MathDB

5

Part of 1995 All-Russian Olympiad

Problems(3)

three numbers with the same digits

Source: All-Russian olympiad 1995, Grade 9, Second Day, Problem 5

10/21/2013
We call natural numbers similar if they are written with the same (decimal) digits. For example, 112, 121, 211 are similar numbers having the digits 1, 1, 2. Show that there exist three similar 1995-digit numbers with no zero digits, such that the sum of two of them equals the third. S. Dvoryaninov
number theory proposednumber theory
GCD

Source:

6/30/2012
The sequence a1,a2,...a_1, a_2, ... of natural numbers satisfies GCD(ai,aj)=GCD(i,j)GCD(a_i, a_j)=GCD(i, j) for all i≠ji \neq j. Prove that ai=ia_i=i for all ii.
number theorygreatest common divisor
a^2_1+a^_22+...+a^2_k is divisible by a_1+a_2+...+a_k

Source: All-Russian olympiad 1995, Grade 11, Second Day, Problem 5

10/21/2013
Prove that for every natural number a1>1a_1>1 there exists an increasing sequence of natural numbers ana_n such that a12+a22+⋯+ak2a^2_1+a^2_2+\cdots+a^2_k is divisible by a1+a2+⋯+aka_1+a_2+\cdots+a_k for all k≥1k \geq 1.
A. Golovanov
algebradifference of squaresspecial factorizationsnumber theory proposednumber theory