MATHCOUNTS 2022 Exercises – 2

Posted on December 3, 2021 by kevin
  1. An ant is traveling along the side of a regular dodecagon with side length as $1$. What is the expected value of minimum distance the ant travels between two different vertices randomly chosen? Express your answer as a common fraction in simplest form.
  2. A best-of-five series ends when one team wins three games. The probability of team A defeating team B in any game is $\frac{1}{5}$. The probability that team A will win the series can be expressed as a fraction in lowest terms as $\frac{m}{n}$. Find $m+n$.
  3. A positive integer less than 100 is randomly chosen. The probability that at least one of its digits is 4 or that the number is divisible by 4 can be expressed as a common fraction as $\frac{a}{b}$. Find $a+b$.
  4. A pair of dice is rolled until the sum of the two numbers obtained is a multiple of $4$. What is the probability that a multiple of $4$ is obtained at the $6th$ toss and not before? Express you answer as a decimal to the nearest hundredth.
  5. A regular dodecagon is inscribed in a unit circle. What is the expected value of the distance between two different vertices of the dodecagon? Express you answer as a decimal to the nearest hundredth.
  6. On a four-by-four grid, the points are $1$ unit apart horizontally and vertically. Two distinct points are randomly selected from this grid. What is the probability that the distance between those two points is less than $3.5$? Express your answer as a common fraction in simplest form.
  7. What is the arithmetic mean of all possible 5-digit numbers which use the digits 1, 2, 3, 5 and 9?
  8. Two standard dice are rolled. What is the probability that the product of the two numbers shown on the top of the dices exceeds $9$? Express your answer as a common fraction in simplest form.
  9. A point E is randomly selected inside a unit square $ABCD$. What is the probability that $\angle{AEB}=90^\circ$?
  10. A point E is randomly selected inside a unit square $ABCD$. What is the probability that the distance from $E$ to each of $A$, $B$, $C$ and $D$ is greater than $\frac{1}{2}$, and the distance from $E$ to each of $AB$, $BC$, $CD$ and $DA$ is greater than $\frac{1}{4}$?
Answers
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