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2024 AMC 8 Problems

Revision as of 14:22, 21 January 2024 by Rainier2020 (talk | contribs) (→‎Problem 42568)
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== You can do it!

==Problem 1== You can do it!

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 second cousin's neighbor's doctor's teacher's pet cat's favourite number in the Egyptian 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 divided by the last digit of pi.

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 multiplied by the amount of force Bob exerted when calculating the last digit of pi assuming Bob went to sleep at $9:00$ PM yesterday?

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

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

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 (F) you want to know how many problems you got correct on the AMC 8 (G) you think problem 15 would be really fun to solve (I) you looked at problem 15 (J) you didn't realize I skipped option H

Problem 9

If you are canoeing down a road, and a tire falls off, how many pancakes do you need to build a doghouse?

(A) 34 (C) purple (B) teaspoon (D) penguin (E) $\infty$

Problem 4

What is the sum of the roots of $\frac{1}{x}$ $+1=x$?

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


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 $1+1=3$, 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 $S$ 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 $S$. (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 $n$ >/ 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) $n=k$ + 3 (B) $n=k$ - 3 (C) $n=k$ + 4 (D) $n=k$ (E) $n=k$ + 5

Problem 15

Let $D$ 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 WITHOUT THE SOLUTION as a punishment for trying to cheat


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 $\text{x=2024}$. 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 funbeast mathkindacool_ Rainier2020

Problem 20

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

$(A) 2+\sqrt2i (B)3+\sqrt2i (C) 4+\sqrt2i (D) 5+\sqrt2i (E) 7+\sqrt2i$

Problem 21

what is 9+10?

(A) 21 (B) 69 (C) 19 (D) nothing (E) something

Problem 22

What is the sum of the cubes of the solutions cubed of $x^5+2x^4+3x^3+3x^2+2x+1=0$?

(A) 1 (B) 8 (C) 27 (D) -1 (E) -27

Solution

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) 

And so am I ;)

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
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