MathDB

3

Part of 2016 India IMO Training Camp

Problems(5)

Combinatorium Ultimatum --- Prove that n is 2 mod 4

Source: India TST 2016 Day 1 Problem 3

7/22/2016
Let nn be a natural number. A sequence x1,x2,⋯ ,xn2x_1,x_2, \cdots ,x_{n^2} of n2n^2 numbers is called n−<spanclass=′latex−italic′>good</span>n-<span class='latex-italic'>good</span> if each xix_i is an element of the set {1,2,⋯ ,n}\{1,2,\cdots ,n\} and the ordered pairs (xi,xi+1)\left(x_i,x_{i+1}\right) are all different for i=1,2,3,⋯ ,n2i=1,2,3,\cdots ,n^2 (here we consider the subscripts modulo n2n^2). Two n−n-good sequences x1,x2,⋯ ,xn2x_1,x_2,\cdots ,x_{n^2} and y1,y2,⋯ ,yn2y_1,y_2,\cdots ,y_{n^2} are called <spanclass=′latex−italic′>similar</span><span class='latex-italic'>similar</span> if there exists an integer kk such that yi=xi+ky_i=x_{i+k} for all i=1,2,⋯ ,n2i=1,2,\cdots,n^2 (again taking subscripts modulo n2n^2). Suppose that there exists a non-trivial permutation (i.e., a permutation which is different from the identity permutation) σ\sigma of {1,2,⋯ ,n}\{1,2,\cdots ,n\} and an n−n- good sequence x1,x2,⋯ ,xn2x_1,x_2,\cdots,x_{n^2} which is similar to σ(x1),σ(x2),⋯ ,σ(xn2)\sigma\left(x_1\right),\sigma\left(x_2\right),\cdots ,\sigma\left(x_{n^2}\right). Show that n≡2(mod4)n\equiv 2\pmod{4}.
combinatorics
All Russian MO 2015 11 th grade, ineq.

Source:

5/23/2015
Let a,b,c,d be real numbers satisfying ∣a∣,∣b∣,∣c∣,∣d∣>1|a|,|b|,|c|,|d|>1 and abc+abd+acd+bcd+a+b+c+d=0abc+abd+acd+bcd+a+b+c+d=0. Prove that 1a−1+1b−1+1c−1+1d−1>0\frac {1} {a-1}+\frac {1} {b-1}+ \frac {1} {c-1}+ \frac {1} {d-1} >0
inequalities
Consecutive Numbers in Equilateral Triangle

Source: India IMO Training Camp 2016, Practice 2, Problem 3

7/22/2016
An equilateral triangle with side length 33 is divided into 99 congruent triangular cells as shown in the figure below. Initially all the cells contain 00. A move consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells by 11 simultaneously. Determine all positive integers nn such that after performing several such moves one can obtain 99 consecutive numbers n,(n+1),⋯ ,(n+8)n,(n+1),\cdots ,(n+8) in some order. [asy] size(3cm); pair A=(0,0),D=(1,0),B,C,E,F,G,H,I; G=rotate(60,A)*D; B=(1/3)*D; C=2*B;I=(1/3)*G;H=2*I;E=C+I-A;F=H+B-A; draw(A--D--G--A^^B--F--H--C--E--I--B,black);[/asy]
combinatorics
Partition of N into two subsets

Source: India TST 2016 Day 4 Problem 3

7/22/2016
Let N\mathbb N denote the set of all natural numbers. Show that there exists two nonempty subsets AA and BB of N\mathbb N such that
[*] A∩B={1};A\cap B=\{1\}; [*] every number in N\mathbb N can be expressed as the product of a number in AA and a number in BB; [*] each prime number is a divisor of some number in AA and also some number in BB; [*] one of the sets AA and BB has the following property: if the numbers in this set are written as x1<x2<x3<⋯x_1<x_2<x_3<\cdots, then for any given positive integer MM there exists k∈Nk\in \mathbb N such that xk+1−xk≥Mx_{k+1}-x_k\ge M. [*] Each set has infinitely many composite numbers.
number theorysetprime numbers
There is a tri-blued unit square

Source: India TST 2016 Day 3 Problem 3

7/22/2016
Let nn be an odd natural number. We consider an n×nn\times n grid which is made up of n2n^2 unit squares and 2n(n+1)2n(n+1) edges. We colour each of these edges either <spanclass=′latex−italic′>red</span>\color{red} <span class='latex-italic'>red</span> or <spanclass=′latex−italic′>blue</span>\color{blue}<span class='latex-italic'>blue</span>. If there are at most n2n^2 <spanclass=′latex−italic′>red</span>\color{red} <span class='latex-italic'>red</span> edges, then show that there exists a unit square at least three of whose edges are <spanclass=′latex−italic′>blue</span>\color{blue}<span class='latex-italic'>blue</span>.
combinatoricssquare grid