MathDB

A4/B6

Part of 2020 Princeton University Math Competition

Problems(3)

2020 PUMaC Algebra A4 / B6

Source:

1/1/2022
Let PP be a 1010-degree monic polynomial with roots r1,r2,...,r10≠r_1, r_2, . . . , r_{10} \ne and let QQ be a 4545-degree monic polynomial with roots 1ri+1rj−1rirj\frac{1}{r_i}+\frac{1}{r_j}-\frac{1}{r_ir_j} where i<ji < j and i,j∈{1,...,10}i, j \in \{1, ... , 10\}. If P(0)=Q(1)=2P(0) = Q(1) = 2, then log⁡2(∣P(1)∣)\log_2 (|P(1)|) can be written as a/ba/b for relatively prime integers a,ba, b. Find a+ba + b.
algebra
2020 PUMaC Geometry A4 / B6

Source:

12/31/2021
Let CC be a circle centered at point OO, and let PP be a point in the interior of CC. Let QQ be a point on the circumference of CC such that PQ⊥OPPQ \perp OP, and let DD be the circle with diameter PQPQ. Consider a circle tangent to CC whose circumference passes through point PP. Let the curve Γ\Gamma be the locus of the centers of all such circles. If the area enclosed by Γ\Gamma is 1/1001/100 the area of CC, then what is the ratio of the area of CC to the area of DD?
geometry
2020 PUMaC NT A4 / B6

Source:

1/1/2022
Given two positive integers a≠ba \ne b, let f(a,b)f(a, b) be the smallest integer that divides exactly one of a,ba, b, but not both. Determine the number of pairs of positive integers (x,y)(x, y), where x≠yx \ne y, 1≤x,y,≤1001\le x, y, \le 100 and gcd⁡(f(x,y),gcd⁡(x,y))=2\gcd(f(x, y), \gcd(x, y)) = 2.
number theory