Two three-digit numbers are made up of six different digits. The first digit of the second number is twice as big as the last digit of the first number. (Note: 0 is also a digit but cannot be the first digit of a number!) How big is the smallest possible sum of the two numbers?

Mathematical Circle

Math Kangaroo (Benjamin) 2016 | Problem 24

537

Let us assume these three digit numbers are $ABC$, $DEF$.

According to the question $D=2C$.

Let's follow the given condition and try to construct the smallest numbers.

So here $ABC=102$.

And if I follow the given condition then $DEF= 435$.

We did this keeping in mind that repetitions are not allowed.

Now calculate the answer.

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Luigi owns a few square tables and some chairs for his little restaurant. If he sets out his tables individually with 4 chairs each, then he is 6 chairs short. If he always puts two tables together to create a bigger table with 6 chairs, then he has 4 chairs left over.
How many tables does Luigi have?

Mathematical Circle

Math Kangaroo (Benjamin) 2016 | Problem 20

10

Let us assume the number of tables and chairs are $x, y$ respectively.

Let's follow the given condition and construct the equations.

For the first case if he set table individually with 4 chairs each then he is 6 chairs short.

So, $4x-6=y$.

Now if he put two tables together with 6 chairs each, then he has 4 chairs left over.

So, $6\frac{x}{2}+4=y$.

Comparing the equation of hint 3 and hint 4 we get,

$4x-6= 3x+4$

So, $ x=10$

Hence the answer is $10$.

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