1. What is the minimum difference between a $12$ digit odd number and a $11$ digit even number?

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2. In a box there are $420$ balls of $20$ different colors. What is the minimum number of balls one need to pick so that he can be sure of having at least two balls of same colors?

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3. $A$ is three digit number all of whose digits are different. $B$ is a three digit number all of whose digits are same. Find the minimum difference between $A$ and $B$.

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4. $A$ is the product of seven odd prime numbers. $A\times B$ is a perfect even square. What is the minimum number of prime factors of $B$?

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5. A

*palindrome*, such as $83438$, is a number that remains the same when its digits are reversed. How many four digit numbers are not palindromes?

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6. Each of Rinku, Tiya and Abir are asked to choose three numbers so that the sum of the third and the first number is twice the second one and the third number is $2$ greater than the second number. It was observed that the second number of Rinku is $2$ smaller than the second number of Tiya and the second number of Tiya is $2$ smaller than the second number of Abir. If the third number chosen by Tiya is $15$, find the first number chosen by Abir.

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7. A four headed monster has 63 children. He wants to give a different name to each of his children. The names will consist of 3 English letters and one letter can be used more than once in a single name. What is the minimum number of letters the monster must use?

Similar: viewtopic.php?f=41&t=478

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8. Abir started at $9:30$ pm from his house to go to Nayel’s house. When he reached there, he saw it was $8:55$ pm in Nayel’s clock. He started back towards his house when it was $9:00$ pm at Nayel’s. He reached at his house when it was $10:15$ pm in his clock. If Abir needed equal time to go and come back, find out the number of minutes Nayel’s clock is slow by.

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9. $ABC$ is a triangle with $AB = 4$, $BC = 5$ and $AC = 3$. $O$ is the midpoint of $BC$. A line from $O$ parallel to $AC$ meets $AB$ at $D$. $E$ is on $DO$ so that $DO = OE$ and $D$ and $E$ are on opposite sides of $BC$. Find $BE^2$.

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10. The diagram above shows the various paths along which Mr. Baizid Bhuiyan Juwel can travel from point Teknaf, where he is released, to point Tetulia, where he is rewarded with a food pellet. How many different paths from Teknaf to Tetulia can Juwel take if it goes directly from Teknaf to Tetulia without retracting any point along a path?

(Unfortunately this was a "known" problem. This appeared in a previous BdMO. )