Found problems: 39
2010 ELMO Shortlist, 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]
2010 Regional Olympiad of Mexico Center Zone, 6
Let $ABC$ be an equilateral triangle and $D$ the midpoint of $BC$. Let $E$ and $F$ be points on $AC$ and $AB$ respectively such that $AF=CE$. $P=BE$ $\cap$ $CF$.
Show that $\angle$$APF=$ $\angle$$BPD$
2010 F = Ma, 10
A block of mass $m_\text{1}$ is on top of a block of mass $m_\text{2}$. The lower block is on a horizontal surface, and a rope can pull horizontally on the lower block. The coefficient of kinetic friction for all surfaces is $\mu$. What is the resulting acceleration of the lower block if a force $F$ is applied to the rope? Assume that $F$ is sufficiently large so that the top block slips on the lower block.
[asy]
size(200);
import roundedpath;
draw((0,0)--(30,0),linewidth(3));
path A=(7,0.5)--(17,0.5)--(17,5.5)--(7,5.5)--cycle;
filldraw(roundedpath(A,1),lightgray);
path B=(10,6)--(15,6)--(15,9)--(10,9)--cycle;
filldraw(roundedpath(B,1),lightgray);
label("1",(12.5,6),1.5*N);
label("2",(12,0.5),3*N);
draw((17,3)--(27,3),EndArrow(size=13));
label(scale(1.2)*"$F$",(22,3),2*N);
[/asy]
(A) $a_\text{2}=(F-\mu g(2m_\text{1}+m_\text{2}))/m_\text{2}$
(B) $a_\text{2}=(F-\mu g(m_\text{1}+m_\text{2}))/m_\text{2}$
(C) $a_\text{2}=(F-\mu g(m_\text{1}+2m_\text{2}))/m_\text{2}$
(D) $a_\text{2}=(F+\mu g(m_\text{1}+m_\text{2}))/m_\text{2}$
(E) $a_\text{2}=(F-\mu g(m_\text{2}-m_\text{1}))/m_\text{2}$
2010 F = Ma, 4
Two teams of movers are lowering a piano from the window of a $10$ floor apartment building. The rope breaks
when the piano is $30$ meters above the ground. The movers on the ground, alerted by the shouts of the movers
above, first notice the piano when it is $14$ meters above the ground. How long do they have to get out of the way
before the piano hits the ground?
(A) $\text{0.66 sec}$
(B) $\text{0.78 sec}$
(C) $\text{1.67 sec}$
(D) $\text{1.79 sec}$
(E) $\text{2.45 sec}$
2010 F = Ma, 11
The three masses shown in the accompanying diagram are equal. The pulleys are small, the string is lightweight, and friction is negligible. Assuming the system is in equilibrium, what is the ratio $a/b$? The figure is not drawn to scale!
[asy]
size(250);
dotfactor=10;
dot((0,0));
dot((15,0));
draw((-3,0)--(25,0),dashed);
draw((0,0)--(0,3),dashed);
draw((15,0)--(15,3),dashed);
draw((0,0)--(0,-15));
draw((15,0)--(15,-10));
filldraw(circle((0,-16),1),lightgray);
filldraw(circle((15,-11),1),lightgray);
draw((0,0)--(4,-4));
filldraw(circle((4.707,-4.707),1),lightgray);
draw((15,0)--(5.62,-4.29));
draw((0.5,3)--(14.5,3),Arrows(size=5));
label(scale(1.2)*"$a$",(7.5,3),1.5*N);
draw((2.707,-4.707)--(25,-4.707),dashed);
draw((25,-0.5)--(25,-4.2),Arrows(size=5));
label(scale(1.2)*"$b$",(25,-2.35),1.5*E);
[/asy]
(A) $1/2$
(B) $1$
(C) $\sqrt{3}$
(D) $2$
(E) $2\sqrt{3}$
2010 F = Ma, 24
A uniform circular disk of radius $R$ begins with a mass $M$; about an axis through the center of the disk and perpendicular to the plane of the disk the moment of inertia is $I_\text{0}=\frac{1}{2}MR^2$. A hole is cut in the disk as shown in the diagram. In terms of the radius $R$ and the mass $M$ of the original disk, what is the moment of inertia of the resulting object about the axis shown?
