MathDB

2

Part of 2010 Indonesia TST

Problems(7)

sum of subsets with k elements from vertices of a tree with n vertices

Source: 2010 Indonesia TST stage 2 test 4 p2

12/16/2020
Let TT be a tree withn n vertices. Choose a positive integer kk where 1≤k≤n1 \le k \le n such that SkS_k is a subset with kk elements from the vertices in TT. For all S∈SkS \in S_k, define c(S)c(S) to be the number of component of graph from SS if we erase all vertices and edges in TT, except all vertices and edges in SS. Determine ∑S∈Skc(S)\sum_{S\in S_k} c(S), expressed in terms of nn and kk.
combinatoricsgraph theory
government’s land with dimensions n x n are going to be sold in phases

Source: 2010 Indonesia TST stage 2 test 5 p2

12/16/2020
A government’s land with dimensions n×nn \times n are going to be sold in phases. The land is divided into n2n^2 squares with dimension 1×11 \times 1. In the first phase, nn farmers bought a square, and for each rows and columns there is only one square that is bought by a farmer. After one season, each farmer could buy one more square, with the conditions that the newly-bought square has a common side with the farmer’s land and it hasn’t been bought by other farmers. Determine all values of n such that the government’s land could not be entirely sold within nn seasons.
combinatorics
1 mod 2009^(2009)

Source:

11/12/2009
Let A\equal{}\{n: 1 \le n \le 2009^{2009},n \in \mathbb{N} \} and let S\equal{}\{n: n \in A,\gcd \left(n,2009^{2009}\right)\equal{}1\}. Let P P be the product of all elements of S S. Prove that P≡1(mod20092009). P \equiv 1 \pmod{2009^{2009}}. Nanang Susyanto, Jogjakarta
modular arithmeticnumber theorynumber theory proposed
functional equation

Source: Indonesia IMO 2010 TST, Stage 1, Test 2, Problem 2

11/12/2009
Find all functions f:R→R f: \mathbb{R} \rightarrow \mathbb{R} satisfying f(x^3\plus{}y^3)\equal{}xf(x^2)\plus{}yf(y^2) for all real numbers x x and y y. Hery Susanto, Malang
functioninductionalgebra proposedalgebrafunctional equation
coloring the points

Source: Indonesia IMO 2010 TST, Stage 1, Test 3, Problem 2

11/12/2009
Given an equilateral triangle, all points on its sides are colored in one of two given colors. Prove that the is a right-angled triangle such that its three vertices are in the same color and on the sides of the equilateral triangle. Alhaji Akbar, Jakarta
combinatorics proposedcombinatorics
O1, O2, O3 are collinear

Source: Indonesia IMO 2010 TST, Stage 1, Test 4, Problem 2

11/12/2009
Circles Γ1 \Gamma_1 and Γ2 \Gamma_2 are internally tangent to circle Γ \Gamma at P P and Q Q, respectively. Let P1 P_1 and Q1 Q_1 are on Γ1 \Gamma_1 and Γ2 \Gamma_2 respectively such that P1Q1 P_1Q_1 is the common tangent of P1 P_1 and Q1 Q_1. Assume that Γ1 \Gamma_1 and Γ2 \Gamma_2 intersect at R R and R1 R_1. Define O1,O2,O3 O_1,O_2,O_3 as the intersection of PQ PQ and P1Q1 P_1Q_1, the intersection of PR PR and P1R1 P_1R_1, and the intersection QR QR and Q1R1 Q_1R_1. Prove that the points O1,O2,O3 O_1,O_2,O_3 are collinear. Rudi Adha Prihandoko, Bandung
geometric transformationprojective geometrygeometrypower of a pointradical axisgeometry proposed
polynomial to be killed

Source: Indonesia IMO 2010 TST, Stage 1, Test 5, Problem 2

11/12/2009
Consider a polynomial with coefficients of real numbers \phi(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d with three positive real roots. Assume that ϕ(0)<0 \phi(0)<0, prove that 2b^3\plus{}9a^2d\minus{}7abc \le 0. Hery Susanto, Malang
algebrapolynomialinequalitiesalgebra proposed