2024 AMC 10A Problems/Problem 1

The following problem is from both the 2024 AMC 10A #1 and 2024 AMC 12A #1, so both problems redirect to this page.

Problem

What is the value of

Solution 1 (Direct Computation)

The likely fastest method will be direct computation. evaluates to and evaluates to . The difference is

Solution by juwushu.

Solution 2 (Distributive Property)

We have ~MRENTHUSIASM

Solution 3 (Solution 1 but Distributive)

Note that and , therefore the answer is .

~Tacos_are_yummy_1

Solution 4 (Modular arithmetic, fast)

Evaluating the given expression yields , so the answer is either (A) or (D). Evaluating yields . Because answer (D) is , that cannot be the answer, so we choose choice .

Solution 5 (Process of Elimination)

We simply look at the units digit of the problem we have (or take mod ) Since the only answer with 2 in the units digit is or We can then continue if you are desperate to use guess and check or a actually valid method to find the answer is . ~mathkiddus

Solution 6 (Faster Distribution)

Observe that and

~laythe_enjoyer211

Solution 7 (cubes)

Let .

Then, we have \begin{align*} 101\cdot 9901=(x+1)\cdot (x^2-x+1)=x^3+1 \\ 99\cdot 10101=(x-1)\cdot (x^2+x+1)=x^3-1 \end{align*}

Then, the answer can be rewritten as

~erics118

Solution 8 (Super fast)

Its not hard to observe and express into , and into .

We then simplify the original expression into , which could then be simplified into , which we can get the answer of .

~RULE101

Video Solution by Pi Academy

https://youtu.be/GPoTfGAf8bc?si=JYDhLVzfHUbXa3DW

Video Solution

https://youtu.be/Z76bafQsqTc

~Thesmartgreekmathdude

Video Solution 1 by Power Solve

https://www.youtube.com/watch?v=j-37jvqzhrg

See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.