MathDB

23

Part of 2017 AMC 10

Problems(2)

Triangle Counting in a Grid

Source: 2017 AMC10A #23

2/8/2017
How many triangles with positive area have all their vertices at points (i,j)(i,j) in the coordinate plane, where ii and jj are integers between 11 and 55, inclusive?
<spanclass=′latex−bold′>(A)</span> 2128<spanclass=′latex−bold′>(B)</span> 2148<spanclass=′latex−bold′>(C)</span> 2160<spanclass=′latex−bold′>(D)</span> 2200<spanclass=′latex−bold′>(E)</span> 2300<span class='latex-bold'>(A)</span>\ 2128 \qquad<span class='latex-bold'>(B)</span>\ 2148 \qquad<span class='latex-bold'>(C)</span>\ 2160 \qquad<span class='latex-bold'>(D)</span>\ 2200 \qquad<span class='latex-bold'>(E)</span>\ 2300
AMCAMC 10AMC 10 A2017 AMC 10Acounting
123456789101112...4344 mod 45

Source: 2017 AMC 10B #23, 12B #19

2/16/2017
Let N=123456789101112…4344N = 123456789101112\dots4344 be the 7979-digit number obtained that is formed by writing the integers from 11 to 4444 in order, one after the other. What is the remainder when NN is divided by 4545?
<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 9<spanclass=′latex−bold′>(D)</span> 18<spanclass=′latex−bold′>(E)</span> 44<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 9 \qquad<span class='latex-bold'>(D)</span>\ 18 \qquad<span class='latex-bold'>(E)</span>\ 44
AMCAMC 10AMC 10 B2017 AMC 10B2017 AMC 12B