This one is the classic jumping frog with state diagrams, this time in the coordinate plane between a fence and river.
Find the least positive integer such that is a product of at least four not necessarily distinct primes.
A strictly increasing sequence of positive integers has the property that for every positive integer , the subsequence is geometric and the subsequence is arithmetic. Suppose that . Find .
A fair coin is tossed repeatedly until either heads comes up three times in a row or tails comes up three times in a row. What is the probability that the coin will be tossed more than times? Express your answer as a common fraction.
Consider the equilateral triangle with sides of length cm. A point in the interior of is said to be “special” if it is a distance of cm from one side of the triangle and a distance of cm from another side. Consider the convex polygon whose vertices consist of the special points. What is the area of this polygon? Express your answer as a decimal to the nearest tenth.
The three coin denominations used in Coinistan have values 7 cents, 12 cents and 23 cents. Fareed and Krzysztof notice that their two Coinistan coin collections have the same number of coins and the same total value, but not the same number of 7-cent coins. What is the smallest possible value of Krzysztof’s collection?
How many four-digit integers , with , have the property that the three two-digit integers form an increasing arithmetic sequence? One such number is , where and .
Four lines are drawn through the figure shown (see problem statement in video). What is the maximum number of non-overlapping regions created inside the figure?
Gretchen labels each of the six faces of a cube with a distinct positive integer so that for each vertex of the cube, the product of the three numbers on the faces touching the vertex is a perfect square. What is the least possible value of the sum of the numbers on this cube?
I track the scores of my students who are prepping for various math contests. Here is an example of student who is hoping to qualify for AIME next year. My method is to assign an old AMC 10 as homework, and the student takes it under timed exam conditions. The student grades her exam at home, and then spends as much time as she likes to solve any remaining problems. We meet to discuss the problems that could not be solved, and I explain the solutions and offer quick proofs and explanations of underlying theorems.
There isn’t a lot of strategy or overthinking here. Old AMCs are freely available online, with 2 per year, there is no end of practice tests. Take an exam, study what you could not solve, take another exam, lather, rinse, repeat.
In just a few months this student has raised her score to very close to the AIME floor most years. Notice she’s more than doubled the number of correctly solved problems.
There’s no magic or secret way to prepare. Time spent researching books and classes is better spent taking as many exams as possible and most importantly, studying the problems you couldn’t solve. Good luck!