MathDB

7

Part of 2005 IMO Shortlist

Problems(3)

Two permutations

Source: Iran prepration exam

4/24/2006
Suppose that a1 a_1, a2 a_2, … \ldots, an a_n are integers such that n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n. Prove that there exist two permutations (b1,b2,…,bn) \left(b_1,b_2,\ldots,b_n\right) and (c1,c2,…,cn) \left(c_1,c_2,\ldots,c_n\right) of (1,2,…,n) \left(1,2,\ldots,n\right) such that for each integer i i with 1≤i≤n 1\leq i\leq n, we have n\mid a_i \minus{} b_i \minus{} c_i
Proposed by Ricky Liu & Zuming Feng, USA
abstract algebragroup theorycombinatoricspermutationsIMO Shortlist
perimeter Inequality [p(ABC) p(PQR) >= (p(DEF))^2]

Source: IMO Shortlist 2005, problem G7, created by Hojoo Lee

7/2/2006
In an acute triangle ABCABC, let DD, EE, FF be the feet of the perpendiculars from the points AA, BB, CC to the lines BCBC, CACA, ABAB, respectively, and let PP, QQ, RR be the feet of the perpendiculars from the points AA, BB, CC to the lines EFEF, FDFD, DEDE, respectively.
Prove that p(ABC)p(PQR)≥(p(DEF))2p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}, where p(T)p\left(T\right) denotes the perimeter of triangle TT .
Proposed by Hojoo Lee, Korea
geometryinequalitiescircumcircleIMO Shortlist
P(m!) is composite

Source: IMO Shortlist 2005, N7

3/19/2007
Let P(x)=anxn+an−1xn−1+…+a0P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0}, where a0,…,ana_{0},\ldots,a_{n} are integers, an>0a_{n}>0, n≥2n\geq 2. Prove that there exists a positive integer mm such that P(m!)P(m!) is a composite number.
polynomialnumber theorycomposite numbersalgebraIMO Shortlist