MathDB

25

Part of 2020 AMC 12/AHSME

Problems(2)

Floors and Fractionals

Source: 2020 AMC 12A #25

1/31/2020
The number a=pqa = \tfrac{p}{q}, where pp and qq are relatively prime positive integers, has the property that the sum of all real numbers xx satisfying ⌊x⌋⋅{x}=a⋅x2\lfloor x \rfloor \cdot \{x\} = a \cdot x^2 is 420420, where ⌊x⌋\lfloor x \rfloor denotes the greatest integer less than or equal to xx and {x}=x−⌊x⌋\{x\} = x - \lfloor x \rfloor denotes the fractional part of xx. What is p+q?p + q?
<spanclass=′latex−bold′>(A)</span>245<spanclass=′latex−bold′>(B)</span>593<spanclass=′latex−bold′>(C)</span>929<spanclass=′latex−bold′>(D)</span>1331<spanclass=′latex−bold′>(E)</span>1332<span class='latex-bold'>(A) </span> 245 \qquad <span class='latex-bold'>(B) </span> 593 \qquad <span class='latex-bold'>(C) </span> 929 \qquad <span class='latex-bold'>(D) </span> 1331 \qquad <span class='latex-bold'>(E) </span> 1332
AMCAMC 122020 AMC 12A2020 AMCAMC 12 Arelatively primenumber theory
Trig Probability

Source: 2020 AMC 12B #25

2/6/2020
For each real number aa with 0≤a≤10 \leq a \leq 1, let numbers xx and yy be chosen independently at random from the intervals [0,a][0, a] and [0,1][0, 1], respectively, and let P(a)P(a) be the probability that sin⁡2(πx)+sin⁡2(πy)>1.\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1. What is the maximum value of P(a)?P(a)?
<spanclass=′latex−bold′>(A)</span> 712<spanclass=′latex−bold′>(B)</span> 2−2<spanclass=′latex−bold′>(C)</span> 1+24<spanclass=′latex−bold′>(D)</span> 5−12<spanclass=′latex−bold′>(E)</span> 58<span class='latex-bold'>(A)</span>\ \frac{7}{12} \qquad<span class='latex-bold'>(B)</span>\ 2 - \sqrt{2} \qquad<span class='latex-bold'>(C)</span>\ \frac{1+\sqrt{2}}{4} \qquad<span class='latex-bold'>(D)</span>\ \frac{\sqrt{5}-1}{2} \qquad<span class='latex-bold'>(E)</span>\ \frac{5}{8}
trigonometryAMCAMC 12AMC 12 Bprobability2020 AMC 12B2020 AMC