Found problems: 39
2010 Philippine MO, 5
Determine, with proof, the smallest positive integer $n$ with the following property: For every choice of $n$ integers, there exist at least two whose sum or difference is divisible by $2009$.
2010 F = Ma, 7
Harry Potter is sitting $2.0$ meters from the center of a merry-go-round when Draco Malfoy casts a spell that glues Harry in place and then makes the merry-go-round start spinning on its axis. Harry has a mass of $\text{50.0 kg}$ and can withstand $\text{5.0} \ g\text{'s}$ of acceleration before passing out. What is the magnitude of Harry's angular momentum when he passes out?
(A) $\text{200 kg} \cdot \text{m}^2\text{/s}$
(A) $\text{330 kg} \cdot \text{m}^2\text{/s}$
(A) $\text{660 kg} \cdot \text{m}^2\text{/s}$
(A) $\text{1000 kg} \cdot \text{m}^2\text{/s}$
(A) $\text{2200 kg} \cdot \text{m}^2\text{/s}$
2010 F = Ma, 19
Consider the following graphs of position [i]vs.[/i] time.
[asy]
size(500);
picture pic;
// Rectangle
draw(pic,(0,0)--(20,0)--(20,15)--(0,15)--cycle);
label(pic,"0",(0,0),S);
label(pic,"2",(4,0),S);
label(pic,"4",(8,0),S);
label(pic,"6",(12,0),S);
label(pic,"8",(16,0),S);
label(pic,"10",(20,0),S);
label(pic,"-15",(0,2),W);
label(pic,"-10",(0,4),W);
label(pic,"-5",(0,6),W);
label(pic,"0",(0,8),W);
label(pic,"5",(0,10),W);
label(pic,"10",(0,12),W);
label(pic,"15",(0,14),W);
label(pic,rotate(90)*"x (m)",(-2,7),W);
label(pic,"t (s)",(11,-2),S);
// Tick Marks
draw(pic,(4,0)--(4,0.3));
draw(pic,(8,0)--(8,0.3));
draw(pic,(12,0)--(12,0.3));
draw(pic,(16,0)--(16,0.3));
draw(pic,(20,0)--(20,0.3));
draw(pic,(4,15)--(4,14.7));
draw(pic,(8,15)--(8,14.7));
draw(pic,(12,15)--(12,14.7));
draw(pic,(16,15)--(16,14.7));
draw(pic,(20,15)--(20,14.7));
draw(pic,(0,2)--(0.3,2));
draw(pic,(0,4)--(0.3,4));
draw(pic,(0,6)--(0.3,6));
draw(pic,(0,8)--(0.3,8));
draw(pic,(0,10)--(0.3,10));
draw(pic,(0,12)--(0.3,12));
draw(pic,(0,14)--(0.3,14));
draw(pic,(20,2)--(19.7,2));
draw(pic,(20,4)--(19.7,4));
draw(pic,(20,6)--(19.7,6));
draw(pic,(20,8)--(19.7,8));
draw(pic,(20,10)--(19.7,10));
draw(pic,(20,12)--(19.7,12));
draw(pic,(20,14)--(19.7,14));
// Path
add(pic);
path A=(0,14)--(20,14);
draw(A);
label("I.",(8,-4),3*S);
path B=(0,6)--(20,6);
picture pic2=shift(30*right)*pic;
draw(shift(30*right)*B);
label("II.",(38,-4),3*S);
add(pic2);
path C=(0,12)--(20,14);
picture pic3=shift(60*right)*pic;
draw(shift(60*right)*C);
label("III.",(68,-4),3*S);
add(pic3);
[/asy]
Which of the graphs could be the motion of a particle in the given potential?
(A) $\text{I}$
(B) $\text{III}$
(C) $\text{I and II}$
(D) $\text{I and III}$
(E) $\text{I, II, and III}$
2010 F = Ma, 14
A $\text{5.0 kg}$ block with a speed of $\text{8.0 m/s}$ travels $\text{2.0 m}$ along a horizontal surface where it makes a head-on, perfectly elastic collision with a $\text{15.0 kg}$ block which is at rest. The coefficient of kinetic friction between both blocks and the surface is $0.35$. How far does the $\text{15.0 kg}$ block travel before coming to rest?
(A) $\text{0.76 m}$
(B) $\text{1.79 m}$
(C) $\text{2.29 m}$
(D) $\text{3.04 m}$
(E) $\text{9.14 m}$
2010 F = Ma, 2
If, instead, the graph is a graph of VELOCITY vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)?
