MathDB

1

Part of 2012 Iran Team Selection Test

Problems(6)

parity of C(n,k)

Source: Iran TST 2012 -first day- problem 1

4/23/2012
Find all positive integers n≥2n \geq 2 such that for all integers i,ji,j that 0≤i,j≤n 0 \leq i,j\leq n , i+ji+j and (ni)+(nj) {n\choose i}+ {n \choose j} have same parity.
Proposed by Mr.Etesami
modular arithmeticpolynomialnumber theorylucas theorem
An equality in a grid sheet

Source: Iran TST 2012-First exam-2nd day-P4

4/24/2012
Consider m+1m+1 horizontal and n+1n+1 vertical lines (m,n≥4m,n\ge 4) in the plane forming an m×nm\times n table. Cosider a closed path on the segments of this table such that it does not intersect itself and also it passes through all (m−1)(n−1)(m-1)(n-1) interior vertices (each vertex is an intersection point of two lines) and it doesn't pass through any of outer vertices. Suppose AA is the number of vertices such that the path passes through them straight forward, BB number of the table squares that only their two opposite sides are used in the path, and CC number of the table squares that none of their sides is used in the path. Prove that A=B−C+m+n−1.A=B-C+m+n-1.
Proposed by Ali Khezeli
combinatorics proposedcombinatorics
Consecutive numbers on edges of the graph

Source: Iran TST 2012-Second exam-1st day-P1

5/12/2012
Is it possible to put (n2)\binom{n}{2} consecutive natural numbers on the edges of a complete graph with nn vertices in a way that for every path (or cycle) of length 33 where the numbers a,ba,b and cc are written on its edges (edge bb is between edges cc and aa), bb is divisible by the greatest common divisor of the numbers aa and cc?
Proposed by Morteza Saghafian
inductiongraph theorygreatest common divisorcombinatorics proposedcombinatorics
Easy inequality with ab+bc+ca=1

Source: Iran TST 2012-Second exam-2nd day-P4

5/13/2012
For positive reals a,ba,b and cc with ab+bc+ca=1ab+bc+ca=1, show that 3(a+b+c)≤aabc+bbca+ccab.\sqrt{3}({\sqrt{a}+\sqrt{b}+\sqrt{c})\le \frac{a\sqrt{a}}{bc}+\frac{b\sqrt{b}}{ca}+\frac{c\sqrt{c}}{ab}.}
Proposed by Morteza Saghafian
inequalitiesinequalities proposed
Regular 2^k-gon and reflections

Source: Iran TST 2012-Third exam-1st day-P1

5/15/2012
Consider a regular 2k2^k-gon with center OO and label its sides clockwise by l1,l2,...,l2kl_1,l_2,...,l_{2^k}. Reflect OO with respect to l1l_1, then reflect the resulting point with respect to l2l_2 and do this process until the last side. Prove that the distance between the final point and OO is less than the perimeter of the 2k2^k-gon.
Proposed by Hesam Rajabzade
geometrygeometric transformationreflectionperimeterrotationinductiongeometry proposed
Polynomial and complete residue system

Source: Iran TST 2012-Third exam-2nd day-P4

5/16/2012
Suppose pp is an odd prime number. We call the polynomial f(x)=∑j=0najxjf(x)=\sum_{j=0}^n a_jx^j with integer coefficients ii-remainder if ∑p−1∣j,j>0aj≡i(modp) \sum_{p-1|j,j>0}a_{j}\equiv i\pmod{p}. Prove that the set {f(0),f(1),...,f(p−1)}\{f(0),f(1),...,f(p-1)\} is a complete residue system modulo pp if and only if polynomials f(x),(f(x))2,...,(f(x))p−2f(x), (f(x))^2,...,(f(x))^{p-2} are 00-remainder and the polynomial (f(x))p−1(f(x))^{p-1} is 11-remainder.
Proposed by Yahya Motevassel
algebrapolynomialmodular arithmeticinductionVietacalculusIran