Olympiad problems

2011 F = Ma, 9

Tags: 2011 , Problem 9

A spring has an equilibrium length of $2.0$ meters and a spring constant of $10$ newtons/meter. Alice is pulling on one end of the spring with a force of $3.0$ newtons. Bob is pulling on the opposite end of the spring with a force of $3.0$ newtons, in the opposite direction. What is the resulting length of the spring? (A) $\text{1.7 m}$ (B) $\text{2.0 m}$ (C) $\text{2.3 m}$ (D) $\text{2.6 m}$ (E) $\text{8.0 m}$

2011 F = Ma, 22

Tags: 2011 , Problem 22

This graph depicts the torque output of a hypothetical gasoline engine as a function of rotation frequency. The engine is incapable of running outside of the graphed range. [asy] size(200); draw((0,0)--(10,0)--(10,7)--(0,7)--cycle); for (int i=1;i<10;++i) { draw((i,0)--(i,7),dashed+linewidth(0.5)); } for (int j=1;j<7;++j) { draw((0,j)--(10,j),dashed+linewidth(0.5)); } draw((0,0)--(0,-0.3)); draw((4,0)--(4,-0.3)); draw((8,0)--(8,-0.3)); draw((0,0)--(-0.3,0)); draw((0,2)--(-0.3,2)); draw((0,4)--(-0.3,4)); draw((0,6)--(-0.3,6)); label("0",(0,-0.5),S); label("1000",(4,-0.5),S); label("2000",(8,-0.5),S); label("0",(-0.5,0),W); label("10",(-0.5,2),W); label("20",(-0.5,4),W); label("30",(-0.5,6),W); label("I",(1,-1.5),S); label("II",(6,-1.5),S); label("III",(9,-1.5),S); label(scale(0.95)*"Engine Revolutions per Minute",(5,-3.5),N); label(scale(0.95)*rotate(90)*"Output Torque (Nm)",(-1.5,3),W); path A=(0.9,2.7)--(1.213, 2.713)-- (1.650, 2.853)-- (2.087, 3)-- (2.525, 3.183)-- (2.963, 3.471)-- (3.403, 3.888)-- (3.823, 4.346)-- (4.204, 4.808)-- (4.565, 5.277)-- (4.945, 5.719)-- (5.365, 6.101)-- (5.802, 6.298)-- (6.237, 6.275)-- (6.670, 6.007)-- (7.101, 5.600)-- (7.473, 5.229)-- (7.766, 4.808)-- (8.019, 4.374)-- (8.271, 3.894)-- (8.476, 3.445)-- (8.568, 2.874)-- (8.668, 2.325)-- (8.765, 1.897)-- (8.794, 1.479)--(8.9,1.2); draw(shift(0.1*right)*shift(0.2*down)*A,linewidth(3)); [/asy] At what engine RPM (revolutions per minute) does the engine produce maximum power? (A) $\text{I}$ (B) At some point between $\text{I}$ and $\text{II}$ (C) $\text{II}$ (D) At some point between $\text{II}$ and $\text{III}$ (E) $\text{III}$

2011 F = Ma, 4

Tags: 2011 , Problem 4

Rank the [i]magnitudes[/i] of the [i]distance[/i] traveled 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) $\text{I = II = III}$

2011 F = Ma, 10

Tags: 2011 , problem 10

Which of the following changes will result in an [i]increase[/i] in the period of a simple pendulum? (A) Decrease the length of the pendulum (B) Increase the mass of the pendulum (C) Increase the amplitude of the pendulum swing (D) Operate the pendulum in an elevator that is accelerating upward (E) Operate the pendulum in an elevator that is moving downward at constant speed.

