MathDB

2023 AMC 12/AHSME

Part of AMC 12/AHSME

Subcontests

(18)
8
1

Cardinality is the Least Element

How many nonempty subsets BB of {0,1,2,3,…,12}\{0, 1, 2, 3, \dots, 12\} have the property that the number of elements in BB is equal to the least element of BB? For example, B={4,6,8,11}B = \{4, 6, 8, 11\} satisfies the condition.
<spanclass=′latex−bold′>(A)</span> 256<spanclass=′latex−bold′>(B)</span> 136<spanclass=′latex−bold′>(C)</span> 108<spanclass=′latex−bold′>(D)</span> 144<spanclass=′latex−bold′>(E)</span> 156<span class='latex-bold'>(A)</span>\ 256 \qquad<span class='latex-bold'>(B)</span>\ 136 \qquad<span class='latex-bold'>(C)</span>\ 108 \qquad<span class='latex-bold'>(D)</span>\ 144 \qquad<span class='latex-bold'>(E)</span>\ 156
21
1

bea from the brawl stars

A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is 333\sqrt{3} inches, its top diameter is 6 inches, and its bottom diameter is 12 inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey?
https://cdn.artofproblemsolving.com/attachments/b/4/23f9bc88ea057cb2676f2b8b373330b0f5df69.png
<spanclass=′latex−bold′>(A)</span>6+3π<spanclass=′latex−bold′>(B)</span>6+6π<spanclass=′latex−bold′>(C)</span>63<spanclass=′latex−bold′>(D)</span>65<spanclass=′latex−bold′>(E)</span>63+π<span class='latex-bold'>(A) </span> 6 + 3\pi\qquad <span class='latex-bold'>(B) </span>6 + 6\pi\qquad <span class='latex-bold'>(C) </span> 6\sqrt3 \qquad <span class='latex-bold'>(D) </span> 6\sqrt5 \qquad <span class='latex-bold'>(E) </span> 6\sqrt3 + \pi
7
1

hot diggity

For how many integers nn does the expression log⁡(n2)−(log⁡n)2log⁡n−3\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}} represent a real number, where log denotes the base 1010 logarithm? <spanclass=′latex−bold′>(A)</span>900<spanclass=′latex−bold′>(B)</span>2<spanclass=′latex−bold′>(C)</span>902<spanclass=′latex−bold′>(D)</span>2<spanclass=′latex−bold′>(E)</span>901 <span class='latex-bold'>(A) </span>900 \qquad <span class='latex-bold'>(B) </span>2\qquad <span class='latex-bold'>(C) </span>902 \qquad <span class='latex-bold'>(D) </span> 2 \qquad <span class='latex-bold'>(E) </span>901
18
1

ocanrna of time

Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda's average on all the quizzes she took during the first semester was 3 points higher than Zelda's average on all the quizzes she took during the first semester. Yolanda's average on all the quizzes she took during the second semester was 18 points higher than her average for the first semester and was again 3 points higher than Zelda's average on all the quizzes Zelda took during her second semester. Which one of the following statements cannot possibly be true?
(A) Yolanda's quiz average for the academic year was 22 points higher than Zelda's. (B) Zelda's quiz average for the academic year was higher than Yolanda's. (C) Yolanda's quiz average for the academic year was 3 points higher than Zelda's. (D) Zelda's quiz average for the academic year equaled Yolanda's. (E) If Zelda had scored 3 points higher on each quiz she took, then she would have had the same average for the academic year as Yolanda.
10
2

More Straightforward Algebra

Positive real numbers xx and yy satisfy y3=x2y^3 = x^2 and (y−x)2=4y2(y-x)^2 = 4y^2. What is x+yx+y?
<spanclass=′latex−bold′>(A)</span> 12<spanclass=′latex−bold′>(B)</span> 18<spanclass=′latex−bold′>(C)</span> 24<spanclass=′latex−bold′>(D)</span> 36<spanclass=′latex−bold′>(E)</span> 42<span class='latex-bold'>(A)</span>\ 12 \qquad <span class='latex-bold'>(B)</span>\ 18 \qquad <span class='latex-bold'>(C)</span>\ 24 \qquad <span class='latex-bold'>(D)</span>\ 36 \qquad <span class='latex-bold'>(E)</span>\ 42

