Found problems: 31
2009 Philippine MO, 2
[b](a)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n + 1 = x^2$.
[b](b)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n = x^2 + 1$.
2009 F = Ma, 7
A bird is flying in a straight line initially at $\text{10 m/s}$. It uniformly increases its speed to $\text{15 m/s}$ while covering a distance of $\text{25 m}$. What is the magnitude of the acceleration of the bird?
(A) $\text{5.0 m/s}^2$
(B) $\text{2.5 m/s}^2$
(C) $\text{2.0 m/s}^2$
(D) $\text{0.5 m/s}^2$
(E) $\text{0.2 m/s}^2$
2009 F = Ma, 9
Through what net angle does the disk turn during the $3$ seconds?
(A) $\text{9 rad}$.
(B) $\text{8 rad}$.
(C) $\text{6 rad}$.
(D) $\text{4 rad}$.
(E) $\text{3 rad}$.
2009 F = Ma, 5
Three equal mass satellites $A$, $B$, and $C$ are in coplanar orbits around a planet as shown in the figure. The magnitudes of the angular momenta of the satellites as measured about the planet are $L_A$, $L_B$, and $L_C$. Which of the following statements is correct?
[asy]
// Code created by riben
size(250);
dotfactor=12;
draw(circle((0,0),1.5),linewidth(2));
draw(circle((0,0),6),dashdotted);
draw(circle((0,0),14),dashed);
draw(ellipse((4,0),10,8),linewidth(1));
pair A,B,C;
A=(-7,12.12);
B=(5,7.9);
C=(5.7,-1.87);
dot(A);
dot(B);
dot(C);
label("A",A,NW*1.5);
label("B",B,NW*1.5);
label("C",C,E*1.5);
filldraw((-1.500, 0.078)--
(-1.428, 0.080)--
(-1.337, 0.094)--
(-1.295, 0.157)--
(-1.246, 0.209)--
(-1.186, 0.227)--
(-1.143, 0.290)--
(-1.148, 0.357)--
(-1.135, 0.469)--
(-1.057, 0.505)--
(-0.996, 0.563)--
(-0.936, 0.526)--
(-0.852, 0.557)--
(-0.773, 0.587)--
(-0.772, 0.716)--
(-0.765, 0.828)--
(-0.781, 0.955)--
(-0.732, 1.035)--
(-0.648, 1.083)--
(-0.605, 1.162)--
(-0.604, 1.246)--
(-0.645, 1.295)--
(-0.736, 1.270)--
(-0.796, 1.229)--
(-0.851, 1.193)--
(-0.941, 1.135)--
(-1.014, 1.076)--
(-1.105, 0.995)--
(-1.154, 0.921)--
(-1.227, 0.841)--
(-1.288, 0.760)--
(-1.349, 0.669)--
(-1.398, 0.556)--
(-1.453, 0.465)--
(-1.485, 0.357)--
(-1.510, 0.239)--cycle,gray);
filldraw((-0.119, 1.245)--
(-0.130, 1.193)--
(-0.146, 1.095)--
(-0.202, 1.056)--
(-0.327, 1.033)--
(-0.262, 1.031)--
(-0.278, 0.979)--
(-0.193, 0.949)--
(-0.108, 0.943)--
(-0.013, 0.941)--
(0.032, 0.915)--
(0.026, 0.840)--
(0.015, 0.779)--
(0.019, 0.705)--
(0.074, 0.646)--
(0.113, 0.582)--
(0.162, 0.533)--
(0.167, 0.463)--
(0.241, 0.400)--
(0.311, 0.412)--
(0.416, 0.410)--
(0.465, 0.342)--
(0.541, 0.410)--
(0.611, 0.347)--
(0.679, 0.242)--
(0.728, 0.132)--
(0.732, 0.048)--
(0.671, -0.037)--
(0.615, -0.104)--
(0.540, -0.172)--
(0.409, -0.209)--
(0.324, -0.244)--
(0.253, -0.293)--
(0.188, -0.314)--
(0.162, -0.389)--
(0.181, -0.486)--
(0.270, -0.534)--
(0.340, -0.537)--
(0.380, -0.596)--
(0.424, -0.688)--
(0.418, -0.772)--
(0.352, -0.825)--
(0.281, -0.883)--
(0.241, -0.926)--
(0.145, -0.981)--
(0.044, -1.044)--
(-0.006, -1.107)--
(-0.007, -1.190)--
(0.077, -1.216)--
(0.162, -1.213)--
(0.253, -1.163)--
(0.323, -1.128)--
(0.404, -1.075)--
(0.510, -1.015)--
(0.605, -0.980)--
(0.671, -0.931)--
(0.731, -0.920)--