[asy]
size(14cm);
pair O=origin;
pair
A=O,
B=(3,0),
C=(6,0);
real
r_1=1,
r_2=.5;
pen my_fill_pen_1=gray(.8);
pen my_fill_pen_2=white;
pen my_fill_pen_3=gray(.7);
pen my_circleline_draw_pen=black+1.5bp;
//fill();
filldraw(circle(A,r_1),my_fill_pen_1,my_circleline_draw_pen);
filldraw(circle(B,r_1),my_fill_pen_1,my_circleline_draw_pen);
// Ellipse
filldraw(yscale(.2)*circle(C,r_1),my_fill_pen_1,my_circleline_draw_pen);
draw((C.x,C.y-.75)--(C.x,C.y-.2), dashed);
draw(C--(C.x,C.y+1),dashed);
label("axis of rotation",(C.x,C.y-.75),3*S);
// small ellipse
pair center_small_ellipse;
center_small_ellipse=midpoint(C--(C.x+r_1,C.y));
//dot(center_small_ellipse);
filldraw(yscale(.15)*circle(center_small_ellipse,r_1/2),white);
pair center_elliptic_arc_arrow;
real gr=(sqrt(5)-1)/2;
center_elliptic_arc_arrow=(C.x,C.y+gr);
//dot(center_elliptic_arc_arrow);
draw(//shift((0*center_elliptic_arc_arrow.x,center_elliptic_arc_arrow.y-.2))*
(
yscale(.2)*
(
arc((center_elliptic_arc_arrow.x,center_elliptic_arc_arrow.y+2.4), .4,120,360+60))
),Arrow);
//dot(center_elliptic_arc_arrow);
// lower_Half-Ellipse
real downshift=1;
pair C_prime=(C.x,C.y-downshift);
path lower_Half_Ellipse=yscale(.2)*arc(C_prime,r_1,180,360);
path upper_Half_Ellipse=yscale(.2)*arc(C,r_1,180,360);
draw(lower_Half_Ellipse,my_circleline_draw_pen);
//draw(upper_Half_Ellipse,red);
// Why here ".2*downshift" instead of downshift seems to be not absolutely clean.
filldraw(upper_Half_Ellipse--(C.x+r_1,C.y-.2*downshift)--reverse(lower_Half_Ellipse)--cycle,gray);
//filldraw(shift(C-.1)*(circle((B+.5),.5)),my_fill_pen_2);//
filldraw(circle((B+.5),.5),my_fill_pen_2);//shift(C-.1)*
/*
filldraw(//shift((C.x,C.y-.45))*
yscale(.2)*circle((C.x,C.y-1),r_1),my_fill_pen_3,my_circleline_draw_pen);
*/
draw("$R$",A--dir(240),Arrow);
draw("$R$",B--shift(B)*dir(240),Arrow);
draw(scale(1)*"$\scriptstyle R/2$",(B+.5)--(B+1),.5*LeftSide,Arrow);
draw(scale(1)*"$\scriptstyle R/2$",(B+.5)--(B+1),.5*LeftSide,Arrow);
[/asy]
(A) $\text{(15/32)}MR^2$
(B) $\text{(13/32)}MR^2$
(C) $\text{(3/8)}MR^2$
(D) $\text{(9/32)}MR^2$
(E) $\text{(15/16)}MR^2$
2010 Philippine MO, 1
Find all primes that can be written both as a sum of two primes and as a difference of two primes.
2010 F = Ma, 23
Two streams of water flow through the U-shaped tubes shown. The tube on the left has cross-sectional area $A$, and the speed of the water flowing through it is $v$; the tube on the right has cross-sectional area $A'=1/2A$. If the net force on the tube assembly is zero, what must be the speed $v'$ of the water flowing through the tube on the right?
Neglect gravity, and assume that the speed of the water in each tube is the same upon entry and exit.
[asy]
// Code by riben
size(300);
draw(arc((0,0),10,90,270));
draw(arc((0,0),7,90,270));
draw((0,10)--(25,10));
draw((0,-10)--(25,-10));
draw((0,7)--(25,7));
draw((0,-7)--(25,-7));
draw(ellipse((25,8.5),0.5,1.5));
draw(ellipse((25,-8.5),0.5,1.5));
draw((20,8.5)--(7,8.5),EndArrow(size=7));
draw((7,-8.5)--(20,-8.5),EndArrow(size=7));
draw(arc((-22,0),12,90,-90));
draw(arc((-22,0),7,90,-90));
draw((-22,12)--(-42,12));
draw((-22,-12)--(-42,-12));
draw((-22,7)--(-42,7));
draw((-22,-7)--(-42,-7));
draw(ellipse((-42,9.5),1.5,2.5));
draw(ellipse((-42,-9.5),1.5,2.5));
draw((-38,9.5)--(-23,9.5),EndArrow(size=7));
draw((-23,-9.5)--(-38,-9.5),EndArrow(size=7));
[/asy]
(A) $1/2v$
(B) $v$
(C) $\sqrt{2}v$
(D) $2v$
(E) $4v$
2010 F = Ma, 22
A balloon filled with helium gas is tied by a light string to the floor of a car; the car is sealed so that the motion of the car does not cause air from outside to affect the balloon. If the car is traveling with constant speed along a circular path, in what direction will the balloon on the string lean towards?