Slope of Radical Axis

In the xyxy-plane, a circle of radius 44 with center on the positive xx-axis is tangent to the yy-axis at the origin, and a circle with radius 1010 with center on the positive yy-axis is tangent to the xx-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?
<spanclass=′latex−bold′>(A)</span> 27<spanclass=′latex−bold′>(B)</span> 37<spanclass=′latex−bold′>(C)</span> 229<spanclass=′latex−bold′>(D)</span> 129<spanclass=′latex−bold′>(E)</span> 25<span class='latex-bold'>(A)</span>\ \dfrac{2}{7} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{3}{7} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{2}{\sqrt{29}} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{1}{\sqrt{29}} \qquad<span class='latex-bold'>(E)</span>\ \dfrac{2}{5}
20
2

Not Pascal&amp;#039;s

Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below:
[asy] size(4.5cm); label("11", (0,0)); label("11", (-0.5,-2/3)); label("11", (0.5,-2/3)); label("11", (-1,-4/3)); label("33", (0,-4/3)); label("11", (1,-4/3)); label("11", (-1.5,-2)); label("55", (-0.5,-2)); label("55", (0.5,-2)); label("11", (1.5,-2)); label("11", (-2,-8/3)); label("77", (-1,-8/3)); label("1111", (0,-8/3)); label("77", (1,-8/3)); label("11", (2,-8/3)); [/asy]
Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digit of the sum of the 2023 numbers in the 2023rd row?
<spanclass=′latex−bold′>(A)</span>1<spanclass=′latex−bold′>(B)</span>3<spanclass=′latex−bold′>(C)</span>5<spanclass=′latex−bold′>(D)</span>7<spanclass=′latex−bold′>(E)</span>9<span class='latex-bold'>(A) </span>1\qquad<span class='latex-bold'>(B) </span>3\qquad<span class='latex-bold'>(C) </span>5\qquad<span class='latex-bold'>(D) </span>7\qquad<span class='latex-bold'>(E) </span>9

How pog, another frog

Cyrus the frog jumps 2 units in a direction, then 2 more in another direction. What is the probability that he lands less than 1 unit away from his starting position?
(I forgot answer choices)
19
1

Obligatory Log

What is the product of all the solutions to the equation log⁡7x2023⋅log⁡289x2023=log⁡2023x2023?\log_{7x}2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?
<spanclass=′latex−bold′>(A)</span>(log⁡20237⋅log⁡2023289)2<spanclass=′latex−bold′>(B)</span>log⁡20237⋅log⁡2023289<spanclass=′latex−bold′>(C)</span>1<spanclass=′latex−bold′>(D)</span>log⁡72023⋅log⁡2892023<spanclass=′latex−bold′>(E)</span>(log⁡72023⋅log⁡2892023)2<span class='latex-bold'>(A) </span>(\log_{2023}7 \cdot \log_{2023}289)^2 \qquad<span class='latex-bold'>(B) </span>\log_{2023}7 \cdot \log_{2023}289\qquad<span class='latex-bold'>(C) </span>1\qquad<span class='latex-bold'>(D) </span>\log_{7}2023 \cdot \log_{289}2023\qquad<span class='latex-bold'>(E) </span>(\log_{7}2023 \cdot \log_{289}2023)^2
15
1

