MathDB

17

Part of 2017 AMC 10

Problems(2)

2017 AMC 10A #17

Source: 2017 AMC 10A #17

2/8/2017
Distinct points PP, QQ, RR, SS lie on the circle x2+y2=25x^2+y^2=25 and have integer coordinates. The distances PQPQ and RSRS are irrational numbers. What is the greatest possible value of the ratio PQRS\frac{PQ}{RS }?
<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 5<spanclass=′latex−bold′>(C)</span> 35<spanclass=′latex−bold′>(D)</span> 7<spanclass=′latex−bold′>(E)</span> 52<span class='latex-bold'>(A)</span>\ 3\qquad<span class='latex-bold'>(B)</span>\ 5\qquad<span class='latex-bold'>(C)</span>\ 3\sqrt{5}\qquad<span class='latex-bold'>(D)</span>\ 7\qquad<span class='latex-bold'>(E)</span>\ 5\sqrt{2}
AMCAMC 10AMC 10 A2017 AMC 10Ageometry
Accounting for an overcount

Source: 2017 AMC 10B #17, AMC 12B #11

2/16/2017
Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. How many monotonous positive integers are there?
<spanclass=′latex−bold′>(A)</span> 1024<spanclass=′latex−bold′>(B)</span> 1524<spanclass=′latex−bold′>(C)</span> 1533<spanclass=′latex−bold′>(D)</span> 1536<spanclass=′latex−bold′>(E)</span> 2048<span class='latex-bold'>(A)</span> \text{ 1024} \qquad <span class='latex-bold'>(B)</span> \text{ 1524} \qquad <span class='latex-bold'>(C)</span> \text{ 1533} \qquad <span class='latex-bold'>(D)</span> \text{ 1536} \qquad <span class='latex-bold'>(E)</span> \text{ 2048}
2017 AMC 10BAMC 10AMCcountingAMC 12AMC 12 B