(A) at B
(B) at C
(C) at D
(D) at both B and D
(E) From C to D
2010 Philippine MO, 2
On a cyclic quadrilateral $ABCD$, there is a point $P$ on side $AD$ such that the triangle $CDP$ and the quadrilateral $ABCP$ have equal perimeters and equal areas. Prove that two sides of $ABCD$ have equal lengths.
2010 F = Ma, 9
A point object of mass $M$ hangs from the ceiling of a car from a massless string of length $L$. It is observed to make an angle $\theta$ from the vertical as the car accelerates uniformly from rest. Find the acceleration of the car in terms of $\theta$, $M$, $L$, and $g$.
[asy]
size(250);
import graph;
// Left
draw((-3,0)--(-23,0),linewidth(1.5));
draw((-13,0)--(-13,-14));
filldraw(circle((-13,-15),2),gray);
draw((-13,-15)--(-21,-15),dashed);
draw((-21,-14)--(-21,-1),EndArrow(size=5));
draw((-21,-1)--(-21,-14),EndArrow(size=5));
label(scale(1.5)*"$L$",(-21,-7.5),2*E);
// Right
draw((3,0)--(23,0),linewidth(1.5));
draw((13,0)--(13,-19),dashed);
draw((13,0)--(5,-12));
filldraw(circle((3.89,-13.66),2),gray);
label(scale(1.5)*"$\theta$",(12,-9),1.5*W);
real f(real x){ return 5x^2/12-95x/12+25; }
draw(graph(f,12,7),Arrows);
[/asy]
(A) $Mg \sin \theta$
(B) $MgL \tan \theta$
(C) $g \tan \theta$
(D) $g \cot \theta$
(E) $Mg \tan \theta$
2010 F = Ma, 1
If the graph is a graph of POSITION vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)?
(A) From A to B
(B) From B to C only
(C) From B to D
(D) From C to D only
(E) From D to E
2010 F = Ma, 13
A ball of mass $M$ and radius $R$ has a moment of inertia of $I=\frac{2}{5}MR^2$. The ball is released from rest and rolls down the ramp with no frictional loss of energy. The ball is projected vertically upward off a ramp as shown in the diagram, reaching a maximum height $y_{max}$ above the point where it leaves the ramp. Determine the maximum height of the projectile $y_{max}$ in terms of $h$.
[asy]
size(250);
import roundedpath;
path A=(0,0)--(5,-12)--(20,-12)--(20,-10);
draw(roundedpath(A,1),linewidth(1.5));
draw((25,-10)--(12,-10),dashed+linewidth(0.5));
filldraw(circle((1.7,-1),1),lightgray);
draw((25,-1)--(-1.5,-1),dashed+linewidth(0.5));
draw((23,-9.5)--(23,-1.5),Arrows(size=5));
label(scale(1.1)*"$h$",(23,-6.5),2*E);
[/asy]
(A) $h$
(B) $\frac{25}{49}h$
(C) $\frac{2}{5}h$
(D) $\frac{5}{7}h$
(E) $\frac{7}{5}h$
2010 F = Ma, 21
The gravitational self potential energy of a solid ball of mass density $\rho$ and radius $R$ is $E$. What is the gravitational self potential energy of a ball of mass density $\rho$ and radius $2R$?
(A) $2E$
(B) $4E$
(C) $8E$
(D) $16E$
(E) $32E$
2010 F = Ma, 25
Spaceman Fred's spaceship (which has negligible mass) is in an elliptical orbit about Planet Bob. The minimum distance between the spaceship and the planet is $R$; the maximum distance between the spaceship and the planet is $2R$. At the point of maximum distance, Spaceman Fred is traveling at speed $v_\text{0}$. He then fires his thrusters so that he enters a circular orbit of radius $2R$. What is his new speed?
