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Difference between revisions of "2024 AMC 8 Problems"

(→‎Problem 5)
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{{AMC8 Problems|year=2024|}}
 
{{AMC8 Problems|year=2024|}}
−
==Problem 1==  You can do it!
 
  
−
==Problem 1==  You can do it!
 
  
 +
==Problem 1== 
 +
What is 2^34930490493049(N0 CALCULAT0RS ALL0WED!!!!!!!!!!!!!!)
 +
(A) 1 (B) 2 (C) idk (D) why are you cheating for amc8 (E) PLEASE LET ME USE A CALCULUSER!
 
==Problem 2==
 
==Problem 2==
 
What is the last 3 digits of "pi"?
 
What is the last 3 digits of "pi"?
Line 23: Line 24:
  
 
==Problem 5==
 
==Problem 5==
−
Count from 1 to 278968797890807 for <math>1000000 by going to </math>\text{freerobux_and_vbucks_not_scam.com}$.
+
Count from 1 to 278968797890807 for <math>1000000 by going to </math>\text{freerobux_and_vbucks_not_scam.com}<math>.
  
 
==Problem 6==
 
==Problem 6==
Line 41: Line 42:
  
 
==Problem 9==
 
==Problem 9==
−
Compute <math>\frac{1}{0}</math>.
+
Compute </math>\frac{1}{0}<math>.
  
 
(A) 1 (C) 5 (B) 2 (D) 6 (E)3
 
(A) 1 (C) 5 (B) 2 (D) 6 (E)3
  
 
==Problem 10==
 
==Problem 10==
−
What is the sum of the roots of <math>\frac{1}{x}</math> <math>+1=x</math>?
+
What is the sum of the roots of </math>\frac{1}{x}<math> </math>+1=x<math>?
  
 
A)0  B)-1  C)1  D)-2  E)2
 
A)0  B)-1  C)1  D)-2  E)2
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==Problem 12==
 
==Problem 12==
−
Assuming that <math>1+1=3</math>, then what does <math>\sqrt{235479^{\sqrt{9472853.23462}\times4912}} + \frac{1}{0}</math> equal?
+
Assuming that </math>1+1=3<math>, then what does </math>\sqrt{235479^{\sqrt{9472853.23462}\times4912}} + \frac{1}{0}<math> equal?
  
−
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 256246\qquad \textbf{(E)}\ 10000</math>
+
</math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 256246\qquad \textbf{(E)}\ 10000<math>
  
 
==Problem 13==
 
==Problem 13==
  
−
A finite set <math>S</math> of positive integers has the property that, for each <math>s\in S</math>, and each positive integer divisor <math>d</math> of <math>s</math>, there exists a unique element <math>t \in S</math> satisfying <math>\gcd(s,t)=d</math> (the elements <math>s</math> and <math>t</math> could be equal).
+
A finite set </math>S<math> of positive integers has the property that, for each </math>s\in S<math>, and each positive integer divisor </math>d<math> of </math>s<math>, there exists a unique element </math>t \in S<math> satisfying </math>\gcd(s,t)=d<math> (the elements </math>s<math> and </math>t<math> could be equal).
  
−
Given this information, find all possible values for the number of elements of <math>S</math>. (source: 2021 USAMO)
+
Given this information, find all possible values for the number of elements of </math>S<math>. (source: 2021 USAMO)
  
 
now that you read this problem you have to do it without looking at the solution or else... let's just say bad things will happen
 
now that you read this problem you have to do it without looking at the solution or else... let's just say bad things will happen
  
 
==Problem 14==
 
==Problem 14==
−
Let k >/ 2 be an integer. Find the smallest integer <math>n</math> >/ k + 1 with the property that
+
Let k >/ 2 be an integer. Find the smallest integer </math>n<math> >/ k + 1 with the property that
 
there exists a set of n distinct real numbers such that each of its elements can be written as a
 
there exists a set of n distinct real numbers such that each of its elements can be written as a
 
sum of k other distinct elements of the set. (Source: IMO Shortlist Slovakia)
 
sum of k other distinct elements of the set. (Source: IMO Shortlist Slovakia)
  
−
(A) <math>n=k</math> + 3  (B) <math>n=k</math> - 3  (C) <math>n=k</math> + 4  (D) <math>n=k</math>  (E) <math>n=k</math> + 5
+
(A) </math>n=k<math> + 3  (B) </math>n=k<math> - 3  (C) </math>n=k<math> + 4  (D) </math>n=k<math>  (E) </math>n=k<math> + 5
  
