This is an initially mind-boggling problem from the 1995 American Invitational Mathematics Exam (AIME).
“Find the last three digits of the product of the positive roots of
”
See Log Lunacy for solution.
Tag Archives: algebra
Fireworks Rocket
This is another physics-based problem from Colin Hughes’s Maths Challenge website (mathschallenge.net) that may take a bit more thought.
“A firework rocket is fired vertically upwards with a constant acceleration of 4 m/s2 until the chemical fuel expires. Its ascent is then slowed by gravity until it reaches a maximum height of 138 metres.
Assuming no air resistance and taking g = 9.8 m/s2, how long does it take to reach its maximum height?”
I can never remember the formulas relating acceleration, velocity, and distance, so I always derive them via integration.
See the Fireworks Rocket for solutions.
Falling Sound Problem
This math problem from Colin Hughes’s Maths Challenge website (mathschallenge.net) hearkens back to basic physics.
“A boy drops a stone down a well and hears the splash from the bottom after three seconds. Given that sound travels at a constant speed of 300 m/s and the acceleration of the stone due to gravity is 10 m/s2, how deep is the well?”
See the Falling Sound Problem for solutions.
A Divine Language
I have just finished reading a most remarkable book by Alec Wilkinson, called A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age. I had read an essay of his in the New Yorker that turned out to be essentially excerpts from the book. I was so impressed with his descriptions of mathematics and intrigued by the premise of a mature adult in his 60s revisiting the nightmare of his high school experience with mathematics that I was eager to see if the book was as good as the essay. It was, and more.
The book is difficult to categorize—it is not primarily a history of mathematics, as suggested by Amazon. But it is fascinating on several levels. There is the issue of a mature perspective revisiting a period of one’s youth; the challenges of teaching a novice mathematics, especially a novice who has a strong antagonism for the subject; and insights into why someone would want to learn a subject that can be of no “use” to them in life, especially their later years.
Wilkinson has a strong philosophical urge; he wanted to understand the role of mathematics in human knowledge and the perspective it brought to life. He was constantly asking the big questions: is mathematics discovered or invented, what is the balance between nature and nurture, why does mathematics seem to describe the world so well, what is the link between memorization and understanding, how do you come to understand anything?
15 Degree Triangle Puzzle
This math problem from Colin Hughes’s Maths Challenge website (mathschallenge.net) is a bit more challenging.
“In the diagram, AB represents the diameter, C lies on the circumference of the circle, and you are given that
(Area of Circle) / (Area of Triangle) = 2π.
Prove that the two smaller angles in the triangle are exactly 15° and 75° respectively.”
See the 15 Degree Triangle Puzzle
Log Jam
Here is a tricky little logarithm problem from the 2021 Math Calendar.
Two Candles
This is another candle burning problem, presented by Presh Talwalkar.
“Two candles of equal heights but different thicknesses are lit. The first burns off in 8 hours and the second in 10 hours. How long after lighting, in hours, will the first candle be half the height of the second candle? The candles are lit simultaneously and each burns at a constant linear rate.”
See Two Candles for solutions.
Remainder Problem
Here is a challenging problem from the 2021 Math Calendar.
“Find the remainder from dividing the polynomial
x20 + x15 + x10 + x5 + x + 1
by the polynomial
x4 + x3 + x2 + x + 1”
Recall that all the answers are integer days of the month.
See the Remainder Problem for a solution.
Hard Time Conundrum
This problem comes from the “Problems Drive” section of the Eureka magazine published in 1955 by the Archimedeans at Cambridge University, England. (“The problems drive is a competition conducted annually by the Archimedeans. Competitors work in pairs and are allowed five minutes per question ….”)
“There are ten times as many seconds remaining in the hour as there are minutes remaining in the day. There are half as many minutes remaining in the day as there will be hours remaining in the week at the end of the day. What time is it on what day?”
One of the hardest parts of the problem is just being able to translate the statements into mathematical terms. Solvable in 5 minutes?!!!
See the Hard Time Conundrum for a solution.
Polynomial Division Problem
Here is a challenging problem from the Polish Mathematical Olympiads published in 1960.
“22. Prove that the polynomial
x44 + x33 + x22 + x11 + 1
is divisible by the polynomial
x4 + x3 + x2 + x + 1.”
See the Polynomial Division Problem
(Update 8/23/2021) The idea expressed in this post that mathematicians are “lazy” and seek short-cuts to solving questions and problems, as I did in this one, was recently the subject of a Numberphile post by Marcus du Sautoy: “Mathematics is all about SHORTCUTS“.