MathDB

3

Part of 2007 Indonesia TST

Problems(5)

sequence of positive integers

Source: Indonesia IMO 2007 TST, Stage 2, Test 1, Problem 3

11/15/2009
Let a1,a2,a3,… a_1,a_2,a_3,\dots be infinite sequence of positive integers satisfying the following conditon: for each prime number p p, there are only finite number of positive integers i i such that p∣ai p|a_i. Prove that that sequence contains a sub-sequence ai1,ai2,ai3,… a_{i_1},a_{i_2},a_{i_3},\dots, with 1≤i1<i2<i3<… 1 \le i_1<i_2<i_3<\dots, such that for each m≠n m \ne n, \gcd(a_{i_m},a_{i_n})\equal{}1.
number theory proposednumber theory
P(a)+P(b)+P(c)&lt;= -1 if P(x)=3^{2005}x^{2007}- 3^{2005}x^{2006}-x^2,

Source: 2007 Indonesia TST stage 2 test 2 p3

12/14/2020
Let a,b,ca, b, c be positive reals such that a+b+c=1a + b + c = 1 and P(x)=32005x2007−32005x2006−x2P(x) = 3^{2005}x^{2007 }- 3^{2005}x^{2006} - x^2. Prove that P(a)+P(b)+P(c)≤−1P(a) + P(b) + P(c) \le -1.
algebrainequalitiespolynomial
(f,p) f is function, p is polynomial

Source: Indonesia IMO 2007 TST, Stage 2, Test 4, Problem 3

11/15/2009
Find all pairs of function f:N→N f: \mathbb{N} \rightarrow \mathbb{N} and polynomial with integer coefficients p p such that: (i) p(mn) \equal{} p(m)p(n) for all positive integers m,n>1 m,n > 1 with \gcd(m,n) \equal{} 1, and (ii) \sum_{d|n}f(d) \equal{} p(n) for all positive integers n n.
functionalgebrapolynomialinductionnumber theory proposednumber theory
inequality

Source: Indonesia IMO 2007 TST, Stage 2, Test 2, Problem 3

11/15/2009
For each real number x x< let ⌊x⌋ \lfloor x \rfloor be the integer satisfying \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1 and let \{x\}\equal{}x\minus{}\lfloor x \rfloor. Let c c be a real number such that {n3}>cn3 \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}} for all positive integers n n. Prove that c≤1 c \le 1.
inequalitiesfloor functionDiophantine equationnumber theory proposednumber theory
crow on regular n-gon

Source: Indonesia IMO 2007 TST, Stage 2, Test 5, Problem 3

11/15/2009
On each vertex of a regular n\minus{}gon there was a crow. Call this as initial configuration. At a signal, they all flew by and after a while, those n n crows came back to the n\minus{}gon, one crow for each vertex. Call this as final configuration. Determine all n n such that: there are always three crows such that the triangle they formed in the initial configuration and the triangle they formed in the final configuration are both right-angled triangle.
combinatorics proposedcombinatorics