[asy]
size(300);
draw(circle((0,0),7));
path A=(1,2)--(1,-2)--(-1,-2)--(-1,2)--cycle;
filldraw(shift(7*left)*A,lightgray);
draw((-7,0)--(-7,5),EndArrow(size=21));
label(scale(1.5)*"A",(-8,2),2*N);
label(scale(1.5)*"B",(-8,0),2*W);
label(scale(1.5)*"C",(-7,-2),3*S);
label(scale(1.5)*"D",(-6,0),2*E);
[/asy]
(A) A
(B) B
(C) C
(D) D
(E) Remains vertical
2010 F = Ma, 16
Following the previous set up, find the speed $v$ of the small block after it leaves the slope.
(A) $v=v_\text{0}$
(B) $v=\frac{m}{m+M}v_\text{0}$
(C) $v=\frac{M}{m+M}v_\text{0}$
(D) $v=\frac{M-m}{m}v_\text{0}$
(E) $v=\frac{M-m}{m+M}v_\text{0}$
2010 Contests, 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 ELMO Shortlist, 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]
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, 6
A projectile is launched across flat ground at an angle $\theta$ to the horizontal and travels in the absence of air resistance. It rises to a maximum height $H$ and lands a horizontal distance $R$ away. What is the ratio $H/R$?
(A) $\tan \theta$
(B) $2 \tan \theta$
(C) $\frac{2}{\tan \theta}$
(D) $\frac{1}{2}\tan \theta$
(E) $\frac{1}{4}\tan \theta$
2010 F = Ma, 5
Two projectiles are launched from a $35$ meter ledge as shown in the diagram. One is launched from a $37$ degree angle above the horizontal and the other is launched from $37$ degrees below the horizontal. Both of the launches are given the same initial speed of $v_\text{0} = \text{50 m/s}$.
[asy]
size(300);
import graph;
draw((-8,0)--(0,0)--(0,-11)--(30,-11));
draw((0,-11)--(-4.5,-11),dashdotted);
draw((0,0)--(12,0),dashdotted);
label(scale(0.75)*"35 m",(0,-5.5),5*W);
draw((-4,-4.5)--(-4,-0.5),EndArrow(size=5));
draw((-4,-6)--(-4,-10.5),EndArrow(size=5));
// Projectiles
real f(real x){ return -11x^2/49; }
draw(graph(f,0,7),dashed+linewidth(1.5));
real g(real x){ return -6x^2/145+119x/145; }
draw(graph(g,0,29),dashed+linewidth(1.5));
// Labels
label(scale(0.75)*"Projectile 1",(20,2),E);
label(scale(0.75)*"Projectile 2",(6,-7),E);
[/asy]
The difference in the times of flight for these two projectiles, $t_1-t_2$, is closest to
(A) $\text{3 s}$
(B) $\text{5 s}$
(C) $\text{6 s}$
(D) $\text{8 s}$
(E) $\text{10 s}$
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 F = Ma, 17
Four masses $m$ are arranged at the vertices of a tetrahedron of side length $a$. What is the gravitational potential energy of this arrangement?
(A) $-2\frac{Gm^2}{a}$
(B) $-3\frac{Gm^2}{a}$
(C) $-4\frac{Gm^2}{a}$
(D) $-6\frac{Gm^2}{a}$
(E) $-12\frac{Gm^2}{a}$
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
2010 Contests, 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, 8
A car attempts to accelerate up a hill at an angle $\theta$ to the horizontal. The coefficient of static friction between the tires and the hill is $\mu > \tan \theta$. What is the maximum acceleration the car can achieve (in the direction upwards along the hill)? Neglect the rotational inertia of the wheels.
(A) $g \tan \theta$
(B) $g(\mu \cos \theta - \sin \theta)$
(C) $g(\mu - \sin \theta)$
(D) $g \mu \cos \theta$
(E) $g(\mu \sin \theta - \cos \theta)$
2010 Philippine MO, 4
There are $2008$ blue, $2009$ red and $2010$ yellow chips on a table. At each step, one chooses two chips of different colors, and recolor both of them using the third color. Can all the chips be of the same color after some steps? Prove your answer.
2010 F = Ma, 15
A small block moving with initial speed $v_\text{0}$ moves smoothly onto a sloped big block of mass $M$. After the small block reaches the height $h$ on the slope, it slides down. Find the height $h$.