[asy]
size(300);
// Shape
draw(circle((0,0),25),dashed+gray);
draw(circle((0,0),3.5),linewidth(2));
draw(ellipse((5,0),20,15));
// Dashed Lines
draw((25,13)--(25,-35),dotted);
draw((0,-35)--(0,-3.3),dotted);
draw((0,3.3)--(0,13),dotted);
draw((-15,13)--(-15,-35),dotted);
// Labels
draw((-14,-35)--(-1,-35),Arrows(size=6,SimpleHead));
label(scale(1.2)*"$R$",(-7.5,-35),N);
draw((24,-35)--(1,-35),Arrows(size=6,SimpleHead));
label(scale(1.2)*"$2R$",(10,-35),N);
// Blobs on Earth
path A=(-1.433, 2.667)--
(-1.433, 2.573)--
(-1.360, 2.478)--
(-1.408, 2.360)--
(-1.493, 2.207)--
(-1.554, 2.160)--
(-1.614, 2.113)--
(-1.675, 2.065)--
(-1.735, 1.959)--
(-1.772, 1.877)--
(-1.723, 1.759)--
(-1.748, 1.676)--
(-1.748, 1.523)--
(-1.772, 1.369)--
(-1.760, 1.240)--
(-1.857, 1.145)--
(-1.941, 1.098)--
(-2.050, 1.122)--
(-2.111, 1.086)--
(-2.244, 1.039)--
(-2.390, 1.004)--
(-2.511, 0.909)--
(-2.486, 0.697)--
(-2.499, 0.555)--
(-2.535, 0.414)--
(-2.668, 0.308)--
(-2.765, 0.237)--
(-2.910, 0.131)--
(-3.068, 0.036)--
(-3.250, 0.024)--
(-3.310, 0.154)--
(-3.274, 0.272)--
(-3.286, 0.402)--
(-3.298, 0.532)--
(-3.250, 0.650)--
(-3.165, 0.768)--
(-3.128, 0.933)--
(-3.068, 1.074)--
(-3.032, 1.204)--
(-2.971, 1.310)--
(-2.886, 1.452)--
(-2.801, 1.558)--
(-2.729, 1.652)--
(-2.656, 1.770)--
(-2.583, 1.912)--
(-2.486, 1.995)--
(-2.365, 2.089)--
(-2.244, 2.207)--
(-2.123, 2.313)--
(-2.014, 2.419)--
(-1.905, 2.478)--
(-1.832, 2.573)--
(-1.687, 2.643)--
(-1.578, 2.714)--cycle;
filldraw(A,gray);
path B=(-0.397, 2.527)--
(-0.468, 2.321)--
(-0.538, 2.154)--
(-0.639, 2.065)--
(-0.760, 2.085)--
(-0.922, 2.085)--
(-0.993, 2.016)--
(-0.770, 1.918)--
(-0.649, 1.829)--
(-0.498, 1.780)--
(-0.367, 1.770)--
(-0.205, 1.751)--
(-0.084, 1.761)--
(-0.104, 1.613)--
(-0.114, 1.495)--
(-0.094, 1.358)--
(0.007, 1.220)--
(0.067, 1.131)--
(0.108, 1.013)--
(0.188, 0.905)--
(0.239, 0.787)--
(0.330, 0.650)--
(0.461, 0.620)--
(0.622, 0.620)--
(0.794, 0.591)--
(0.905, 0.610)--
(0.956, 0.689)--
(1.026, 0.591)--
(1.097, 0.483)--
(1.198, 0.374)--
(1.258, 0.276)--
(1.339, 0.188)--
(1.319, -0.009)--
(1.309, -0.166)--
(1.198, -0.343)--
(1.077, -0.432)--
(0.935, -0.520)--
(0.814, -0.589)--
(0.633, -0.677)--
(0.481, -0.727)--
(0.350, -0.776)--
(0.229, -0.894)--
(0.229, -1.041)--
(0.229, -1.228)--
(0.340, -1.346)--
(0.522, -1.415)--
(0.643, -1.513)--
(0.693, -1.651)--
(0.784, -1.798)--
(0.723, -1.936)--
(0.612, -2.044)--
(0.471, -2.123)--
(0.350, -2.201)--
(0.249, -2.270)--
(0.108, -2.339)--
(-0.013, -2.418)--
(-0.124, -2.535)--
(-0.135, -2.673)--
(-0.175, -2.811)--
(-0.084, -2.840)--
(0.067, -2.840)--
(0.209, -2.830)--
(0.350, -2.742)--
(0.522, -2.653)--
(0.582, -2.604)--
(0.713, -2.545)--
(0.845, -2.457)--
(0.935, -2.408)--
(1.057, -2.388)--
(1.228, -2.280)--
(1.329, -2.191)--
(1.460, -2.132)--
(1.581, -2.093)--
(1.692, -2.044)--
(1.793, -2.005)--
(1.844, -1.906)--
(1.844, -1.828)--
(1.904, -1.749)--
(2.005, -1.621)--
(1.955, -1.454)--
(1.894, -1.287)--
(1.773, -1.189)--
(1.632, -0.992)--
(1.592, -0.874)--
(1.491, -0.736)--
(1.410, -0.569)--
(1.460, -0.412)--
(1.561, -0.274)--
(1.592, -0.078)--
(1.622, 0.168)--
(1.551, 0.306)--
(1.440, 0.404)--
(1.420, 0.561)--
(1.551, 0.620)--
(1.703, 0.630)--
(1.824, 0.532)--
(1.955, 0.365)--
(2.046, 0.453)--
(2.116, 0.551)--
(2.167, 0.689)--
(2.096, 0.807)--
(1.965, 0.905)--
(1.834, 0.935)--
(1.743, 0.994)--
(1.622, 1.131)--
(1.531, 1.249)--
(1.430, 1.348)--
(1.359, 1.515)--
(1.420, 1.702)--
(1.511, 1.839)--
(1.571, 2.016)--
(1.672, 2.134)--
(1.592, 2.232)--
(1.440, 2.291)--
(1.289, 2.350)--
(1.178, 2.252)--
(1.127, 2.134)--
(1.067, 1.997)--
(0.986, 1.898)--
(0.845, 1.839)--
(0.693, 1.839)--
(0.522, 1.859)--
(0.471, 1.977)--
(0.380, 2.124)--
(0.289, 2.203)--
(0.188, 2.291)--
(0.047, 2.311)--
(-0.074, 2.370)--
(-0.195, 2.508)--cycle;
filldraw(B,gray);
[/asy]
(A) $\sqrt{3/2}v_\text{0}$
(B) $\sqrt{5}v_\text{0}$
(C) $\sqrt{3/5}v_\text{0}$
(D) $\sqrt{2}v_\text{0}$
(E) $2v_\text{0}$
2021 Indonesia TST, N
For a three-digit prime number $p$, the equation $x^3+y^3=p^2$ has an integer solution. Calculate $p$.
