MathDB

25

Part of 2016 AMC 10

Problems(2)

Three LCM equations

Source: Problem #25 2016 AMC 10A

2/3/2016
How many ordered triples (x,y,z)(x,y,z) of positive integers satisfy lcm(x,y)=72,lcm(x,z)=600\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600 and lcm(y,z)=900\text{lcm}(y,z)=900?
<spanclass=′latex−bold′>(A)</span> 15<spanclass=′latex−bold′>(B)</span> 16<spanclass=′latex−bold′>(C)</span> 24<spanclass=′latex−bold′>(D)</span> 27<spanclass=′latex−bold′>(E)</span> 64<span class='latex-bold'>(A)</span>\ 15\qquad<span class='latex-bold'>(B)</span>\ 16\qquad<span class='latex-bold'>(C)</span>\ 24\qquad<span class='latex-bold'>(D)</span>\ 27\qquad<span class='latex-bold'>(E)</span>\ 64
number theoryleast common multipleAMCAMC 10AMC 10 A2016 AMC 10A
A 'Sum'what Tricky Problem

Source: Problem #25 2016 AMC 10B

2/21/2016
Let f(x)=∑k=210(⌊kx⌋−k⌊x⌋)f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor), where ⌊r⌋\lfloor r \rfloor denotes the greatest integer less than or equal to rr. How many distinct values does f(x)f(x) assume for x≥0x \ge 0?
<spanclass=′latex−bold′>(A)</span> 32<spanclass=′latex−bold′>(B)</span> 36<spanclass=′latex−bold′>(C)</span> 45<spanclass=′latex−bold′>(D)</span> 46<spanclass=′latex−bold′>(E)</span> infinitely many<span class='latex-bold'>(A)</span>\ 32\qquad<span class='latex-bold'>(B)</span>\ 36\qquad<span class='latex-bold'>(C)</span>\ 45\qquad<span class='latex-bold'>(D)</span>\ 46\qquad<span class='latex-bold'>(E)</span>\ \text{infinitely many}
AMC10AMCAMC 10AMC 10 Bfloor function