**Bangladesh National Mathematical Olympiad 2011: Primary**

**Problem 1:**

Find the value of \[ 1+\frac{1}{1+\dfrac{1}{1+1}} \]

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**Problem 2:**

$2, 11$ and $2011$ are prime numbers. What will be the GCD of $2\times 2011$ and $11 \times 2011$?

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**Problem 3:**

There is a basket in front of you that has $100$ balls numbered from $1$ to $100$ (each ball has a different number). You are asked to pick balls from this basket randomly without looking. After picking how many balls can you be sure that you have chosen at least one even-numbered ball?

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**Problem 4:**

To form a triangle, the sum of lengths of any two sides has to be greater than the length of the other. Suppose that we want to form a triangle with sides $a, 31$ and $a + 1$ where a is an integer (or whole number) greater than $1$. Find the minimum value of $a$.

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**Problem 5:**

We say that a number has a 'square root' if we find another integer which gives us the first number if multiplied by itself. For example, $4$ has a square root since , but $5$ has no square root. Out of the $100$ numbers from $1$ to $100$, how many integers have square roots?

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**Problem 6:**

In the diagram, the same square has been represented in two ways. If $a = 5$ and $b = 12$ in this diagram, find the value of $c$. (the white rectangle in the second diagram is a square)

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**Problem 7:**

You can very easily check that sum of three consecutive integers is three times the second number. Similarly, sum of five consecutive numbers is five times the third number, sum of seven consecutive numbers is seven times the fourth number and so on. What is the smallest even number that can be expressed as sum of three consecutive integers, sum of five consecutive integers and also sum of eleven consecutive integers?

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**Problem 8:**

If $3$ and $4$ individually divides a number, then also divides that number. For example, all of $3, 4$ and $12$ divide $48$. But if a number is divisible by $3$ and $6$ individually it may or may not be divisible by $3 \times 6=18$ . For example, both of $54$ and $60$ are divisible by $3$ and $6$ individually, though only $54$ is divisible by $18$. Can you explain why this happens?

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**Problem 9:**

In the first round of a chess tournament, each player plays against every other player exactly once. A player gets $3, 1$ or $-1$ points respectively for winning, drawing or losing a match. After the end of the first round, it is found that the sum of the scores of all the players is $20$. How many players were there in the tournament?

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**Problem 10:**

$A, B, C, D, E, F$ are six children of different ages in the range of 11 to 16. It is known that C and F always speak truth whereas among the rest two are truthful and the other two lie. When they are asked about their ages, they replied as follows-

A: The sum of the ages of the other five is an even number.

B: A is the eldest.

C: The sum of the ages of the other five is divisible by 5.

D: E is elder than A by two years.

E: The sum of the ages of A, B, D, E is an odd number.

F: The sum of the ages of the other five is divisible by 5.

Find out: (i) What is the sum of the ages of C and F? (ii) Which two among these six children lie? (iii) What are the ages of A and E?

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