**Bangladesh National Mathematical Olympiad 2012: Primary**

**Problem 1:**

Find a three digit number so that when its digits are arranged in reverse order and added with the original number, the result is a three digit number with all of its digits being equal. In case of two digit numbers, here is an example: $23+32=55 $

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**Problem 2:**

Subrata writes a letter to Ruponti every day (in successive intervals of $24$ hours) from Korea. But Ruponti receives the letters in intervals of $25$ hours. What is the number of the letter Ruponti receives on the $25^{th}$ day?

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**Problem 3:**

Prove that, the difference between two prime numbers larger than $2$ can’t be a prime number other than $2$.

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**Problem 4:**

Write a number in a paper and hold the paper upside down. If what you get is exactly same as the number before rotation then that number is called

*beautiful*. Example: $986$ is a

*beautiful*number. Find out the largest $5$ digit

*beautiful*number.

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**Problem 5:**

If a number is multiplied with itself thrice, the resultant is called its cube. For example: $3 × 3 × 3 = 27$, hence $27$ is the cube of $3$. If $1,\ 170$ and $387$ are added with a positive integer, cubes of three consecutive integers are obtained. What are those three consecutive integers?

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**Problem 6:**

Consider the given diagram. There are three rectangles shown here. Their lengths are $3,\ 4$ and $5$ units respectively, widths respectively $2,\ 3$ and $4$ units. Each small grid represents a square $1$ unit long and $1$ unit wide. Use these diagrams to find out the sum of the consecutive numbers from $1$ to $500$. (If you use some direct formula for doing so, you must provide its proof) viewtopic.php?f=13&t=1734

**Problem 7:**

When Tanvir climbed the Tajingdong mountain, on his way to the top he saw it was raining $11$ times. At Tajindong, on a rainy day, it rains either in the morning or in the afternoon; but it never rains twice in the same day. On his way, Tanvir spent $16$ mornings and $13$ afternoons without rain. How many days did it take for Tanvir to climb the Tajindong mountain in total?

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**Problem 8:**

A magic box takes two numbers. If we can obtain the first numbers by multiplying the second number with itself several times then a green light on the box turns on. Otherwise, a red light turns on. For example, if you enter $16$ and $2$ then the green light turns on because $2×2×2×2 = 16$. But if you enter $18$ and $9$ then the red light turns on. If the two numbers are equal then the green light turns on. If the first number is $256$ then for how many different second numbers will the green light turn on?

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**Problem 9:**

Each room of the Magic Castle has exactly one door. The rooms are designed such that when you can go from one room to the next one through a door, the second room's length is equal to the first room's width, and the second room's width is half of the first room's width (see the figure). Each door can be used only once. Magic Prince has entered the castle and now needs to get out. To get out of each room, the prince needs time equal to the width of the room. The prince has to use each door to get out of the castle. Due to the blessings of a Sufi, the prince can become as small as he wants (so that he can go into even very small rooms). If the castle is a square of side length $20$ meters then how long will it take for the prince to get out? viewtopic.php?f=13&t=1727

**Problem 10:**

Tusher chose some consecutive numbers starting from $1$. He noticed that the least common multiple of those numbers is divisible by $100$. What is the minimum number of numbers he chose?

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