Usain Bolt moment

Usain is walking for exercise by zigzagging across a 100100-meter by 3030-meter rectangular field, beginning at point AA and ending on the segment BC‾\overline{BC}. He wants to increase the distance walked by zigzagging as shown in the figure below (APQRS)(APQRS). What angle θ=∠PAB=∠QPC=∠RQB=⋯\theta=\angle PAB=\angle QPC=\angle RQB=\cdots will produce in a length that is 120120 meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.)
[asy] import olympiad; draw((-50,15)--(50,15)); draw((50,15)--(50,-15)); draw((50,-15)--(-50,-15)); draw((-50,-15)--(-50,15)); draw((-50,-15)--(-22.5,15)); draw((-22.5,15)--(5,-15)); draw((5,-15)--(32.5,15)); draw((32.5,15)--(50,-4.090909090909)); label("θ\theta", (-41.5,-10.5)); label("θ\theta", (-13,10.5)); label("θ\theta", (15.5,-10.5)); label("θ\theta", (43,10.5)); dot((-50,15)); dot((-50,-15)); dot((50,15)); dot((50,-15)); dot((50,-4.09090909090909)); label("DD",(-58,15)); label("AA",(-58,-15)); label("CC",(58,15)); label("BB",(58,-15)); label("SS",(58,-4.0909090909)); dot((-22.5,15)); dot((5,-15)); dot((32.5,15)); label("PP",(-22.5,23)); label("QQ",(5,-23)); label("RR",(32.5,23)); [/asy]
<spanclass=′latex−bold′>(A)</span> arccos⁡56<spanclass=′latex−bold′>(B)</span> arccos⁡45<spanclass=′latex−bold′>(C)</span> arccos⁡310<spanclass=′latex−bold′>(D)</span> arcsin⁡45<spanclass=′latex−bold′>(E)</span> arcsin⁡56<span class='latex-bold'>(A)</span>~\arccos\frac{5}{6}\qquad<span class='latex-bold'>(B)</span>~\arccos\frac{4}{5}\qquad<span class='latex-bold'>(C)</span>~\arccos\frac{3}{10}\qquad<span class='latex-bold'>(D)</span>~\arcsin\frac{4}{5}\qquad<span class='latex-bold'>(E)</span>~\arcsin\frac{5}{6}
12
2

cube cube cube cube cube

What is the value of 23−12+43−33+63−53+⋯+183−173? 2^3 - 1^2 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?
<spanclass=′latex−bold′>(A)</span>2023<spanclass=′latex−bold′>(B)</span>2679<spanclass=′latex−bold′>(C)</span>2941<spanclass=′latex−bold′>(D)</span>3159<spanclass=′latex−bold′>(E)</span>3235<span class='latex-bold'>(A) </span> 2023 \qquad<span class='latex-bold'>(B) </span> 2679 \qquad<span class='latex-bold'>(C) </span> 2941 \qquad<span class='latex-bold'>(D) </span> 3159 \qquad<span class='latex-bold'>(E) </span> 3235

Operation Complex Numbers

For complex numbers u=a+biu=a+bi and v=c+div=c+di, define the binary operation ⊗\otimes by u⊗v=ac+bdi.u\otimes v=ac+bdi. Suppose zz is a complex number such that z⊗z=z2+40z\otimes z=z^{2}+40. What is ∣z∣|z|?
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 32<spanclass=′latex−bold′>(C)</span> 26<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 52<span class='latex-bold'>(A)</span>~\sqrt{10}\qquad<span class='latex-bold'>(B)</span>~3\sqrt{2}\qquad<span class='latex-bold'>(C)</span>~2\sqrt{6}\qquad<span class='latex-bold'>(D)</span>~6\qquad<span class='latex-bold'>(E)</span>~5\sqrt{2}
17
2

a new addition to the frog family

Flora the frog starts at 00 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance mm with probability 12m\frac{1}{2^m}. What is the probability that Flora will eventually land at 1010?
<spanclass=′latex−bold′>(A)</span>5512<spanclass=′latex−bold′>(B)</span>451024<spanclass=′latex−bold′>(C)</span>1271024<spanclass=′latex−bold′>(D)</span>5111024<spanclass=′latex−bold′>(E)</span>12<span class='latex-bold'>(A) </span> \frac{5}{512} \qquad <span class='latex-bold'>(B) </span> \frac{45}{1024} \qquad <span class='latex-bold'>(C) </span> \frac{127}{1024} \qquad <span class='latex-bold'>(D) </span> \frac{511}{1024} \qquad <span class='latex-bold'>(E) </span> \frac{1}{2}