(0.817, -0.852)--
(0.898, -0.798)--
(0.963, -0.777)--
(0.964, -0.708)--
(1.024, -0.645)--
(1.025, -0.571)--
(0.976, -0.488)--
(0.912, -0.425)--
(0.878, -0.347)--
(0.823, -0.289)--
(0.779, -0.225)--
(0.744, -0.193)--
(0.756, -0.100)--
(0.816, -0.033)--
(0.837, 0.047)--
(0.838, 0.122)--
(0.824, 0.200)--
(0.800, 0.307)--
(0.796, 0.381)--
(0.872, 0.416)--
(0.967, 0.414)--
(1.016, 0.360)--
(1.096, 0.381)--
(1.117, 0.428)--
(1.058, 0.506)--
(0.998, 0.564)--
(0.954, 0.591)--
(0.914, 0.617)--
(0.860, 0.676)--
(0.800, 0.716)--
(0.751, 0.775)--
(0.757, 0.859)--
(0.797, 0.921)--
(0.823, 0.987)--
(0.889, 1.096)--
(0.850, 1.160)--
(0.780, 1.176)--
(0.700, 1.183)--
(0.645, 1.125)--
(0.579, 1.039)--
(0.518, 0.986)--
(0.438, 0.956)--
(0.343, 0.967)--
(0.289, 1.049)--
(0.249, 1.117)--
(0.195, 1.176)--
(0.125, 1.192)--
(0.030, 1.208)--
(-0.040, 1.220)--cycle,gray);
[/asy]
(A) $L_\text{A} > L_\text{B} > L_\text{C}$
(B) $L_\text{C} > L_\text{B} > L_\text{A}$
(C) $L_\text{B} > L_\text{C} > L_\text{A}$
(D) $L_\text{B} > L_\text{A} > L_\text{C}$
(E) The relationship between the magnitudes is different at various instants in time.
2009 F = Ma, 18
A simple pendulum of length $L$ is constructed from a point object of mass $m$ suspended by a massless string attached to a fixed pivot point. A small peg is placed a distance $2L/3$ directly below the fixed pivot point so that the pendulum would swing as shown in the figure below. The mass is displaced $5$ degrees from the vertical and released. How long does it take to return to its starting position?
[asy]
// Code by riben
size(275);
draw(circle((0,0),1),linewidth(2));
filldraw(circle((0,0),1),gray);
draw((0,0)--(0,-70.8));
draw(circle((0,-71.8),3));
filldraw(circle((0,-71.8),3),gray);
draw(circle((0,-45),1));
filldraw(circle((0,-45),1),gray);
filldraw(circle((15,-70),3),gray,linewidth(0.2));
filldraw(circle((-15,-67),3),gray,linewidth(0.2));
draw((0,0)--(14.5,-66.5),dashed);
draw((0,-45)--(-13,-65),dashed);
// Labels
label("Fixed Pivot Point",(0,0),4*E);
label("Small Peg",(0,-45),12*E);
label("Point Object of mass m",(0,-70),17*E);
draw((-40,1)--(-40,-76.8),EndArrow(size=5));
draw((-40,-76.8)--(-40,1),EndArrow(size=5));
label("L",(-40,-37.9),E*2);
[/asy]
(A) $\pi \sqrt{\frac{L}{g}} \left(1+\sqrt{\frac{2}{3}}\right)$
(B) $\pi \sqrt{\frac{L}{g}} \left(2+\frac{2}{\sqrt{3}}\right)$
(C) $\pi \sqrt{\frac{L}{g}} \left(1+\frac{1}{3}\right)$
(D) $\pi \sqrt{\frac{L}{g}} \left(1+\sqrt{3}\right)$
(E) $\pi \sqrt{\frac{L}{g}} \left(1+\frac{1}{\sqrt{3}}\right)$
2009 F = Ma, 17
You are given a standard kilogram mass and a tuning fork that is calibrated in Hz. You are also provided with a complete collection of laboratory equipment, but none of it is calibrated in SI units. You do not know the values of any fundamental constants. Which of the following quantities could you measure in SI units?
(A) The acceleration due to gravity.
(B) The speed of light in a vacuum.
(C) The density of room temperature water.
(D) The spring constant of a given spring.
(E) The air pressure in the room.