MathDB

5

Part of 2006 Iran MO (3rd Round)

Problems(5)

L(n),T(n)

Source: Iranian National Olympiad (3rd Round) 2006

8/26/2006
For each nn, define L(n)L(n) to be the number of natural numbers 1≤a≤n1\leq a\leq n such that n∣an−1n\mid a^{n}-1. If p1,p2,…,pkp_{1},p_{2},\ldots,p_{k} are the prime divisors of nn, define T(n)T(n) as (p1−1)(p2−1)⋯(pk−1)(p_{1}-1)(p_{2}-1)\cdots(p_{k}-1). a) Prove that for each n∈Nn\in\mathbb N we have n∣L(n)T(n)n\mid L(n)T(n). b) Prove that if gcd⁡(n,T(n))=1\gcd(n,T(n))=1 then φ(n)∣L(n)T(n)\varphi(n) | L(n)T(n).
abstract algebragroup theorynumber theory proposednumber theory
Biggest k

Source: Iranian National Olympiad (3rd Round) 2006

9/19/2006
Find the biggest real number k k such that for each right-angled triangle with sides a a, b b, c c, we have a^{3}\plus{}b^{3}\plus{}c^{3}\geq k\left(a\plus{}b\plus{}c\right)^{3}.
functiongeometryinradiuscircumcircleinequalitiesincenterinequalities unsolved
Extremal Set Theory

Source: Iranian National Olympiad (3rd Round) 2006

9/11/2006
Let EE be a family of subsets of {1,2,…,n}\{1,2,\ldots,n\} with the property that for each A⊂{1,2,…,n}A\subset \{1,2,\ldots,n\} there exist B∈FB\in F such that n−d2≤∣A△B∣≤n+d2\frac{n-d}2\leq |A \bigtriangleup B| \leq \frac{n+d}2. (where A△B=(A∖B)∪(B∖A)A \bigtriangleup B = (A\setminus B) \cup (B\setminus A) is the symmetric difference). Denote by f(n,d)f(n,d) the minimum cardinality of such a family. a) Prove that if nn is even then f(n,0)≤nf(n,0)\leq n. b) Prove that if n−dn-d is even then f(n,d)≤⌈nd+1⌉f(n,d)\leq \lceil \frac n{d+1}\rceil. c) Prove that if nn is even then f(n,0)=nf(n,0) = n
ceiling functioncombinatorics proposedcombinatorics
Calculating ruler

Source: Iranian National Math Olympiad (Final exam) 2006

9/19/2006
A calculating ruler is a ruler for doing algebric calculations. This ruler has three arms, two of them are sationary and one can move freely right and left. Each of arms is gradient. Gradation of each arm depends on the algebric operation ruler does. For eaxample the ruler below is designed for multiplying two numbers. Gradations are logarithmic. http://aycu05.webshots.com/image/5604/2000468517162383885_rs.jpg For working with ruler, (e.g for calculating x.yx.y) we must move the middle arm that the arrow at the beginning of its gradation locate above the xx in the lower arm. We find yy in the middle arm, and we will read the number on the upper arm. The number written on the ruler is the answer. 1) Design a ruler for calculating xyx^{y}. Grade first arm (xx) and (yy) from 1 to 10. 2) Find all rulers that do the multiplication in the interval [1,10][1,10]. 3) Prove that there is not a ruler for calculating x2+xy+y2x^{2}+xy+y^{2}, that its first and second arm are grade from 0 to 10.
functionvectoralgebrafunctional equationalgebra proposed
Triangle

Source: Iranian National Olympiad (3rd Round) 2006

9/21/2006
MM is midpoint of side BCBC of triangle ABCABC, and II is incenter of triangle ABCABC, and TT is midpoint of arc BCBC, that does not contain AA. Prove that cos⁡B+cos⁡C=1⟺MI=MT\cos B+\cos C=1\Longleftrightarrow MI=MT
geometryincentertrigonometrycircumcircleparallelogramgeometry proposed