The yearly LoC

Triangle ABCABC has side lengths in arithmetic progression, and the smallest side has length 6.6. If the triangle has an angle of 120∘,120^\circ, what is the area of ABCABC?
<spanclass=′latex−bold′>(A)</span> 123<spanclass=′latex−bold′>(B)</span> 86<spanclass=′latex−bold′>(C)</span> 142<spanclass=′latex−bold′>(D)</span> 202<spanclass=′latex−bold′>(E)</span> 153<span class='latex-bold'>(A)</span>\ 12\sqrt 3 \qquad<span class='latex-bold'>(B)</span>\ 8\sqrt 6 \qquad<span class='latex-bold'>(C)</span>\ 14\sqrt 2 \qquad<span class='latex-bold'>(D)</span>\ 20\sqrt 2 \qquad<span class='latex-bold'>(E)</span>\ 15\sqrt 3
25
1

Weird DeMoivre Stuff

There is a unique sequence of integers a1,a2,⋯a2023a_1, a_2, \cdots a_{2023} such that tan⁡2023x=a1tan⁡x+a3tan⁡3x+a5tan⁡5x+⋯+a2023tan⁡2023x1+a2tan⁡2x+a4tan⁡4x⋯+a2022tan⁡2022x \tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x} whenever tan⁡2023x\tan 2023x is defined. What is a2023?a_{2023}?
<spanclass=′latex−bold′>(A)</span>−2023<spanclass=′latex−bold′>(B)</span>−2022<spanclass=′latex−bold′>(C)</span>−1<spanclass=′latex−bold′>(D)</span>1<spanclass=′latex−bold′>(E)</span>2023<span class='latex-bold'>(A) </span> -2023 \qquad<span class='latex-bold'>(B) </span> -2022 \qquad<span class='latex-bold'>(C) </span> -1 \qquad<span class='latex-bold'>(D) </span> 1 \qquad<span class='latex-bold'>(E) </span> 2023
24
2

suspiciously sequenced sub-sets

Let KK be the number of sequences A1A_1, A2A_2, …\dots, AnA_n such that nn is a positive integer less than or equal to 1010, each AiA_i is a subset of {1,2,3,…,10}\{1, 2, 3, \dots, 10\}, and Ai−1A_{i-1} is a subset of AiA_i for each ii between 22 and nn, inclusive. For example, {}\{\}, {5,7}\{5, 7\}, {2,5,7}\{2, 5, 7\}, {2,5,7}\{2, 5, 7\}, {2,5,6,7,9}\{2, 5, 6, 7, 9\} is one such sequence, with n=5n = 5. What is the remainder when KK is divided by 1010?
<spanclass=′latex−bold′>(A)</span>1<spanclass=′latex−bold′>(B)</span>3<spanclass=′latex−bold′>(C)</span>5<spanclass=′latex−bold′>(D)</span>7<spanclass=′latex−bold′>(E)</span>9<span class='latex-bold'>(A) </span> 1 \qquad <span class='latex-bold'>(B) </span> 3 \qquad <span class='latex-bold'>(C) </span> 5 \qquad <span class='latex-bold'>(D) </span> 7 \qquad <span class='latex-bold'>(E) </span> 9

Misplaced much?

Integers a,b,c,da, b, c, d satisfy the following: abcd=26⋅39⋅57abcd=2^6\cdot 3^9\cdot 5^7 lcm(a,b)=23⋅32⋅53\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3 lcm(a,c)=23⋅33⋅53\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3 lcm(a,d)=23⋅33⋅53\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3 lcm(b,c)=21⋅33⋅52\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2 lcm(b,d)=22⋅33⋅52\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2 lcm(c,d)=22⋅33⋅52\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2
Find gcd(a,b,c,d)\text{gcd}(a,b,c,d) <spanclass=′latex−bold′>(A)</span> 30<spanclass=′latex−bold′>(B)</span> 45<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 15<spanclass=′latex−bold′>(E)</span> 6<span class='latex-bold'>(A)</span>~30\qquad<span class='latex-bold'>(B)</span>~45\qquad<span class='latex-bold'>(C)</span>~3\qquad<span class='latex-bold'>(D)</span>~15\qquad<span class='latex-bold'>(E)</span>~6
11
2

