Difference between revisions of "2023 AMC 10A Problems/Problem 8"

(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...")
 
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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?

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