2024 PUMaC Team Round, Question 14 Inquiry
by A22-4, Apr 23, 2025, 6:39 PM
2024 PUMaC Team Round Question 14 reads as follows:
What is the largest value of
for which I can find nonnegative integers 
such that for all indices 
, 
divides 
?
(Note: This should say "... nonnegative integers
...")
I interpreted this correction to mean the following:
What is the largest value of for which I there exists nonnegative integers such that for all indices , divides ?
The official answer (https://static1.squarespace.com/static/570450471d07c094a39efaed/t/67421bd74806e80a7ab11c7d/1732385751115/PUMaC_2024_Team__Final_.pdf) is 107. However, I believe I have a construction with
integers - take the set of all integers with a digit sum of 
in base , then append 
to the list.
I checked this with Python using the following code:
Am I wrong or are they wrong? Any insight would be appreciated!
What is the largest value of
for which I can find nonnegative integers 
such that for all indices 
, 
divides 
?(Note: This should say "... nonnegative integers
...")I interpreted this correction to mean the following:
What is the largest value of
for which I there exists nonnegative integers such that for all indices , divides ?The official answer (https://static1.squarespace.com/static/570450471d07c094a39efaed/t/67421bd74806e80a7ab11c7d/1732385751115/PUMaC_2024_Team__Final_.pdf) is 107. However, I believe I have a construction with
integers - take the set of all integers with a digit sum of 
in base , then append 
to the list.I checked this with Python using the following code:
def digit_sum_base(n, base):
total = 0
while n > 0:
total += n % base
n //= base
return total
target_sum = 19
base = 17
limit = 2024
qualified_numbers = [n for n in range(limit) if digit_sum_base(n, base) == target_sum]
qualified_numbers.append(2023)
from math import comb
all_divisible = True
for i in range(len(qualified_numbers)):
for j in range(i):
a, b = qualified_numbers[i], qualified_numbers[j]
if comb(a, b) % 17 != 0:
all_divisible = False
break
if not all_divisible:
break
print(len(qualified_numbers), all_divisible)Am I wrong or are they wrong? Any insight would be appreciated!
This post has been edited 1 time. Last edited by A22-4, Yesterday at 6:40 PM
Reason: Changed square brackets to parentheses on the note
Reason: Changed square brackets to parentheses on the note
be a positive integer such that 
,![\[
A = \{a_1, a_2, a_3, \ldots, a_m\}
\]](http://latex.artofproblemsolving.com/1/7/9/179687a134d5495ffff0b44fe763a2bb8b3ddd9c.png)
consist of distinct positive integers not exceeding 2025. Suppose that for every 
,
,
,
as well. Prove that![\[
\frac{a_1 + a_2 + a_3 + \cdots + a_m}{m} \geq 1013.
\]](http://latex.artofproblemsolving.com/c/0/d/c0d190f78d94deac1b947ba57a2de51d0b06f0bd.png)
and ![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
be real numbers such that 
Prove that
.
are there such that 
and 
is an integer.
be reals such that 
. Prove that
Let 
. Prove that
and 
for each positive integer 
,
?
,
and 
meet at a point 
.
and the points 
lie on the same straight line.
March 2025
November 2024