2020 AMC 12B/7 moment

What is the degree measure of the acute angle formed by lines with slopes 22 and 13\tfrac{1}{3}?
<spanclass=′latex−bold′>(A)</span> 30<spanclass=′latex−bold′>(B)</span> 37.5<spanclass=′latex−bold′>(C)</span> 45<spanclass=′latex−bold′>(D)</span> 52.5<spanclass=′latex−bold′>(E)</span> 60<span class='latex-bold'>(A)</span>~30\qquad<span class='latex-bold'>(B)</span>~37.5\qquad<span class='latex-bold'>(C)</span>~45\qquad<span class='latex-bold'>(D)</span>~52.5\qquad<span class='latex-bold'>(E)</span>~60

lana del rey

What is the maximum area of an isosceles trapezoid that has legs of length 11 and one base twice as long as the other? <spanclass=′latex−bold′>(A)</span>54<spanclass=′latex−bold′>(B)</span>87<spanclass=′latex−bold′>(C)</span>524<spanclass=′latex−bold′>(D)</span>32<spanclass=′latex−bold′>(E)</span>334 <span class='latex-bold'>(A) </span>\frac 54 \qquad <span class='latex-bold'>(B) </span> \frac 87 \qquad <span class='latex-bold'>(C)</span> \frac{5\sqrt2}4 \qquad <span class='latex-bold'>(D) </span> \frac 32 \qquad <span class='latex-bold'>(E) </span> \frac{3\sqrt3}4
23
2

BAW-algmanip

How many ordered pairs of positive real numbers (a,b)(a,b) satisfy the equation (1+2a)(2+2b)(2a+b)=32ab?(1+2a)(2+2b)(2a+b) = 32ab?
<spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>1<spanclass=′latex−bold′>(C)</span>2<spanclass=′latex−bold′>(D)</span>3<spanclass=′latex−bold′>(E)</span>an infinite number<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>1\qquad<span class='latex-bold'>(C) </span>2\qquad<span class='latex-bold'>(D) </span>3\qquad<span class='latex-bold'>(E) </span>\text{an infinite number}

Die Problem

When nn standard six-sided dice are rolled, the product of the numbers rolled can be any of 936936 possible values. What is nn?
<spanclass=′latex−bold′>(A)</span> 6<spanclass=′latex−bold′>(B)</span> 8<spanclass=′latex−bold′>(C)</span> 9<spanclass=′latex−bold′>(D)</span> 10<spanclass=′latex−bold′>(E)</span> 11<span class='latex-bold'>(A)</span>~6\qquad<span class='latex-bold'>(B)</span>~8\qquad<span class='latex-bold'>(C)</span>~9\qquad<span class='latex-bold'>(D)</span>~10\qquad<span class='latex-bold'>(E)</span>~11
14
2

Roots of Unity

How many complex numbers satisfy the equation z5=z‾z^{5}=\overline{z}, where z‾\overline{z} is the conjugate of the complex number zz?
<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 7<span class='latex-bold'>(A)</span>~2\qquad<span class='latex-bold'>(B)</span>~3\qquad<span class='latex-bold'>(C)</span>~5\qquad<span class='latex-bold'>(D)</span>~6\qquad<span class='latex-bold'>(E)</span>~7

3 Distinct Integer Roots

For how many ordered pairs (a,b)(a,b) of integers does the polynomial x3+ax2+bx+6x^3+ax^2+bx+6 have 33 distinct integer roots?
<spanclass=′latex−bold′>(A)</span> 5<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 8<spanclass=′latex−bold′>(D)</span> 7<spanclass=′latex−bold′>(E)</span> 4<span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ 8 \qquad<span class='latex-bold'>(D)</span>\ 7 \qquad<span class='latex-bold'>(E)</span>\ 4
16
2

