Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
During AMC testing, the AoPS Wiki is in read-only mode and no edits can be made.

Difference between revisions of "1957 AHSME Problems/Problem 43"

m (typo fix)
(diagram)
Line 11: Line 11:
  
 
== Solution ==
 
== Solution ==
 +
 +
<asy>
 +
 +
path p = (0,0){right}..(1,1)..(2,4)..(3,9)..(4,16);
 +
 +
// Shaded Region
 +
fill(p--(4,0)--cycle,lightred);
 +
 +
// x-Axis
 +
draw((-4,0)--(16,0), arrow=Arrows);
 +
label("$x$",(18,0));
 +
 +
// y-Axis
 +
draw((0,-4)--(0,16), arrow=Arrows);
 +
label("$y$",(0,18));
 +
 +
// y=x^2
 +
draw(p);
 +
 +
// x=4
 +
draw((4,-5)--(4,20), arrow=Arrows(TeXHead));
 +
 +
</asy>
 +
 
<math>\boxed{\textbf{(B) }35}</math>.
 
<math>\boxed{\textbf{(B) }35}</math>.
  

Revision as of 09:38, 27 July 2024

Problem

We define a lattice point as a point whose coordinates are integers, zero admitted. Then the number of lattice points on the boundary and inside the region bounded by the $x$-axis, the line $x = 4$, and the parabola $y = x^2$ is:

$\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 35\qquad \textbf{(C)}\ 34\qquad \textbf{(D)}\ 30\qquad \textbf{(E)}\ \infty$

Solution

[asy] path p = (0,0){right}..(1,1)..(2,4)..(3,9)..(4,16); // Shaded Region fill(p--(4,0)--cycle,lightred); // x-Axis draw((-4,0)--(16,0), arrow=Arrows); label("$x$",(18,0)); // y-Axis draw((0,-4)--(0,16), arrow=Arrows); label("$y$",(0,18)); // y=x^2 draw(p); // x=4 draw((4,-5)--(4,20), arrow=Arrows(TeXHead)); [/asy]

$\boxed{\textbf{(B) }35}$.

See Also

1957 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 42 		Followed by
Problem 44
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50
All AHSME Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1957_AHSME_Problems/Problem_43&oldid=223768"