4

Part of 2024 AIME

Problems(2)

Jen, Subset of 10

Source: 2024 AIME I #4

Jen randomly picks

4

distinct elements from

\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}

. The lottery machine also picks

4

distinct elements. If the lottery machine picks at least

2

of Jen’s numbers, Jen wins a prize. If the lottery machine’s numbers are all

4

of Jen’s, Jen wins the Grand Prize. Given that Jen wins a prize, what is the probability she wins a Grand Prize?

Another early logarithm

Source: AIME II 2024 Problem 4

x,y

z

be positive real numbers that satisfy the following system of equations:

\log_2\left({x \over yz}\right) = {1 \over 2}

\log_2\left({y \over xz}\right) = {1 \over 3}

\log_2\left({z \over xy}\right) = {1 \over 4}

Then the value of

\left|\log_2(x^4y^3z^2)\right|

{m \over n}

m

n

are relatively prime positive integers. Find

m+n

AMCAIMEalgebrasystem of equationsnumber theoryrelatively primeAIME II