2010 F = Ma, 18
Which of the following represents the force corresponding to the given potential?
[asy]
// Code by riben
size(400);
picture pic;
// Rectangle
draw(pic,(0,0)--(22,0)--(22,12)--(0,12)--cycle);
label(pic,"-15",(2,0),S);
label(pic,"-10",(5,0),S);
label(pic,"-5",(8,0),S);
label(pic,"0",(11,0),S);
label(pic,"5",(14,0),S);
label(pic,"10",(17,0),S);
label(pic,"15",(20,0),S);
label(pic,"-2",(0,2),W);
label(pic,"-1",(0,4),W);
label(pic,"0",(0,6),W);
label(pic,"1",(0,8),W);
label(pic,"2",(0,10),W);
label(pic,rotate(90)*"F (N)",(-2,6),W);
label(pic,"x (m)",(11,-2),S);
// Tick Marks
draw(pic,(2,0)--(2,0.3));
draw(pic,(5,0)--(5,0.3));
draw(pic,(8,0)--(8,0.3));
draw(pic,(11,0)--(11,0.3));
draw(pic,(14,0)--(14,0.3));
draw(pic,(17,0)--(17,0.3));
draw(pic,(20,0)--(20,0.3));
draw(pic,(0,2)--(0.3,2));
draw(pic,(0,4)--(0.3,4));
draw(pic,(0,6)--(0.3,6));
draw(pic,(0,8)--(0.3,8));
draw(pic,(0,10)--(0.3,10));
draw(pic,(2,12)--(2,11.7));
draw(pic,(5,12)--(5,11.7));
draw(pic,(8,12)--(8,11.7));
draw(pic,(11,12)--(11,11.7));
draw(pic,(14,12)--(14,11.7));
draw(pic,(17,12)--(17,11.7));
draw(pic,(20,12)--(20,11.7));
draw(pic,(22,2)--(21.7,2));
draw(pic,(22,4)--(21.7,4));
draw(pic,(22,6)--(21.7,6));
draw(pic,(22,8)--(21.7,8));
draw(pic,(22,10)--(21.7,10));
// Paths
path A=(0,6)--(5,6)--(5,4)--(11,4)--(11,8)--(17,8)--(17,6)--(22,6);
path B=(0,6)--(5,6)--(5,2)--(11,2)--(11,10)--(17,10)--(17,6)--(22,6);
path C=(0,6)--(5,6)--(5,5)--(11,5)--(11,7)--(17,7)--(17,6)--(22,6);
path D=(0,6)--(5,6)--(5,7)--(11,7)--(11,5)--(17,5)--(17,6)--(22,6);
path E=(0,6)--(5,6)--(5,8)--(11,8)--(11,4)--(17,4)--(17,6)--(22,6);
draw(A);
label("(A)",(9.5,-3),4*S);
draw(shift(35*right)*B);
label("(B)",(45.5,-3),4*S);
draw(shift(20*down)*C);
label("(C)",(9.5,-23),4*S);
draw(shift(35*right)*shift(20*down)*D);
label("(D)",(45.5,-23),4*S);
draw(shift(40*down)*E);
label("(E)",(9.5,-43),4*S);
add(pic);
picture pic2=shift(35*right)*pic;
picture pic3=shift(20*down)*pic;
picture pic4=shift(35*right)*shift(20*down)*pic;
picture pic5=shift(40*down)*pic;
add(pic2);
add(pic3);
add(pic4);
add(pic5);
[/asy]
2010 Contests, 3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$.
[i]George Xing.[/i]