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-
An Ancient Russian Riddle
Two woodcutters, Ivan and Prokhor, were working together in the forest. They settled down to eat lunch. Ivan had `4` pitas, and Prokhor had `8`. At that moment, a hunter passed by them and asked to join the meal, so they divided the `12` pitas equally among the three of them. After the meal, the hunter really wanted to thank the woodcutters and gave them `15` kopecks. After he left, Ivan and Prokhor began to discuss how to divide the money.
Prokhor suggested simply dividing it equally. But Ivan argued that this would not be fair, because he had twice as few pitas, therefore, in his opinion, Prokhor should receive `10` kopecks and Ivan himself - `5`. How do you think the money should be divided fairly?
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Question
Three travelers need to cross a river. They have a boat that can hold a load of up to `100` kilograms. The travelers weigh `45`, `50`, and `80` kilograms. How should the travelers proceed to cross to the other side?
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Question
In the magical land, there are only two types of coins: `16` LC (Magical Pounds) and `27` LC. Is it possible to buy a notebook that costs one Magical Pound and receive exact change?
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Question
The numbers `1`, `2`, `3`, ..., `9` are divided into `3` sets. Prove that there is a set where the product of the numbers is greater than or equal to `72`.
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Question
In the following arithmetic puzzle, different digits have been replaced by different letters, and identical digits – by identical letters. Reconstruct the puzzle:
`BAOxxBAxxB=2002`
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Question
The company "The Diligent Builder" is engaged in stockpiling trees in a magical forest in Canada. A nature protection association called "The Green Avenger" wants to protect the forest and opposes the company's activity. As a result, the company's CEO said the following sentence:
"`99%` of the trees in the forest are maple trees. In the coming year, we are going to cut down only maple trees, and as a result, the percentage of maple trees in the forest in a year will become `98%`."
What percentage of the trees in the forest do the Diligent Builders intend to cut down?
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Where Did the Extra Thaler Come From?
A cobbler named Karl made boots and sent his young son Hans to the market to sell them for `25` thalers. When Hans arrived at the market, two disabled men approached him, one without a left leg, the other without a right leg, and asked to buy one boot each. Hans agreed and sold each of them a boot for `12.5` thalers.
When Hans returned home and told his father what had happened, Karl decided that these people should have been sold boots at a lower price - `10` thalers per boot. So he gave Hans `5` thalers and asked him to return `2.5` thalers to each of them.
On his way to the market, Hans saw a sweets stall, couldn't resist, and spent `3` thalers of what his father had given him there. After that, he found the two disabled men and gave each of them one thaler, because that's all he had left. When Hans returned home, he regretted what he had done and told his father everything. The cobbler Karl was very angry and locked his son in the pantry as punishment.
Thus, Hans sits in the pantry and analyzes what happened that day: "I returned one thaler to each of the disabled men, which means that each of them ultimately paid `12.5-1=11.5` thalers for his boot. So in total they paid `11.5*2=23` thalers. And I spent three thalers on sweets. That's a total of `26` thalers, but there were `25`! Where did one more thaler come from?"
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Camel Division (Ancient Question)
An old Arab merchant had three sons. He bequeathed them 17 camels, and in his will, he requested that the eldest son receive half of the camels, the middle son receive a third, and the youngest a ninth. The sons could not divide the camels among themselves as stated in the will without slaughtering some of the camels – and they did not want to do that. So they turned to the Qadi for help.
The Qadi added one of his own camels to the 17 camels, and divided the 18 camels as follows: the eldest son received 9 camels, which is half of the amount, the middle son received 6 camels, which is a third of the amount, and the youngest son received 2 camels, which is a ninth of the amount, for a total of 17 camels divided, and the extra camel was returned to the Qadi.
The brothers were amazed by the wisdom of the Qadi and began to think: how did it happen that each received even more than he was supposed to receive according to the will?
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Question
Given the sequence `1 , 1/2 ,1/3 ,1/4 ,1/5,...`, does there exist an arithmetic sequence composed of terms from the aforementioned sequence?
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Of length 5
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Of any length
Sources:Topics:Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Arithmetic -> Fractions Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) -
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Question
Positive numbers a, b, c, d satisfy `a^3 + b^3 +c ^3 + d^3 >= 3` and also `a^5 + b^5 +c ^5 + d^5 <= 5`
Prove that `a + b +c + d >= 3 / 2`
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