Two Thousand Twenty Five

  1. What positive integer can be expressed any of these ways?
    \(5 \times 5 \times 9 \times 9\)
    \(5^2(2^3+1)\)
    \(45^2\)
    \(\binom{10}{2}^2\)
    \(\bigl(3^3\bigr)^2 + \bigl(6^2\bigr)^2\)
    \(11 + 12 + \dotsb + 64\)
    \(69 + 70 + \dotsb + 93\)
    \(403 + 404 + \dotsb + 407\)
    \((1+2 + \dotsb + 9)^2\)
    \(1^3+2^3 + \dotsb + 9^3\)
  2. What positive integer has the following expressions in bases two, fifteen, and twenty-five respectively?
    \(11111101001_2\)
    \(900_{15}\)
    \(360_{25}\)
  3. What is the smallest positive integer that has exactly fifteen odd divisors?

  4. Politeness of a Number

    What is the smallest positive integer that can be written as the sum of one or more consecutive positive integers in exactly fifteen different ways? (Does this have to do with its number of odd divisors?)

  5. Centered Octahedral Number

    Starting with a single central marble placed on a flat table, suppose you were to start arranging rings of marbles in octagons nested around that central marble. At first you would need 9 marbles — the central marble plus 8 new marbles — to build the first octagon which has side-lengths that are two marbles long, with each marble at a corner being a member of both its adjoining sides. For the next octagon you’d need 25 marbles — the 9 marbles from before plus 16 new marbles — to build an octagon with side-lengths that are three marbles long. How many marbles would you need to build such an octagon with a side-length of 23 marbles?

  6. Determinants and Permanants

    Recall what the determinant (\(\operatorname{det}\)) of a matrix is. The permanant (\(\operatorname{perm}\)) of a matrix is like the determinant, but without any of the negative signs. For example: \[ \operatorname{det}\begin{pmatrix}a & b \\ c & d \end{pmatrix} = ad - bc \qquad \operatorname{perm}\begin{pmatrix}a & b \\ c & d \end{pmatrix} = ad + bc \] Considering all \(2 \times 2\) matrices that have positive integer entries in the range \(0, 1, 2, \dotsc, 23,\) how many have the property that their determinant is twice their permanent? (Does this have anything to do with the previous question?)

References