MathDB

3

Part of 2005 Iran MO (3rd Round)

Problems(5)

Holder

Source: Iran 2005

8/27/2005
Find all α>0\alpha>0 and β>0\beta>0 that for each (x1,…,xn)(x_1,\dots,x_n) and (y1,…,yn)∈R+n(y_1,\dots,y_n)\in\mathbb {R^+}^n that:(∑xiα)(∑yiβ)≥∑xiyi(\sum x_i^\alpha)(\sum y_i^\beta)\geq\sum x_iy_i
functioninequalitiesalgebra proposedalgebra
Inequality

Source: Iran 2005

8/27/2005
Prove that in acute-angled traingle ABC if rr is inradius and RR is radius of circumcircle then: a2+b2+c2≥4(R+r)2a^2+b^2+c^2\geq 4(R+r)^2
inequalitiesgeometryinradiuscircumcircletrigonometrygeometry proposed
p,q

Source: Iran 2005

8/29/2005
p(x)p(x) is an irreducible polynomial in Q[x]\mathbb Q[x] that \mbox{deg}\ p is odd. q(x),r(x)q(x),r(x) are polynomials with rational coefficients that p(x)∣q(x)2+q(x).r(x)+r(x)2p(x)|q(x)^2+q(x).r(x)+r(x)^2. Prove that p(x)2∣q(x)2+q(x).r(x)+r(x)2p(x)^2|q(x)^2+q(x).r(x)+r(x)^2
algebrapolynomialnumber theory proposednumber theory
n-mino

Source: Iran 2005

9/1/2005
f(n)f(n) is the least number that there exist a f(n)−f(n)-mino that contains every n−n-mino. Prove that 10000≤f(1384)≤96000010000\leq f(1384)\leq960000. Find some bound for f(n)f(n)
geometryrectanglelogarithmscombinatorics proposedcombinatorics
Abc conjecture

Source: Iran 2005

9/21/2005
For each m∈Nm\in \mathbb N we define rad (m)=∏pirad\ (m)=\prod p_i, where m=∏piαim=\prod p_i^{\alpha_i}.
abc Conjecture Suppose ϵ>0\epsilon >0 is an arbitrary number, then there exist KK depinding on ϵ\epsilon that for each 3 numbers a,b,c∈Za,b,c\in\mathbb Z that gcd(a,b)=1gcd (a,b)=1 and a+b=ca+b=c then: max{∣a∣,∣b∣,∣c∣}≤K(rad (abc))1+ϵ max\{|a|,|b|,|c|\}\leq K(rad\ (abc))^{1+\epsilon}
Now prove each of the following statements by using the abcabc conjecture : a) Fermat's last theorem for n>Nn>N where NN is some natural number. b) We call n=∏piαin=\prod p_i^{\alpha_i} strong if and only αi≥2\alpha_i\geq 2. c) Prove that there are finitely many nn such that n, n+1, n+2n,\ n+1,\ n+2 are strong. d) Prove that there are finitely many rational numbers pq\frac pq such that: ∣23−pq∣<21384q3 \Big| \sqrt[3]{2}-\frac pq \Big|<\frac{2^ {1384}}{q^3}
number theorygreatest common divisorlogarithmsgeometrynumber theory proposed