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    Mock Aiome

    Uploaded by

    Sancho Pancho
    1. The document provides instructions for submitting responses to a mock AIME problem set. Students should submit their 3-digit answers to a Google Form and include leading zeros even though they are not strictly necessary. 2. The problem set was written by a group of 5 individuals and edited by the same group. It contains 15 multiple choice problems requiring computational reasoning and mathematical skills. 3. The final problem can be expressed as an equation relating positive integers a, b, and c, and students are asked to compute the remainder when the sum of these integers is divided by 1000.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Mock Aiome

    Uploaded by

    Sancho Pancho
    75 views3 pages
    1. The document provides instructions for submitting responses to a mock AIME problem set. Students should submit their 3-digit answers to a Google Form and include leading zeros even though they are not strictly necessary. 2. The problem set was written by a group of 5 individuals and edited by the same group. It contains 15 multiple choice problems requiring computational reasoning and mathematical skills. 3. The final problem can be expressed as an equation relating positive integers a, b, and c, and students are asked to compute the remainder when the sum of these integers is divided by 1000.

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    1. The document provides instructions for submitting responses to a mock AIME problem set. Students should submit their 3-digit answers to a Google Form and include leading zeros even though they are not strictly necessary. 2. The problem set was written by a group of 5 individuals and edited by the same group. It contains 15 multiple choice problems requiring computational reasoning and mathematical skills. 3. The final problem can be expressed as an equation relating positive integers a, b, and c, and students are asked to compute the remainder when the sum of these integers is divided by 1000.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    75 views3 pages

    Mock Aiome

    Uploaded by

    Sancho Pancho
    1. The document provides instructions for submitting responses to a mock AIME problem set. Students should submit their 3-digit answers to a Google Form and include leading zeros even though they are not strictly necessary. 2. The problem set was written by a group of 5 individuals and edited by the same group. It contains 15 multiple choice problems requiring computational reasoning and mathematical skills. 3. The final problem can be expressed as an equation relating positive integers a, b, and c, and students are asked to compute the remainder when the sum of these integers is divided by 1000.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 3

    Mock AIOME

    trumpeter
    14 September 2015
    Submit all submissions to the Google Form here. As usual, all answers are three digit
    integers from 000-999, inclusive. As Google Forms apparently doesnt know the difference
    between 001 and 1, it doesnt matter if you type 001 or 1 in. Always good practice to
    type out 001 to remember format for the actual AIME, though.
    Problem Writer: trumpeter
    Problem Selection Committee: anduril, MathStudent2002, qwerty137, suli, trumpeter
    Problem Editors: anduril, MathStudent2002, qwerty137, suli, trumpeter

    1. Compute the sum of the distinct prime factors of a4 ((a b3 ) b + 1) + 1 if a = 9 and


    b = 2.
    2. Let n be the smallest positive integer such that
    n100
    100!
    is an integer. Let d be the number of positive factors of n. Compute the number
    of positive factors of d.
    3. ABCD is a rectangle with side lengths 5 and 12. A circle centered at C has radius 2.
    Let E be a point on this circle. Compute the largest possible value of the distance
    AE.
    4. The sum of all positive integers n such that the base 2 representation of n has
    exactly 8 1s and 8 0s (the first digit cannot be 0) is S. Compute the remainder
    when S is divided by 1000.
    5. Compute the sum of the three smallest positive integers n such that n4 + 4 is a
    product of 2 distinct primes.
    6. Let a, b, and c be positive real numbers such that a + b + c = 12288. What is the
    smallest value of n such that the maximum possible value of
    !
    !
    !
    n
    n
    n
    X
    X
    X
    1
    1
    1
    ck
    bj +
    ai +
    i=1

    j=1

    k=1

    is not an integer?
    7. Let ABC be a triangle and D the midpoint of BC. Let X and Y be the circumcenters of 4ABD and 4ACD, respectively. Furthermore, let E be the midpoint
    of XY . Given that AB = 39, BC = 56, and AC = 25, the length of AE can be
    for relatively prime positive integers m and n. Compute m + n.
    expressed as m
    n
    8. Define the inverse of a modulo n as the integer b such that ab 1 is divisible by
    n. If there is no such integer b, the inverse does not exist. Now, define f (x) as a
    function that outputs 1 if x has an inverse modulo 2015 and 1 if x does not have
    an inverse modulo 2015. Compute the remainder when


    2015
    X



    f (n) n

    n=1

    is divided by 1000.
    9. A, B, C, and D are points in a plane such that B, C, and D are collinear and
    BAD + DAC = BAC = 126 . Given that AD = 1, the minimum possible
    value of
    BA
    CA
    +
    sin BAD sin CAD

    can be expressed as a b for positive integers a and b. Compute a + b.


    2

    10. Let P (x) = x3 2x + 3 and Q(x) = x7 6x5 + 9x4 + 12x3 36x2 + 17x + 36. Let
    z be a randomly selected root of P (x). Compute the expected value of Q(z).
    11. Let n be the smallest positive integer such that the last 2015 digits of 21n are
    000 . . . 0001, where there are 2014 0s followed by one 1. Compute the remainder
    when the number of positive factors of n is divided by 1000.
    12. Let d (x) be the number of distinct digits in the decimal representation of x and
    e (n), for positive integers n, be the expected value of d (x) for x = 1, 2, 3, . . . , 10n .
    Then,
    a bn c
    e (n) = 10
    d 10n
    for positive integers a, b, c, and d such that GCD (a, c, d) = 1. Compute a+b+c+d.
    13. Let x and n be integers such that n is positive and
    
    x2 = (10n + 12) x + 4n3 + 4n2 + 6n + 9 .
    In addition, x is the larger solution to the quadratic in x above. Compute the
    smallest possible value of x 2n.
    14. Let ABCD be a scalene quadrilateral that has both a circumcircle and incircle. If
    AC BD = 280 and AB, BC, CD, and AD are integers, compute the minimum
    possible perimeter of quadrilateral ABCD. Note: a scalene quadrilateral is one
    that has no two sides that are equal.
    15. We can show that

    
    

    
    
    X
    i
    i1
    1
    cb

    =
    ln
    ae
    ,
    2i i + 1 i + 2
    i=1

    where a, b, and c are positive integers such that GCD (b, c) = 1. Compute the
    remainder when a + b + c is divided by 1000.

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