Two Thousand Twenty Four

  1. What positive integer can be expressed any of these ways?
    \(2^3 \times 11 \times 23\)
    \(4\times22^2 + 4\times22\)
    \(2^3\bigl(2^8 - 3\bigr)\)
    \(3^4\bigl(3^3-2\bigr) - 1\)
    \(45^2 - 1\)
    \(2(11) + 2(1001)\)
    \(\binom{24}{3}\)
    \(2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3\)
    \(2^2 + 4^2 + 6^2 + 8^2 + 10^2 + 12^2 + 14^2 + 16^2 + 18^2 + 20^2 + 22^2\)
    \(77 + 78 + \dotsb + 98 + 99\)
    \(119 + 120 + \dotsb + 133 + 134\)
  2. What positive integer has the following expressions in bases two, three, and twenty-two respectively?
    \(11111101000_2\)
    \(2202222_3\)
    \(440_{22}\)

  3. Tetrahedral Numbers

    Suppose you were to stack balls in the shape of a triangle-based pyramid — a tetrahedron. If you want the pyramid’s base to be two balls long, you’d need 4 balls total to build the pyramid. If you want the pyramid’s base to be three balls long, you’d need 10 balls total to build the pyramid. If you want the pyramid’s base to be four balls long, you’d need 20 balls total to build the pyramid. How many balls would be required to build a pyramid with a base that is 22 balls long?

  4. Dodecahedral Numbers

    Similar to the last prompt, if you were building a “stack” of balls into the shape of a dodecahedron such that the length of a side was 8 balls long, how many balls would you need total?

    In general, why does a length \(n\) dodecahedron and a length \(3n-2\) tetrahedron require the same number of balls for any \(n?\)

  5. Chebyshev Polynomials

    The \(n\)th Chebyshev polynomial \(T_n\) can be defined in terms of the formula \(T_n\bigl(\cos(\theta)\bigr) = \cos(n\theta).\) What is the value of \(T_3(8)?\)

  6. Harschad Numbers

    What number is not only divisible by the sum of its digits when represented in decimal (base-10), but also when represented in the following bases? 4, 12, 17, 22, 23, 24, and 30.

  7. Betrothed Pair

    Two numbers are a betrothed pair if the sum of the nontrivial divisors of each of them equals the other one. What number forms a betrothed pair with 2295?

References