(A) $h=\frac{v_\text{0}^2}{2g}$
(B) $h=\frac{1}{g}\frac{Mv_\text{0}^2}{m+M}$
(C) $h=\frac{1}{2g}\frac{Mv_\text{0}^2}{m+M}$
(D) $h=\frac{1}{2g}\frac{mv_\text{0}^2}{m+M}$
(e) $h=\frac{v_\text{0}^2}{g}$
2010 F = Ma, 20
Consider the following graph of position vs. time, which represents the motion of a certain particle in the given potential.
[asy]
import roundedpath;
size(300);
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=(0.102, 6.163)--
(0.192, 6.358)--
(0.369, 6.500)--
(0.526, 6.642)--
(0.643, 6.712)--
(0.820, 6.830)--
(0.938, 6.901)--
(1.075, 7.043)--
(1.193, 7.185)--
(1.369, 7.256)--
(1.506, 7.374)--
(1.644, 7.445)--
(1.840, 7.515)--
(1.958, 7.586)--
(2.134, 7.657)--
(2.291, 7.752)--
(2.468, 7.846)--
(2.625, 7.846)--
(2.899, 7.893)--
(3.095, 8.035)--
(3.350, 8.035)--
(3.586, 8.106)--
(3.860, 8.106)--
(4.135, 8.106)--
(4.371, 8.035)--
(4.606, 8.035)--
(4.881, 8.012)--
(5.155, 7.917)--
(5.391, 7.823)--
(5.665, 7.728)--
(5.960, 7.563)--
(6.175, 7.468)--
(6.332, 7.374)--
(6.528, 7.232)--
(6.725, 7.161)--
(6.882, 6.996)--
(7.117, 6.854)--
(7.333, 6.712)--
(7.509, 6.523)--
(7.666, 6.358)--
(7.902, 6.146)--
(8.098, 5.980)--
(8.274, 5.791)--
(8.451, 5.649)--
(8.647, 5.484)--
(8.882, 5.248)--
(9.196, 5.059)--
(9.392, 4.894)--
(9.628, 4.752)--
(9.824, 4.634)--
(10.118, 4.516)--
(10.452, 4.350)--
(10.785, 4.232)--
(11.001, 4.185)--
(11.315, 4.138)--
(11.648, 4.114)--
(12.002, 4.114)--
(12.257, 4.091)--
(12.610, 4.067)--
(12.825, 4.161)--
(13.081, 4.185)--
(13.316, 4.279)--
(13.492, 4.327)--
(13.689, 4.445)--
(13.826, 4.516)--
(14.022, 4.587)--
(14.159, 4.705)--
(14.316, 4.823)--
(14.532, 4.964)--
(14.669, 5.059)--
(14.866, 5.177)--
(15.062, 5.248)--
(15.278, 5.461)--
(15.474, 5.697)--
(15.650, 5.838)--
(15.847, 6.004)--
(16.043, 6.169)--
(16.258, 6.334)--
(16.415, 6.523)--
(16.592, 6.736)--
(16.788, 6.830)--
(17.063, 7.067)--
(17.357, 7.232)--
(17.573, 7.397)--
(17.808, 7.515)--
(18.063, 7.634)--
(18.358, 7.704)--
(18.573, 7.870)--
(18.887, 7.941)--
(19.142, 8.012)--
(19.358, 8.035)--
(19.574, 8.082)--
(19.770, 8.130);
draw(shift(1.8*up)*roundedpath(A,0.09),linewidth(1.5));
[/asy]
What is the total energy of the particle?
(A) $\text{-5 J}$
(B) $\text{0 J}$
(C) $\text{5 J}$
(D) $\text{10 J}$
(E) $\text{15 J}$
2010 F = Ma, 12
A ball with mass $m$ projected horizontally off the end of a table with an initial kinetic energy $K$. At a time $t$ after
it leaves the end of the table it has kinetic energy $3K$. What is $t$? Neglect air resistance.
(A) $(3/g)\sqrt{K/m}$
(B) $(2/g)\sqrt{K/m}$
(C) $(1/g)\sqrt{8K/m}$
(D) $(K/g)\sqrt{6/m}$
(E) $(2K/g)\sqrt{1/m}$
2010 Philippine MO, 3
Let $\mathbb{R}^*$ be the set of all real numbers, except $1$. Find all functions $f:\mathbb{R}^* \rightarrow \mathbb{R}$ that satisfy the functional equation $$x+f(x)+2f\left(\frac{x+2009}{x-1}\right)=2010$$.