Found problems: 6
2022 USAMTS Problems, 3
A positive integer $N$ is called [i]googolicious[/i] if there are exactly $10^{100}$ positive integers $x$ that satisfy \[\left\lfloor \frac{N}{\left\lfloor \frac{N}{x} \right\rfloor } \right\rfloor = x,\] where $z$ denotes the greatest integer less than $z.$ Find, with proof, all googolicious integers $N.$
2009 F = Ma, 3
Suppose, instead, that all collisions are instantaneous and perfectly inelastic. After a long time, which of the following is true?
(A) The center block is moving to the left.
(B) The center block is moving to the right.
(C) The center block is at rest somewhere to the left of its initial position.
(D) The center block is at rest at its initial position.
(E) The center block is at rest somewhere to the right of its initial position.
2010 F = Ma, 3
If, instead, the graph is a graph of ACCELERATION vs. TIME and the squirrel starts from rest, then the squirrel has the greatest speed at what time(s) or during what time interval?
(A) at B
(B) at C
(C) at D
(D) at both B and D
(E) From C to D
2010 Contests, 3
If, instead, the graph is a graph of ACCELERATION vs. TIME and the squirrel starts from rest, then the squirrel has the greatest speed at what time(s) or during what time interval?
(A) at B
(B) at C
(C) at D
(D) at both B and D
(E) From C to D
2008 F = Ma, 3
The position [i]vs.[/i] time graph for an object moving in a straight line is shown below. What is the instantaneous velocity at $\text{t = 2 s}$?
[asy]
size(300);
// x-axis
draw((0,0)--(12,0));
// x-axis tick marks
draw((4,0)--(4,-0.2));
draw((8,0)--(8,-0.2));
draw((12,0)--(12,-0.2));
// x-axis labels
label("1",(4,0),2*S);
label("2",(8,0),2*S);
label("3",(12,0),2*S);
//y-axis
draw((0,-3)--(0,5));
//y-axis tick marks
draw((-0.2,-2)--(0,-2));
draw((-0.2,0)--(0,0));
draw((-0.2,2)--(0,2));
draw((-0.2,4)--(0,4));
// y-axis labels
label("-2",(0,-2),2*W);
label("0",(0,0),2*W);
label("2",(0,2),2*W);
label("4",(0,4),2*W);
// Axis Labels
label("Time (s)",(10,0),N);
label(rotate(90)*"Position (m)",(0,1),6*W);
//Line
draw((0,4)--(12,-2));
[/asy]
(a) $-\text{2 m/s}$
(b) $-\frac{1}{2} \ \text{m/s}$
(c) $\text{0 m/s}$
(d) $\text{2 m/s}$
(e) $\text{4 m/s}$
2011 F = Ma, 3
Rank the [i]magnitudes[/i] of the maximum velocity achieved during the ten second interval.
(A) $\text{I} > \text{II} > \text{III}$
(B) $\text{II} > \text{I} > \text{III}$
(C) $\text{III} > \text{II} > \text{I}$
(D) $\text{I} > \text{II = III}$
(E) $\text{I = II = III}$