 
==Problem 15==
 
==Problem 15==
  
−
Let <math>D</math> be an interior point of the acute triangle <math>ABC</math> with <math>AB > AC</math> so that <math>\angle DAB= \angle CAD</math>. The point <math>E</math> on the segment <math>AC</math> satisfies <math>\angle ADE= \angle BCD</math>, the point <math>F</math> on the segment <math>AB</math> satisfies <math>\angle FDA= \angle DBC</math>, and the point <math>X</math> on the line <math>AC</math> satisfies <math>CX=BX</math>. Let <math>O_1</math> and <math>O_2</math> be the circumcentres of the triangles <math>ADC</math> and <math>EXD</math> respectively. Prove that the lines <math>BC</math>, <math>EF</math>, and <math>O_1 O_2</math> are concurrent. (source: 2021 IMO)
+
Let </math>D<math> be an interior point of the acute triangle </math>ABC<math> with </math>AB > AC<math> so that </math>\angle DAB= \angle CAD<math>. The point </math>E<math> on the segment </math>AC<math> satisfies </math>\angle ADE= \angle BCD<math>, the point </math>F<math> on the segment </math>AB<math> satisfies </math>\angle FDA= \angle DBC<math>, and the point </math>X<math> on the line </math>AC<math> satisfies </math>CX=BX<math>. Let </math>O_1<math> and </math>O_2<math> be the circumcentres of the triangles </math>ADC<math> and </math>EXD<math> respectively. Prove that the lines </math>BC<math>, </math>EF<math>, and </math>O_1 O_2<math> are concurrent. (source: 2021 IMO)
  
 
now go do this problem as a punishment for trying to cheat
 
now go do this problem as a punishment for trying to cheat
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==Problem 17==
 
==Problem 17==
−
  Let <math>\text{x=2024}</math>. Compute the last three digits of <math>((x^3-(x-8)^3)^4-(x-69)^2)^5</math>?
+
  Let </math>\text{x=2024}<math>. Compute the last three digits of </math>((x^3-(x-8)^3)^4-(x-69)^2)^5<math>?
  
 
NO CALCULATORS ARE ALLOWED.
 
NO CALCULATORS ARE ALLOWED.
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==Problem 22==
 
==Problem 22==
−
What is the sum of the cubes of the solutions cubed of <math>x^5+2x^4+3x^3+3x^2+2x+1=0</math>?
+
What is the sum of the cubes of the solutions cubed of </math>x^5+2x^4+3x^3+3x^2+2x+1=0$?
  
 
==Problem 23==
 
==Problem 23==

Revision as of 12:32, 21 January 2024

2024 AMC 8 (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 1 point for each correct answer. There is no penalty for wrong answers.
  3. No aids are permitted other than plain scratch paper, writing utensils, ruler, and erasers. In particular, graph paper, compass, protractor, calculators, computers, smartwatches, and smartphones are not permitted. Rules
  4. Figures are not necessarily drawn to scale.
  5. You will have 40 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25


Problem 1

What is 2^34930490493049(N0 CALCULAT0RS ALL0WED!!!!!!!!!!!!!!) (A) 1 (B) 2 (C) idk (D) why are you cheating for amc8 (E) PLEASE LET ME USE A CALCULUSER!

Problem 2

What is the last 3 digits of "pi"?


$\textbf{(A)}\ 911 \qquad \textbf{(B)}\ 643 \qquad \textbf{(C)}\ 356 \qquad \textbf{(D)}\ 190\qquad \textbf{(E)}\ 1234$


-Multpi12

Problem 3

If Bartholomew's pet cat's favourite number in the alphabet is purple, then what is the square root of the combined time in years Bartholomew's father went to get the milk and the time it takes for you to count to 1 million out loud.

Problem 4

Bob has $5$ friends that go to school, only on Tuesday and Wednesday. The chance of rain on Tuesday and Wednesday is $50\%$. Assuming Bob and his friends are all humans, what is the street number of Bob's house?

$\textbf{(A)}\ 75423 \qquad \textbf{(B)}\ 5987 \qquad \textbf{(C)}\ 491 \qquad \textbf{(D)}\ 653\qquad \textbf{(E)}\ 21574$

Problem 5

Count from 1 to 278968797890807 for $1000000 by going to$\text{freerobux_and_vbucks_not_scam.com}$.

==Problem 6==

4 random points are chosen on a sphere. What is the probability that the tetrahedron with vertices of the 4 points contains the center of the sphere?

(A) 1/2 (B) 1/4 (C) 3/8 (D) 1/8 (E) 3/10 (Source: Putnam) lmao

[[2024 AMC 8 Problems/Problem 6|Solution]]

==Problem 8== Bob has a magical nuclear button that will explode one out of the 12,500 nukes on our planet. One nuke is called the "Tsar Bomb". At random, what is the chance that Bob will explode the Tsar Bomba after detonating another special nuke called "OPO"?

