The suffering program Quiz Document
Elliot Albano
June-July 2023
1 Problems
1. (2020 AIME II/3) The value of x that satisfies log2x320 = log2x+3 32020
can be written as m
n, where m and n are relatively prime integers. Find
m+n
2. Let x and y be positive real numbers such that logx64 + logy216 = 5
3and
logy64 + logx216 = 1. What is the value of log2(xy)?
3. (2019 AIME II/6) In a martian civilization, all logarithms whose bases
are not specified can be assumed to be base b, for some b fixed b≥2. A
martian student writes down
3(log(√xlog x)) = 56
loglog xx= 54
and finds that this system of equations only has a single real number
solution x > 1. Find b.
4. (2019 PMO Qualifying/17) Evaluate the sum P2019
n=1 cos(n2π
3)
5. Suppose a real number x > 1 satisfies
log 3
√3(log3x) + log3(log27 x) + log27 (log 3
√3x) = 1
Compute log3(log3x)
6. (2021 AMC 12A/14) What is the value of (P20
k=1(logk
53k2)(P100
k=1 logk
925k)
1
