MathDB

A1/B3

Part of 2020 Princeton University Math Competition

Problems(3)

2020 PUMaC Algebra A1 / B3

Source:

1/1/2022
Let f(x)=x+ax+bf(x) =\frac{x+a}{x+b} satisfy f(f(f(x)))=xf(f(f(x))) = x for real numbers a,ba, b. If the maximum value of a is p/qp/q, where p,qp, q are relatively prime integers, what is ∣p∣+∣q∣|p| + |q|?
algebra
2020 PUMaC Geometry A1 / B3

Source:

12/31/2021
Let γ1\gamma_1 and γ2\gamma_2 be circles centered at OO and P P respectively, and externally tangent to each other at point QQ. Draw point DD on γ1\gamma_1 and point EE on γ2\gamma_2 such that line DEDE is tangent to both circles. If the length OQ=1OQ = 1 and the area of the quadrilateral ODEPODEP is 520520, then what is the value of length PQPQ?
geometry
2020 PUMaC NT A1 / B3

Source:

1/1/2022
Compute the last two digits of 92020+920202+...+9202020209^{2020} + 9^{2020^2}+ ... + 9^{2020^{2020}}
number theory