(A) 1/12500 (B) People will die (C) The FBI will come (D) you don't know (E) you want to cheat for the AMC8

==Problem 9== Compute$ (Error compiling LaTeX. Unknown error_msg)\frac{1}{0}$.

(A) 1 (C) 5 (B) 2 (D) 6 (E)3

==Problem 10== What is the sum of the roots of$ (Error compiling LaTeX. Unknown error_msg)\frac{1}{x}$$ (Error compiling LaTeX. Unknown error_msg)+1=x$?

A)0 B)-1 C)1 D)-2 E)2


[[2024 AMC 8 Problems/Problem 10|Solution]]

==Problem 11== The equation (2^(333x-2))+(2^(111x+2))=(2^(222x+1))+1 has three real roots. Find their sum. (Source: AIME)


(A) 4/113 (B) 2/111 (C) 6/11 (D) 5/111 (E) 14/113


You thought we could let you cheat?

==Problem 12== Assuming that$ (Error compiling LaTeX. Unknown error_msg)1+1=3$, then what does$\sqrt{235479^{\sqrt{9472853.23462}\times4912}} + \frac{1}{0}$equal?$\textbf{(A)}\ -1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 256246\qquad \textbf{(E)}\ 10000$==Problem 13==

A finite set$ (Error compiling LaTeX. Unknown error_msg)S$of positive integers has the property that, for each$s\in S$, and each positive integer divisor$d$of$s$, there exists a unique element$t \in S$satisfying$\gcd(s,t)=d$(the elements$s$and$t$could be equal).

Given this information, find all possible values for the number of elements of$ (Error compiling LaTeX. Unknown error_msg)S$. (source: 2021 USAMO)

now that you read this problem you have to do it without looking at the solution or else... let's just say bad things will happen

==Problem 14== Let k >/ 2 be an integer. Find the smallest integer$ (Error compiling LaTeX. Unknown error_msg)n$>/ k + 1 with the property that there exists a set of n distinct real numbers such that each of its elements can be written as a sum of k other distinct elements of the set. (Source: IMO Shortlist Slovakia)

(A)$ (Error compiling LaTeX. Unknown error_msg)n=k$+ 3 (B)$n=k$- 3 (C)$n=k$+ 4 (D)$n=k$(E)$n=k$+ 5

==Problem 15==

Let$ (Error compiling LaTeX. Unknown error_msg)D$be an interior point of the acute triangle$ABC$with$AB > AC$so that$\angle DAB= \angle CAD$. The point$E$on the segment$AC$satisfies$\angle ADE= \angle BCD$, the point$F$on the segment$AB$satisfies$\angle FDA= \angle DBC$, and the point$X$on the line$AC$satisfies$CX=BX$. Let$O_1$and$O_2$be the circumcentres of the triangles$ADC$and$EXD$respectively. Prove that the lines$BC$,$EF$, and$O_1 O_2$are concurrent. (source: 2021 IMO)

now go do this problem as a punishment for trying to cheat

[[2024 AMC 8 Problems/Problem 15|Solution]]

What is the value of 1, assuming that 1=1, but 1= 3(4x^2-7x+5) - 2(5x^2-9x-3)=6x

==Problem 17== 

Let$ (Error compiling LaTeX. Unknown error_msg)\text{x=2024}$. Compute the last three digits of$((x^3-(x-8)^3)^4-(x-69)^2)^5$?

NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED. NO CALCULATORS ARE ALLOWED.

==Problem 18==

hi guys. trying to cheat? im ashamed of you code: nsb

==Problem 19== Write your AoPS name here if you took the AMC 8.

Probablity NapoleonicAviator Multpi12

==Problem 20== Find the sum of the square root of -2 and the last digit of pi.

==Problem 21== this question = 9+10, bc 9+10 = 21

==Problem 22== What is the sum of the cubes of the solutions cubed of$ (Error compiling LaTeX. Unknown error_msg)x^5+2x^4+3x^3+3x^2+2x+1=0$?

Problem 23

lol we are the defenders against the cheaters... get outta here and study


SubText: and im here writing soulutions for these joke problems. (Multpi12)

Problem 24

wait when are the questions coming tho I think it's 1/25 for official answers since all tests end at 1/24

Problem 25

Did you think you could cheat the AMC ;)

and why did you scroll all the way here lol

See Also

2024 AMC 8 (Problems • Answer Key • Resources)
Preceded by
2023 AMC 8 		Followed by
2025 AMC 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2024_AMC_8_Problems&oldid=211844"