Algebra
Algebra is a broad branch of mathematics that uses symbols (usually letters) to represent numbers and to state rules and relationships. It involves manipulating expressions, solving equations and inequalities, and studying functions and structures. Questions cover a wide range of these topics.
Algebraic Techniques Equations Inequalities Word Problems Sequences-
Question
Calculate without a calculator using the most efficient method:
\( { (2001 \cdot 2021+100) \cdot (1991 \cdot 2031+400) \over 2011^4}\)
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Question
Which of these two numbers is larger:
`A=2011*20122012*201320132013 ` or `B=2013*20112011*201220122012` ?
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Question
Answer without using pen and paper (and without a calculator, of course
):Which number is larger: `12345678^2` or `12345677*12345679` ?
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Movie Buffs
Avi, Beni, and Gili love movies. Avi goes to the movies every `3` days, Beni – every `5` days, and Gili goes every `7` days. Today, they all went to the movies together. In how many days can this happen again?
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Question
The "Sweet Math" candies are sold in boxes of `12` units, and the "Geometry with Nuts" candies – in boxes of `15` units.
What is the minimum number of boxes that must be purchased so that there are equal quantities of candies of both types?
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Question
Shmuel's salary increased by `20%`, and then decreased by `20%`. Is Shmuel's current salary equal to his initial salary?
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Question
I went to the market to buy vegetables. I paid for tomatoes half of the money I had plus one shekel. After that, I paid for cucumbers half of the remaining money plus one shekel, and then I paid for onions half of the remaining money plus one shekel, and that used up all the money I had.
What is the amount of money I came to the market with?
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Question
The following numbers are written on the board:
`1,2^1,2^2,2^3,2^4,2^5`
In one operation, you are allowed to erase two numbers written on the board and write their (non-negative) difference in their place.
Is it possible to reach a state, through such operations, where only the number `15` is written on the board?
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Question
The sum of several numbers is equal to `1`. Is it possible that the sum of their squares is less than one-tenth?
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The Sophisticated Task
Hannah has a basket with `13` apples. Hannah wants to know the total weight of all these apples. Rachel has a digital scale, and she is willing to help Hannah, but only under the following conditions: In each weighing, Hannah can weigh exactly `2` apples, and the number of weighings cannot exceed `8`.
Explain how, under these conditions, Hannah can know the total weight of the apples.
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