MathDB

4

Part of 2011 Iran MO (3rd Round)

Problems(5)

ongoing points in a permutation

Source: Iran 3rd round 2011-combinatorics exam-p4

9/4/2011
We say the point ii in the permutation σ\sigma ongoing if for every j<ij<i we have σ(j)<σ(i)\sigma (j)<\sigma (i).
a) prove that the number of permutations of the set {1,....,n}\{1,....,n\} with exactly rr ongoing points is s(n,r)s(n,r).
b) prove that the number of nn-letter words with letters {a1,....,ak},a1<.....<ak\{a_1,....,a_k\},a_1<.....<a_k. with exactly rr ongoing points is ∑m(km)S(n,m)s(m,r)\sum_{m}\dbinom{k}{m} S(n,m) s(m,r).
combinatorics proposedcombinatorics
having at least 2d(n) prime factors

Source: Iran 3rd round 2011-number theory exam-p4

9/5/2011
Suppose that nn is a natural number and nn is not divisible by 33. Prove that (n2n+nn+n+1)2n+(n2n+nn+n+1)n+1(n^{2n}+n^n+n+1)^{2n}+(n^{2n}+n^n+n+1)^n+1 has at least 2d(n)2d(n) distinct prime factors where d(n)d(n) is the number of positive divisors of nn. proposed by Mahyar Sefidgaran
algebrapolynomialnumber theory proposednumber theory
fixed incircle and circumcircle

Source: Iran 3rd round 2011-geometry exam-p4

9/6/2011
A variant triangle has fixed incircle and circumcircle. Prove that the radical center of its three excircles lies on a fixed circle and the circle's center is the midpoint of the line joining circumcenter and incenter.
proposed by Masoud Nourbakhsh
geometrycircumcircleincentergeometric transformationreflectionradical axisgeometry proposed
strange long inequality

Source: Iran 3rd round 2011-algebra exam-p4

9/7/2011
For positive real numbers a,ba,b and cc we have a+b+c=3a+b+c=3. Prove a1+(b+c)2+b1+(a+c)2+c1+(a+b)2≤3(a2+b2+c2)a2+b2+c2+12abc\frac{a}{1+(b+c)^2}+\frac{b}{1+(a+c)^2}+\frac{c}{1+(a+b)^2}\le \frac{3(a^2+b^2+c^2)}{a^2+b^2+c^2+12abc}.
proposed by Mohammad Ahmadi
inequalitiesfunctionLaTeXCauchy Inequalityinequalities proposed
smart escalator

Source: Iran 3rd round 2011-final exam-p4

9/12/2011
The escalator of the station champion butcher has this property that if mm persons are on it, then it's speed is m−αm^{-\alpha} where α\alpha is a fixed positive real number. Suppose that nn persons want to go up by the escalator and the width of the stairs is such that all the persons can stand on a stair. If the length of the escalator is ll, what's the least time that is needed for these persons to go up? Why?
proposed by Mohammad Ghiasi
algebra proposedalgebra