2011 F = Ma, 13

Tags: 2011 , problem 13

The apparatus in the diagram consists of a solid cylinder of radius $\text{1}$ cm attached at the center to two disks of radius $\text{2}$ cm. It is placed on a surface where it can roll, but will not slip. A thread is wound around the central cylinder. When the thread is pulled at the angle $\theta = \text{90}^{\circ}$ to the horizontal (directly up), the apparatus rolls to the right. Which below is the largest value of $\theta$ for which it will not roll to the right when pulling on the thread? [asy] size(300); import roundedpath; path A=(1.148, -0.266)-- (-2.564, 0.563)-- (-2.804, 0.832)-- (-3.043, 0.988)-- (-3.283, 1.145)-- (-3.522, 1.414)-- (-3.642, 1.571)-- (-3.809, 1.862)-- (-4.025, 2.131)-- (-4.193, 2.355)-- (-4.264, 2.646)-- (-4.336, 2.982)-- (-4.456, 3.408)-- (-4.432, 3.654)-- (-4.456, 4.013)-- (-4.456, 4.326)-- (-4.408, 4.595)-- (-4.384, 4.752)-- (-4.288, 4.953)-- (-4.240, 5.110)-- (-4.121, 5.312)-- (-4.025, 5.513)-- (-3.929, 5.737)-- (-3.809, 5.872)-- (-3.642, 6.096)-- (-3.450, 6.297)-- (-3.354, 6.522)-- (-3.163, 6.656)-- (-2.899, 6.835)-- (-2.684, 6.925)-- (-2.420, 7.082)-- (-2.157, 7.216)-- (-1.822, 7.283)-- (-1.606, 7.395)-- (-1.199, 7.462)-- (-0.888, 7.440)-- (-0.601, 7.485)-- (2.920, 6.522)-- (2.752, 6.275)-- (2.584, 6.096)-- (2.393, 5.894)-- (2.273, 5.693)-- (2.081, 5.401)-- (1.986, 5.110)-- (1.818, 4.752)-- (1.626, 4.393)-- (1.483, 4.035)-- (1.363, 3.766)-- (1.291, 3.453)-- (1.243, 3.184)-- (1.195, 2.803)-- (1.052, 2.400)-- (1.052, 2.019)-- (1.028, 1.616)-- (1.076, 1.235)-- (1.100, 0.787)-- (1.171, 0.496)-- (1.219, 0.159)-- (1.387, -0.221)--cycle; path B=(10.298, 10.612)-- (10.030, 10.756)-- (9.671, 10.848)-- (9.313, 10.800)-- (8.977, 10.848)-- (8.484, 10.838)-- (8.215, 10.820)-- (7.767, 10.744)-- (7.409, 10.766)-- (6.804, 10.633)-- (6.468, 10.472)-- (6.109, 10.309)-- (5.706, 10.143)-- (5.303, 9.978)-- (5.056, 9.845)-- (4.720, 9.615)-- (4.384, 9.384)-- (4.071, 9.201)-- (3.869, 8.840)-- (3.645, 8.640)-- (3.354, 8.389)-- (3.175, 8.215)-- (2.973, 7.878)-- (2.682, 7.419)-- (2.480, 7.104)-- (2.346, 6.841)-- (2.211, 6.623)-- (2.099, 6.500)-- (1.920, 6.118)-- (1.808, 5.879)-- (1.674, 5.569)-- (1.584, 5.193)-- (1.517, 4.910)-- (1.405, 4.486)-- (1.315, 4.064)-- (1.203, 3.732)-- (1.181, 3.383)-- (1.181, 2.944)-- (1.114, 2.314)-- (1.136, 1.876)-- (1.203, 