{"id":2346,"date":"2021-12-03T17:41:00","date_gmt":"2021-12-03T21:41:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=2346"},"modified":"2024-10-25T11:26:24","modified_gmt":"2024-10-25T15:26:24","slug":"mathcounts-2022-exercises-2","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=2346","title":{"rendered":"MATHCOUNTS 2022 Exercises – 2"},"content":{"rendered":"\n
  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. <\/span><\/li>
  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 $m+n$. <\/span><\/li>
  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 $a+b$. <\/span><\/li>
  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.<\/span><\/li>
  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.<\/span><\/li>
  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. <\/span><\/li>
  7. What is the arithmetic mean of all possible 5-digit numbers which use the digits 1, 2, 3, 5 and 9? <\/span><\/li>
  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. <\/span><\/li>
  9. A point E is randomly selected inside a unit square $ABCD$. What is the probability that $\\angle{AEB}=90^\\circ$?<\/span><\/li>
  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 $\\frac{1}{4}$?<\/span><\/li><\/ol>\n\n\n\nAnswers<\/a>
    \n\n\n\n
    1. $\\dfrac{36}{11}$<\/li>
    2. 3306<\/li>
    3. $\\dfrac{4}{11}$<\/li>
    4. 0.06<\/li>
    5. 1.38<\/li>
    6. $\\dfrac{11}{12}$<\/li>
    7. 44444<\/li>
    8. $\\dfrac{19}{36}$<\/li>
    9. 0<\/li>
    10. $\\dfrac{\\sqrt{3}}{4}-\\dfrac{\\pi}{12}$<\/li><\/ol>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"

      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 … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[18,11,8],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2346"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2346"}],"version-history":[{"count":20,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2346\/revisions"}],"predecessor-version":[{"id":4607,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2346\/revisions\/4607"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2346"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2346"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2346"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}