Tag Archives: algebra

Language Students Problem

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This is a problem from the 2001 American Invitational Mathematics Exam (AIME).

“Each of the 2000 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let m be the smallest number of students who could study both languages, and let M be the largest number of students who could study both languages. Find M – m.”

Answer.

See Language Students Problem for a solution.

Smart Money

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This is a simple puzzle from Futility Closet.

“Mr. Smith goes to Atlantic City to gamble for a weekend. To guard against bad luck, he sets a policy at the start: In every game he plays, he’ll bet exactly half the money he has at the time, and he’ll make all his bets at even odds, so he’ll have an equal chance of winning and of losing this amount. In the end he wins the same number of games that he loses. Does he break even?”

Answer.

See Smart Money for solutions.

Mixed Emotions

This is a brainteaser by S. Ageyev from the November-December 1991 issue of Quantum given in Futility Closet.

“The numbers 1, 2, …, 100 are arranged in a 10 x 10 square table in their natural order (1 in the top left comer, 100 in the bottom right comer). The signs of 50 of these numbers are changed in such a way that exactly half of the numbers in each line and each column get the minus sign. Prove that the sum of all the numbers in the table after this change is zero.”

See Mixed Emotions for solutions.

Hangover Clock Reading

This is another clock puzzle from the 1978 Eureka magazine.

“The hands on my alarm clock are indistinguishable, and there are no numbers around the outside. Accidentally woken up by it one morning, I observed with a snarl that the hands were both pointing at minute divisions, and that they were 9 minutes apart.

Had it not been for my hangover, what could I have deduced?”

Answer.

See the Hangover Clock Reading for a solution.

Functional Equation Puzzle

This is a math Olympiad problem from Puzzle Sphere where Muhammad Zain Sarwar claims it is at Harvard entrance exam level.

“Given the functional relationship  f(x + y) = f(x) + f(y) + xy with the known value  f(4) = 10, determine the value of f(2023).”

Just try some examples and detect the pattern that defines the function.

Answer.

See the Functional Equation Puzzle for a solution.

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