1.626)-- (1.203, 1.394)-- (1.203, 0.977)-- (1.248, 0.656)-- (1.383, 0.318)-- (1.539, -0.158)-- (1.539, -0.412)-- (1.763, -0.721)-- (1.853, -1.086)-- (1.920, -1.405)-- (2.144, -1.645)-- (2.346, -1.863)-- (2.503, -2.246)-- (2.659, -2.560)-- (2.839, -2.826)-- (3.152, -3.013)-- (3.466, -3.316)-- (3.668, -3.557)-- (3.959, -3.816)-- (4.340, -4.091)-- (4.676, -4.323)-- (4.900, -4.424)-- (5.303, -4.536)-- (5.616, -4.770)-- (5.953, -4.863)-- (6.468, -4.922)-- (6.938, -5.029)-- (7.207, -4.826)-- (6.893, -4.615)-- (6.692, -4.397)-- (6.468, -4.297)-- (6.221, -4.035)-- (6.042, -3.885)-- (5.773, -3.764)-- (5.482, -3.460)-- (5.280, -3.241)-- (5.034, -2.911)-- (4.922, -2.664)-- (4.676, -2.310)-- (4.564, -1.970)-- (4.429, -1.701)-- (4.250, -1.412)-- (4.048, -1.125)-- (3.936, -0.692)-- (3.824, -0.306)-- (3.780, -0.031)-- (3.668, 0.285)-- (3.600, 0.582)-- (3.600, 0.859)-- (3.600, 1.253)-- (3.578, 1.645)-- (3.578, 2.039)-- (3.556, 2.407)-- (3.556, 2.778)-- (3.556, 3.032)-- (3.623, 3.407)-- (3.690, 3.713)-- (3.712, 4.061)-- (3.914, 4.445)-- (4.004, 4.752)-- (4.138, 5.062)-- (4.250, 5.463)-- (4.407, 5.774)-- (4.586, 6.041)-- (4.743, 6.329)-- (4.877, 6.570)-- (5.101, 6.885)-- (5.325, 7.132)-- (5.549, 7.494)-- (5.796, 7.765)-- (5.975, 7.985)-- (6.266, 8.259)-- (6.468, 8.412)-- (6.826, 8.713)-- (7.095, 8.917)-- (7.409, 9.169)-- (7.655, 9.255)-- (7.969, 9.438)-- (8.439, 9.608)-- (8.685, 9.856)-- (9.044, 10.042)-- (9.358, 10.109)-- (9.716, 10.249)-- (10.052, 10.317)-- (10.276, 10.425)--cycle; filldraw(roundedpath(A,0.08),gray); filldraw(roundedpath(B,0.09),lightgray); path C=(3.501, 2.540)-- (3.581, 1.437)-- (3.682, 0.389)-- (3.948, -0.294)-- (4.182, -1.097)-- (4.416, -1.892)-- (4.816, -2.628)-- (5.417, -3.408)-- (6.117, -4.040)-- (6.851, -4.565)-- (7.585, -4.989)-- (8.320, -5.216)-- (9.054, -5.396)-- (9.788, -5.570)-- (10.522, -5.530)-- (11.256, -5.459)-- (11.990, -5.348)-- (12.724, -5.048)-- (13.458, -4.751)-- (14.192, -4.320)-- (14.926, -3.785)-- (15.660, -3.141)-- (16.361, -2.375)-- (16.861, -1.500)-- (17.262, -0.698)-- (17.629, 0.196)-- (17.849, 1.069)-- (18.029, 2.015)-- (17.983, 3.064)-- (17.896, 4.377)-- (17.646, 5.110)-- (17.462, 5.960)-- (17.195, 6.611)-- (16.761, 7.339)-- (16.161, 8.007)-- (15.460, 8.669)-- (14.726, 9.153)-- (13.992, 9.709)-- (13.258, 10.055)-- (12.524, 10.185)-- (11.790, 10.409)-- (11.056, 