RIP perfect score :/

Consider the set of complex numbers zz satisfying ∣1+z+z2∣=4|1+z+z^{2}|=4. The maximum value of the imaginary part of zz can be written in the form mn\tfrac{\sqrt{m}}{n}, where mm and nn are relatively prime positive integers. What is m+nm+n?
<spanclass=′latex−bold′>(A)</span> 20<spanclass=′latex−bold′>(B)</span> 21<spanclass=′latex−bold′>(C)</span> 22<spanclass=′latex−bold′>(D)</span> 23<spanclass=′latex−bold′>(E)</span> 24<span class='latex-bold'>(A)</span>~20\qquad<span class='latex-bold'>(B)</span>~21\qquad<span class='latex-bold'>(C)</span>~22\qquad<span class='latex-bold'>(D)</span>~23\qquad<span class='latex-bold'>(E)</span>~24

Triple Chicken McNugget

In Coinland, there are three types of coins, each worth 6,6, 10,10, and 15.15. What is the sum of the digits of the maximum amount of money that is impossible to have?
<spanclass=′latex−bold′>(A)</span>11<spanclass=′latex−bold′>(B)</span>6<spanclass=′latex−bold′>(C)</span>8<spanclass=′latex−bold′>(D)</span>9<spanclass=′latex−bold′>(E)</span>10<span class='latex-bold'>(A) </span>11\qquad<span class='latex-bold'>(B) </span>6\qquad<span class='latex-bold'>(C) </span>8\qquad<span class='latex-bold'>(D) </span>9\qquad<span class='latex-bold'>(E) </span>10 (I forgot the order)
22
2

Integer Divisor Function

Let ff be the unique function defined on the positive integers such that ∑d∣nd⋅f(nd)=1\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1 for all positive integers nn, where the sum is taken over all positive divisors of nn. What is f(2023)f(2023)?
<spanclass=′latex−bold′>(A)</span> −1536<spanclass=′latex−bold′>(B)</span> 96<spanclass=′latex−bold′>(C)</span> 108<spanclass=′latex−bold′>(D)</span> 116<spanclass=′latex−bold′>(E)</span> 144<span class='latex-bold'>(A)</span>~-1536\qquad<span class='latex-bold'>(B)</span>~96\qquad<span class='latex-bold'>(C)</span>~108\qquad<span class='latex-bold'>(D)</span>~116\qquad<span class='latex-bold'>(E)</span>~144

funtcion

A real-valued function ff has the property that for all real numbers aa and b,b, f(a+b)+f(a−b)=2f(a)f(b).f(a + b) + f(a - b) = 2f(a) f(b). Which one of the following cannot be the value of f(1)?f(1)? <spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>1<spanclass=′latex−bold′>(C)</span>−1<spanclass=′latex−bold′>(D)</span>2<spanclass=′latex−bold′>(E)</span>−2 <span class='latex-bold'>(A) </span> 0 \qquad <span class='latex-bold'>(B) </span> 1 \qquad <span class='latex-bold'>(C) </span> -1 \qquad <span class='latex-bold'>(D) </span> 2 \qquad <span class='latex-bold'>(E) </span> -2
6
1

Log Midpoint

Points AA and BB lie on the graph of y=log⁡2xy=\log_{2}x. The midpoint of AB‾\overline{AB} is (6,2)(6, 2). What is the positive difference between the xx-coordinates of AA and BB?
<spanclass=′latex−bold′>(A)</span> 211<spanclass=′latex−bold′>(B)</span> 43<spanclass=′latex−bold′>(C)</span> 8<spanclass=′latex−bold′>(D)</span> 45<spanclass=′latex−bold′>(E)</span> 9<span class='latex-bold'>(A)</span>~2\sqrt{11}\qquad<span class='latex-bold'>(B)</span>~4\sqrt{3}\qquad<span class='latex-bold'>(C)</span>~8\qquad<span class='latex-bold'>(D)</span>~4\sqrt{5}\qquad<span class='latex-bold'>(E)</span>~9