Problems

Jane wrote a number on the board. Then, she looked at it and she noticed it lacks her favourite digit: $5$. So she wrote $5$ at the end of it. She then realized the new number is larger than the original one by exactly $1661$. What is the number written on the board?

Find the biggest 6-digit number such that each of its digits, except for the last two, is equal to the sum of its two right neighbours.

Replace the letters with digits in a way that makes the following sum as big as possible: $$SEND+MORE+MONEY.$$

Jane wrote another number on the board. This time it was a two-digit number and again it did not include digit $5$. Jane then decided to include it, but the number was written too close to the edge, so she decided to fit the $5$ in between the two digits. She noticed that the resulting number is $11$ times larger than the original. What is the sum of digits of the new number?

a) Find the biggest 6-digit integer number such that each digit, except for the two on the left, is equal to the sum of its two left neighbours.

b) Find the biggest integer number such that each digit, except for the first two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the “6-digit number” restriction.)

If $R+RR=BOW$, what is the last digit of the number below? $$F\times A\times I\times N\times T\times I\times N\times G.$$

Shmerlin the magician found the door to the Cave of Wisdom. The door is guarded by Drago the Math Dragon, and also locked with a 4-digit lock. Drago agrees to put Shmerlin to the test: Shmerlin has to choose four integer numbers: $x,y,z$ and $w$, and the dragon will tell him the value of $A\times x+B\times y+C\times z+D\times w$, where $A,B,C$ and $D$ are the four secret digits that open the lock. After that, Shmerlin should come up with a guess of the secret digits. If the guess is correct, Drago will let the magician into the cave. Otherwise, Shmerlin will perish. Does Shmerlin have a way to succeed?

Solve: $HE\times HE=SHE$.

Shmerlin managed to enter the cave and explore it. On his way back, he was once again stopped by Drago. He learns that the door out of the cave is locked again, this time with a more powerful lock. The key required to open it now includes four positive integers, which are no longer digits – they can be much larger. Shmerlin once again can choose four integer numbers: $x,y,z$ and $w$, and the dragon will tell him the value of $A\times x+B\times y+C\times z+D\times w$, where $A,B,C$ and $D$ are the four secret integer numbers that open the lock. Because the lock is much more difficult to crack now, Drago agrees to let Shmerlin try twice. He can choose his four integer numbers and then, basing on what he learns from the dragon, choose again. Will he be able to leave the cave or is he doomed to stay inside forever?

Integer numbers $a,b$ and $c$ are such that the sum of digits of a number $a+b$ is less than $5$, the sum of digits of a number $b+c$ is less than $5$, the sum of digits of a number $a+c$ is less than $5$, but the sum of digits of a number $a+b+c$ is greater than $50$. Can you find such three numbers $a,b$ and $c$?