MATHCOUNTS 2022 Exercises – 4

1. Given $n>1$, what is the smallest positive integer $n$ whose positive divisors have a product of $n^6$?
2. What is the largest integer value of $n$ for which $8^n$ evenly divides $100!$?
3. A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a prime number. How many $3$-primable positive integers are there that are less than 1000?
4. How many whole numbers n, such that $100\le n \le1000$, have the same number of odd factors as even factors?
5. A right triangle has a hypotenuse of $10m$ and a perimeter of $22m$. In square meters, what is the area of the triangle?
6. Circle $O$ has radius $10$ units. Point $P$ is on radius $OQ$ and $OP=6$ units. How many chords containing $P$, including the diameter, have integer lengths?
7. What is the total surface area of the largest regular tetrahedron that can be inscribed inside of a cube of edge length $1 cm$. Express your answer in simplest radical form.
8. The diameter, in inches, of a sphere with twice the volume of a spbere of radius 9 inches can be expressed in the form of $a\sqrt[3]{b}$, where $a$ and $b$ are positive integers and $b$ contains no perfect cube factors. Compute $a+b$.
9. A circular garden is surrounded by a sidewalk with a uniform width of $25$ foot. The total area of the sidewalk equals the total area of the garden. How many feet are in the diameter of the garden? Round your answer to the nearest whole number.
10. Pedro stood at the center of a circular field that had a radius of 120 feet. He walked due north halfway to the circle. He then turned and walked due east halfway to the circle. He turned again and walked due south hallway to the circle. Finally he turned and walked due west halfway to the circle. When he stopped, how many feet was Pedro from the center of the circle? Express your answer to the nearest foot.

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