Let $S$ be the set of all $5^6$ positive integers, whose decimal representation consists of exactly 6 odd digits. Find the number of solutions $(x,y,z)$ of the equation $x+y=10z$, where $x\in S$, $y\in S$, $z\in S$.
Problem
Source: VII Caucasus Mathematical Olympiad
Tags: combinatorics
02.05.2022 13:22
Who can send solution of this problem
06.05.2022 20:42
Chevrolet23 wrote: Who can send solution of this problem İ dont have any idea
06.05.2022 21:44
Let $x=x_1x_2x_3x_4x_5x_6$ where $x_i$ are digits We have $x_6+y_6=10$ $x_5+y_5 \geq 10$ because if not than $z_5=x_4+y_4$ is even $x_4+y_4 \geq 10$ $x_3+y_3 \geq 10$ $x_2+y_2 \geq 10$ $x_1+y_1 \geq 10$ $x_i+y_i=10$ have $5$ solutions and $x_i+y_i \geq 10$ has $15$ solutions So total we have $5*15^5$ solutions.
04.06.2022 09:22
RagvaloD wrote: Let $x=x_1x_2x_3x_4x_5x_6$ where $x_i$ are digits We have $x_6+y_6=10$ $x_5+y_5 \geq 10$ because if not than $z_5=x_4+y_4$ is even $x_4+y_4 \geq 10$ $x_3+y_3 \geq 10$ $x_2+y_2 \geq 10$ $x_1+y_1 \geq 10$ $x_i+y_i=10$ have $5$ solutions and $x_i+y_i \geq 10$ has $15$ solutions So total we have $5*15^5$ solutions. Hi,good solution but i think that x1+y1 can take any value,what about you?
01.03.2024 19:56
Telman wrote: RagvaloD wrote: Let $x=x_1x_2x_3x_4x_5x_6$ where $x_i$ are digits We have $x_6+y_6=10$ $x_5+y_5 \geq 10$ because if not than $z_5=x_4+y_4$ is even $x_4+y_4 \geq 10$ $x_3+y_3 \geq 10$ $x_2+y_2 \geq 10$ $x_1+y_1 \geq 10$ $x_i+y_i=10$ have $5$ solutions and $x_i+y_i \geq 10$ has $15$ solutions So total we have $5*15^5$ solutions. Hi,good solution but i think that x1+y1 can take any value,what about you? Hi, if $x_1+y_1$ less than 10, x+y can't be 7 digit number but 10z is 7 digit number