(x)=0q(x)=0.
Proposed by Mohammad Ahmadi
algebrapolynomialcomplex analysisalgebra unsolved
a natural number with exactly n prime factors

Source: Iranian 3rd round Number Theory exam P3

9/22/2014
Let nn be a positive integer. Prove that there exists a natural number mm with exactly nn prime factors, such that for every positive integer dd the numbers in {1,2,3,…,m}\{1,2,3,\ldots,m\} of order dd modulo mm are multiples of ϕ(d)\phi (d).
(15 points)
number theory proposednumber theory
Subrectangles of a 10*10 Table

Source: Iran 3rd round 2014 - Combinatorics exam problem 3

9/27/2014
We have a 10×1010 \times 10 table. TT is a set of rectangles with vertices from the table and sides parallel to the sides of the table such that no rectangle from the set is a subrectangle of another rectangle from the set. tt is the maximum number of elements of TT. (a) Prove that t>300t>300. (b) Prove that t<600t<600.
Proposed by Mir Omid Haji Mirsadeghi and Kasra Alishahi
geometryrectanglecombinatorics unsolvedcombinatorics
Parallel lines

Source: Iranian 3rd round Geometry exam P3 - 2014

9/28/2014
Distinct points B,B′,C,C′B,B',C,C' lie on an arbitrary line ℓ\ell. AA is a point not lying on ℓ\ell. A line passing through BB and parallel to AB′AB' intersects with ACAC in EE and a line passing through CC and parallel to AC′AC' intersects with ABAB in FF. Let XX be the intersection point of the circumcircles of △ABC\triangle{ABC} and △AB′C′\triangle{AB'C'}(A≠XA \neq X). Prove that EF∥AXEF \parallel AX.
geometrycircumcircleIran
Iranian Non-Remainder theorem!

Source: Iran 3rd round 2014 - final exam problem 3

9/16/2014
(a) nn is a natural number. d1,…,dn,r1,…,rnd_1,\dots,d_n,r_1,\dots ,r_n are natural numbers such that for each i,ji,j that 1≤i<j≤n1\leq i < j \leq n we have (di,dj)=1(d_i,d_j)=1 and di≥2d_i\geq 2. Prove that there exist an xx such that (i) 1≤x≤3n1 \leq x \leq 3^n (ii)For each 1≤i≤n1 \leq i \leq n x≢dirix \overset{d_i}{\not{\equiv}} r_i (b) For each ϵ>0\epsilon >0 prove that there exists natural NN such that for each n>Nn>N and each d1,…,dn,r1,…,rnd_1,\dots,d_n,r_1,\dots ,r_n satisfying the conditions above there exists an xx satisfying (ii) such that 1≤x≤(2+ϵ)n1\leq x \leq (2+\epsilon )^n.
Time allowed for this exam was 75 minutes.
number theory unsolvednumber theory