Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
During AMC 10A/12A testing, the AoPS Wiki is in read-only mode and no edits can be made.

2023 AMC 10A Problems/Problem 8: Difference between revisions

Newer edit →
Huangjialan (talk | contribs)
editor
16 edits
Created page with "or every real number x, define bxc to be equal to the greatest integer less than or equal to x. (We call this the “floor” of x.) For example, b4.2c = 4, b5.7c = 5, b−3.4..."
 
Scrabbler94 (talk | contribs)
editor
290 edits
replace with correct problem 8
Line 1: Line 1:
or every real number x, define bxc to be equal to the greatest integer less than
Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at 110 degrees Fahrenheit, which is 0 degrees on the Breadus scale. Bread is baked at 350 degrees Fahrenheit, which is 100 degrees on the Breadus scale. Bread is done when its internal temperature is 200 degrees Fahrenheit. What is this in degrees on the Breadus scale?
or equal to x. (We call this the “floor” of x.) For example, b4.2c = 4, b5.7c = 5,
 
b−3.4c = −4, b0.4c = 0, and b2c = 2.
<math>\textbf{(A) }33\qquad\textbf{(B) }34.5\qquad\textbf{(C) }36\qquad\textbf{(D) }37.5\qquad\textbf{(E) }39</math>
(a) Determine the integer equal to �
1
3
�
+
�
2
3
�
+
�
3
3
�
+ . . . +
�
59
3
�
+
�
60
3
�
.
(The sum has 60 terms.)
(b) Determine a polynomial p(x) so that for every positive integer m > 4,
bp(m)c =
�
1
3
�
+
�
2
3
�
+
�
3
3
�
+ . . . +
�
m − 2
3
�
+
�
m − 1
3
�
(The sum has m − 1 terms.)
A polynomial f(x) is an algebraic expression of the form
f(x) = anx
n + an−1x
n−1 + · · · + a1x + a0 for some integer n ≥ 0 and for
some real numbers an, an−1, . . . , a1, a0.
(c) For each integer n ≥ 1, define f(n) to be equal to an infinite sum:
f(n) = �
n
1
2 + 1�
+
�
2n
2
2 + 1�
+
�
3n
3
2 + 1�
+
�
4n
4
2 + 1�
+
�
5n
5
2 + 1�
+ · · ·
(The sum contains the terms �
kn
k
2 + 1�
for all positive integers k, and no other
terms.)
Suppose f(t + 1) − f(t) = 2 for some odd positive integer t. Prove that t is a
prime number.

Revision as of 16:18, 9 November 2023

Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at 110 degrees Fahrenheit, which is 0 degrees on the Breadus scale. Bread is baked at 350 degrees Fahrenheit, which is 100 degrees on the Breadus scale. Bread is done when its internal temperature is 200 degrees Fahrenheit. What is this in degrees on the Breadus scale?

$\textbf{(A) }33\qquad\textbf{(B) }34.5\qquad\textbf{(C) }36\qquad\textbf{(D) }37.5\qquad\textbf{(E) }39$

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_AMC_10A_Problems/Problem_8&oldid=201088"