MathDB

1

Part of 2012 Indonesia TST

Problems(8)

Complex Numbers Inequality

Source: 2012 Indonesia Round 2 TST 1 Problem 1

2/26/2012
Let a,b,c∈Ca,b,c \in \mathbb{C} such that a∣bc∣+b∣ca∣+c∣ab∣=0a|bc| + b|ca| + c|ab| = 0. Prove that ∣(a−b)(b−c)(c−a)∣≥33∣abc∣|(a-b)(b-c)(c-a)| \ge 3\sqrt{3}|abc|.
inequalitiesinequalities proposed
Cycling pairs and groups

Source: 2012 Indonesia Round 2 TST 2 Problem 1

3/4/2012
A cycling group that has 4n4n members will have several cycling events, such that: a) Two cycling events are done every week; once on Saturday and once on Sunday. b) Exactly 2n2n members participate in any cycling event. c) No member may participate in both cycling events of a week. d) After all cycling events are completed, the number of events where each pair of members meet is the same for all pairs of members. Prove that after all cycling events are completed, the number of events where each group of three members meet is the same value tt for all groups of three members, and that for n≥2n \ge 2, tt is divisible by n−1n-1.
combinatorics proposedcombinatorics
Functional equation

Source: 2012 Indonesia Round 2 TST 3 Problem 1

3/18/2012
Find all functions f:R→Rf : \mathbb{R} \rightarrow \mathbb{R} such that f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y)f(x+y) + f(x)f(y) = f(xy) + (y+1)f(x) + (x+1)f(y) for all x,y∈Rx,y \in \mathbb{R}.
functionalgebra unsolvedalgebra
Function difference is a polynomial

Source: 2012 Indonesia Round 2 TST 4 Problem 1

3/18/2012
Let PP be a polynomial with real coefficients. Find all functions f:R→Rf : \mathbb{R} \rightarrow \mathbb{R} such that there exists a real number tt such that f(x+t)−f(x)=P(x)f(x+t) - f(x) = P(x) for all x∈Rx \in \mathbb{R}.
functionalgebrapolynomialinductionalgebra unsolved
Homogenous polynomial on sin t and cos t

Source: 2012 Indonesia Round 2.5 TST 1 Problem 1

5/10/2012
Suppose P(x,y)P(x,y) is a homogenous non-constant polynomial with real coefficients such that P(sin⁡t,cos⁡t)=1P(\sin t, \cos t) = 1 for all real tt. Prove that P(x,y)=(x2+y2)kP(x,y) = (x^2+y^2)^k for some positive integer kk.
(A polynomial A(x,y)A(x,y) with real coefficients and having a degree of nn is homogenous if it is the sum of aixiyn−ia_ix^iy^{n-i} for some real number aia_i, for all integer 0≤i≤n0 \le i \le n.)
algebrapolynomialtrigonometryalgebra unsolved
LCM is larger than argument/result

Source: 2012 Indonesia Round 2.5 TST 2 Problem 1

5/21/2012
Given a positive integer nn.
(a) If PP is a polynomial of degree nn where P(x)∈ZP(x) \in \mathbb{Z} for every x∈Zx \in \mathbb{Z}, prove that for every a,b∈Za,b \in \mathbb{Z} where P(a)≠P(b)P(a) \neq P(b), lcm(1,2,…,n)≥∣a−bP(a)−P(b)∣\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|
(b) Find one PP (for each nn) such that the equality case above is achieved for some a,b∈Za,b \in \mathbb{Z}.
number theoryleast common multiplealgebrapolynomialalgebra unsolved
Prove that [1612,2012] doesn't appear in a_n

Source: 2012 Indonesia Round 2.5 TST 3 Problem 1

5/21/2012
The sequence aia_i is defined as a1=2,a2=3a_1 = 2, a_2 = 3, and an+1=2an−1a_{n+1} = 2a_{n-1} or an+1=3an−2an−1a_{n+1} = 3a_n - 2a_{n-1} for all integers n≥2n \ge 2. Prove that no term in aia_i is in the range [1612,2012][1612, 2012].
inductionalgebra proposedalgebrabinary representationSequencerecurrence relation
f(f(n)) + f(n+1) = n+2; prove f(f(n)+n) = n+1

Source: 2012 Indonesia Round 2.5 TST 4 Problem 1

5/31/2012
Suppose a function f:Z+→Z+f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+ satisfies f(f(n))+f(n+1)=n+2f(f(n)) + f(n+1) = n+2 for all positive integer nn. Prove that f(f(n)+n)=n+1f(f(n)+n) = n+1 for all positive integer nn.
functioninductionfloor functionalgebra unsolvedalgebra