A base \(10\) over-expansion of a positive integer \(N\)
is an expression of the form
\[ N = d_k 10^k + d_{k-1} 10^{k-1} + \dotsb + d_0 10^0 \]
with \(d_k \neq 0\) and \(d_i \in \{0,1,2,\dots,10\}\) for all \(i\).
For instance, the integer \(N = 10\) has two base 10 over-expansions: \(10 = 10 \cdot 10^0\)
and the usual base 10 expansion \(10 = 1 \cdot 10^1 + 0 \cdot 10^0\).
Which positive integers have a unique base \(10\) over-expansion?
Suppose that \(f(x) = \sum_{i=0}^{\infty} c_ix^i\)
is a power series for which each coefficient \(c_i\) is 0 or 1.
Show that if \(f(2/3) =3/2,\) then \(f(1/2)\) must be irrational.
How many positive integers \(N\) satisfy all of the following three conditions:
(1) \(N\) is divisible by 2020,
(2) \(N\) has at most 2020 decimal digits,
and (3) the decimal digits of \(N\) are a string of consecutive ones
followed by a string of consecutive zeros.
For a positive integer \(n\), define \(d(n)\) to be the sum of the
digits of \(n\) when written in binary.
For example \(d(13) = 1+1+0+1=3.\)
Determine the value of this number modulo \(2020.\)
\[\sum_{k=1}^{2020} (-1)^{d(k)}k^3 \,. \]
Prove that for each positive integer \(n\), this number isn’t prime.
\[10^{10^{10^n}}\!\!\! + 10^{10^n}\!\! + 10^n - 1\]
Putnam and Beyond §1.5
The 2023-digit number \(99\dots 99\) is written on a blackboard.
Each minute, a robot performs the following operation,
making random decisions where necessary:
a number on the blackboard is chosen
and factored into two factors \(x\) and \(y\),
the chosen number is erased,
and the numbers \(x+2\) and \(y-2\) are written on the board.
Is it possible that at some point
all of the numbers on the blackboard are equal to \(9\)?
A palindrome in base \(b\) is a positive integer
whose base \(b\) digits read the same backwards and forwards;
for example, 2002 is a 4-digit palindrome in base 10.
Note that 200 is not a palindrome in base 10,
but it is the 3-digit palindrome 242 in base 9, and 404 in base 7.
Prove that there is an integer which is a 3-digit palindrome in base \(b\)
for at least 2002 different values of \(b.\)
The sequence of digits
\[ 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 \dots \]
is obtained by writing the positive integers in order.
If the \(10^n\)-th digit in this sequence occurs
in the part of the sequence in which the \(m\)-digit numbers are placed,
define \(f(n)\) to be \(m\).
For example, \(f(2)=2\) because the 100th digit enters the sequence
in the placement of the two-digit integer 55.
Find, with proof, \(f(1987)\).
For each positive integer \(n,\)
let \(k(n)\) be the number of ones
in the binary representation of \(2023 \cdot n.\)
What is the minimum value of \(k(n)?\)
Let \(R_n^{(m)}\) denote the repunit
with \(n\) digits expressed in base \(m.\)
Prove that for distinct primes \(p\) and \(q,\)
\(R_p^{(q)}\) never divides \(R_q^{(p)}.\)