10.429)-- (10.322, 10.298)-- (9.587, 10.105)-- (8.853, 9.855)-- (8.106, 9.497)-- (7.385, 9.140)-- (6.651, 8.590)-- (5.917, 7.995)-- (5.250, 7.164)-- (4.649, 6.335)-- (4.215, 5.576)-- (3.982, 4.552)-- (3.748, 3.692); draw(roundedpath(C,0.1)); path D=(4.583, 10.037)-- (3.966, 10.689)-- (3.349, 11.313)-- (2.571, 11.843)-- (1.713, 12.284)-- (0.560, 12.678)-- (-0.701, 12.899)-- (-1.988, 12.841)-- (-3.195, 12.428)-- (-4.402, 11.877)-- (-5.421, 11.199)-- (-6.198, 10.371)-- (-6.923, 9.408)-- (-7.566, 8.368)-- (-8.049, 7.116)-- (-8.371, 5.709)-- (-8.398, 4.571)-- (-8.371, 3.409)-- (-8.210, 2.284)-- (-7.754, 1.067)-- (-7.218, 0.105)-- (-6.574, -0.767)-- (-5.743, -1.598)-- (-4.965, -2.156)-- (-4.321, -2.529)-- (-3.517, -2.808)-- (-2.685, -2.946)-- (-1.612, -3.124)-- (-0.647, -3.032)-- (0.211, -2.836)-- (1.150, -2.496)-- (2.008, -2.051); draw(roundedpath(D,0.2)); path E=(-10.886, 4.077)-- (-10.800, 4.972)-- (-10.766, 5.931)-- (-10.549, 6.963)-- (-10.299, 7.943)-- (-9.899, 8.672)-- (-9.532, 9.427)-- (-9.065, 10.225)-- (-8.364, 11.002)-- (-7.630, 11.672)-- (-6.896, 12.181)-- (-6.162, 12.553)-- (-5.428, 12.890)-- (-4.694, 13.096)-- (-3.959, 13.170)-- (-3.225, 13.200)-- (-2.491, 12.999)-- (-2.251, 12.835)-- (-1.857, 12.981)-- (-3.025, 12.367)-- (-3.766, 12.037)-- (-4.493, 11.752)-- (-5.227, 11.208)-- (-5.961, 10.692)-- (-6.562, 9.934)-- (-7.063, 9.078)-- (-7.563, 8.330)-- (-7.930, 7.560)-- (-8.164, 6.642)-- (-8.297, 5.656)-- (-8.431, 4.578)-- (-8.447, 3.448)-- (-8.230, 2.649)-- (-7.997, 1.814)-- (-7.663, 1.021)-- (-7.263, 0.289)-- (-6.796, -0.519)-- (-6.162, -1.144)-- (-5.503, -1.666)-- (-5.733, -2.477)-- (-5.038, -2.491)-- (-6.495, -2.251)-- (-7.229, -1.924)-- (-7.963, -1.594)-- (-8.664, -0.964)-- (-9.231, -0.218)-- (-9.699, 0.534)-- (-10.132, 1.298)-- (-10.466, 2.124)-- (-10.700, 3.002)--cycle; filldraw(roundedpath(E,0.35),lightgray); draw((0,0)--(-11.830, 8.804),EndArrow(size=13)); picture pic; draw(pic,circle((0,0),3)); draw(pic,circle((0,0),7)); draw(pic,(-2.5,-1.66)--(-8,9),EndArrow(size=9)); draw(pic,(-2.5,-1.66)--(-7.5,-1.66),dashed); label(pic,scale(0.75)*"$\theta$",(-4.5,-1.5),N); add(shift(35*right)*shift(3*up)*pic); [/asy] (A) $\theta = \text{15}^{\circ}$ (B) $\theta = \text{30}^{\circ}$ (C) $\theta = \text{45}^{\circ}$ (D) $\theta = \text{60}^{\circ}$ (E) None, the apparatus will always roll to the right

