MathDB

3

Part of 2012 Iran MO (3rd Round)

Problems(7)

1000 points with distinct pairwise distances

Source: Iran 3rd round 2012-Special Lesson exam-Part1-P3

7/27/2012
Prove that if nn is large enough, among any nn points of plane we can find 10001000 points such that these 10001000 points have pairwise distinct distances. Can you prove the assertion for nαn^{\alpha} where α\alpha is a positive real number instead of 10001000?
inductionpigeonhole principlegeometryperpendicular bisectoralgebrabinomial theoremcombinatorics proposed
Three sets having the same color

Source: Iran 3rd round 2012-Special Lesson exam-Part 2-P3

9/15/2012
Prove that if nn is large enough, then for each coloring of the subsets of the set {1,2,...,n}\{1,2,...,n\} with 13911391 colors, two non-empty disjoint subsets AA and BB exist such that AA, BB and A∪BA\cup B are of the same color.
graph theorycombinatorics proposedcombinatorics
Primitive roots!

Source: Iran 3rd round 2011-Number Theory exam-P3

9/19/2012
pp is an odd prime number. Prove that there exists a natural number xx such that xx and 4x4x are both primitive roots modulo pp.
Proposed by Mohammad Gharakhani
modular arithmeticnumber theory proposednumber theory
An ellipse, a lower bound for a ratio

Source: Iran 3rd round 2012-Geometry exam-P3

9/20/2012
Cosider ellipse ϵ\epsilon with two foci AA and BB such that the lengths of it's major axis and minor axis are 2a2a and 2b2b respectively. From a point TT outside of the ellipse, we draw two tangent lines TPTP and TQTQ to the ellipse ϵ\epsilon. Prove that TPTQ≥ba.\frac{TP}{TQ}\ge \frac{b}{a}.
Proposed by Morteza Saghafian
conicsellipseratioanalytic geometrygeometrygeometric transformationdilation
Longest paths in a tree

Source: Iran 3rd round 2012-Combinatorics exam-P3

9/20/2012
In a tree with nn vertices, for each vertex xix_i, denote the longest paths passing through it by li1,li2,...,likil_i^1,l_i^2,...,l_i^{k_i}. xix_i cuts those longest paths into two parts with (ai1,bi1),(ai2,bi2),...,(aiki,biki)(a_i^1,b_i^1),(a_i^2,b_i^2),...,(a_i^{k_i},b_i^{k_i}) vertices respectively. If max⁡j=1,...,ki{aij×bij}=pi\max_{j=1,...,k_i} \{a_i^j\times b_i^j\}=p_i, find the maximum and minimum values of ∑i=1npi\sum_{i=1}^{n} p_i.
Proposed by Sina Rezaei
combinatorics proposedcombinatorics
p-th root of rational numbers

Source: Iran 3rd round 2012-Algebra exam-P3

9/20/2012
Suppose pp is a prime number and a,b,c∈Q+a,b,c \in \mathbb Q^+ are rational numbers;
a) Prove that Q(ap+bp)=Q(ap,bp)\mathbb Q(\sqrt[p]{a}+\sqrt[p]{b})=\mathbb Q(\sqrt[p]{a},\sqrt[p]{b}).
b) If bp∈Q(ap)\sqrt[p]{b} \in \mathbb Q(\sqrt[p]{a}), prove that for a nonnegative integer kk we have bakp∈Q\sqrt[p]{\frac{b}{a^k}}\in \mathbb Q.
c) If ap+bp+cp∈Q\sqrt[p]{a}+\sqrt[p]{b}+\sqrt[p]{c} \in \mathbb Q, then prove that numbers ap,bp\sqrt[p]{a},\sqrt[p]{b} and cp\sqrt[p]{c} are rational.
algebrapolynomialalgebra proposed
Increasing sequence, Decreasing Euler's totient function

Source: Iran 3rd round 2012-Final exam-P3

9/24/2012
Prove that for each n∈Nn \in \mathbb N there exist natural numbers a1<a2<...<ana_1<a_2<...<a_n such that ϕ(a1)>ϕ(a2)>...>ϕ(an)\phi(a_1)>\phi(a_2)>...>\phi(a_n).
Proposed by Amirhossein Gorzi
functionceiling functionlogarithmsinductionnumber theory proposednumber theory