2011 F = Ma, 5

Tags: 2011 , Problem 5

A crude approximation is that the Earth travels in a circular orbit about the Sun at constant speed, at a distance of $\text{150,000,000 km}$ from the Sun. Which of the following is the closest for the acceleration of the Earth in this orbit? (A) $\text{exactly 0 m/s}^2$ (B) $\text{0.006 m/s}^2$ (C) $\text{0.6 m/s}^2$ (D) $\text{6 m/s}^2$ (E) $\text{10 m/s}^2$

2011 F = Ma, 7

Tags: 2011 , Problem 7

An ice skater can rotate about a vertical axis with an angular velocity $\omega_\text{0}$ by holding her arms straight out. She can then pull in her arms close to her body so that her angular velocity changes to $2\omega_\text{0}$, without the application of any external torque. What is the ratio of her final rotational kinetic energy to her initial rotational kinetic energy? (A)$\sqrt{2}$ (B) $2$ (C) $2\sqrt{2}$ (D) $4$ (E) $8$

2011 F = Ma, 23

Tags: 2011 , problem 23

A particle is launched from the surface of a uniform, stationary spherical planet at an angle to the vertical. The particle travels in the absence of air resistance and eventually falls back onto the planet. Spaceman Fred describes the path of the particle as a parabola using the laws of projectile motion. Spacewoman Kate recalls from Kepler’s laws that every bound orbit around a point mass is an ellipse (or circle), and that the gravitation due to a uniform sphere is identical to that of a point mass. Which of the following best explains the discrepancy? (A) Because the experiment takes place very close to the surface of the sphere, it is no longer valid to replace the sphere with a point mass. (B) Because the particle strikes the ground, it is not in orbit of the planet and therefore can follow a nonelliptical path. (C) Kate disregarded the fact that motions around a point mass may also be parabolas or hyperbolas. (D) Kepler’s laws only hold in the limit of large orbits. (E) The path is an ellipse, but is very close to a parabola due to the short length of the flight relative to the distance from the center of the planet.

2011 F = Ma, 2

Tags: 2011 , Problem 2

Rank the [i]magnitudes[/i] of the average acceleration 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) $\text{I = II = III}$

2011 F = Ma, 20

Tags: 2011 , problem 20

What is the maximum distance between the particle and the origin? (A) $\text{2.00 m}$ (B) $\text{2.50 m}$ (C) $\text{3.50 m}$ (D) $\text{5.00 m}$ (E) $\text{7.00 m}$

2011 F = Ma, 14

Tags: 2011 , Problem 14

You have $\text{5}$ different strings with weights tied at various point, all hanging from the ceiling, and reaching down to the floor. The string is released at the top, allowing the weights to fall. Which one will create a regular, uniform beating sound as the weights hit the floor? [asy] size(300); // (A) bar picture bar; draw(bar,(0,0)--(0,42)); for (int i=0;i<43;i+=2) { draw(bar,(-2,i)--(-3,i)); } add(bar); picture ball; filldraw(ball,circle((0,0),0.5),gray); add(ball); add(shift(12*up)*ball); add(shift(22*up)*ball); add(shift(30*up)*ball); add(shift(36*up)*ball); add(shift(40*up)*ball); add(shift(42*up)*ball); label(scale(0.75)*"(A)",(-1,0),2*S); // (B) bar add(shift(15*right)*bar); add(shift(15*right)*ball); add(shift(6*up)*shift(15*right)*ball); add(shift(12*up)*shift(15*right)*ball); add(shift(18*up)*shift(15*right)*ball); add(shift(24*up)*shift(15*right)*ball); add(shift(30*up)*shift(15*right)*ball); add(shift(36*up)*shift(15*right)*ball); add(shift(42*up)*shift(15*right)*ball); label(scale(0.75)*"(B)",(14,0),2*S); // (C) bar add(shift(30*right)*bar); add(shift(30*right)*ball); add(shift(2*up)*shift(30*right)*ball); add(shift(6*up)*shift(30*right)*ball); add(shift(12*up)*shift(30*right)*ball); add(shift(20*up)*shift(30*right)*ball); add(shift(30*up)*shift(30*right)*ball); add(shift(42*up)*shift(30*right)*ball); label(scale(0.75)*"(C)",(29,0),2*S); // (D) bar add(shift(45*right)*bar); add(shift(45*right)*ball); add(shift(2*up)*shift(45*right)*ball); add(shift(8*up)*shift(45*right)*ball); add(shift(18*up)*shift(45*right)*ball); add(shift(32*up)*shift(45*right)*ball); label(scale(0.75)*"(D)",(44,0),2*S); // (E) bar add(shift(60*right)*bar); add(shift(60*right)*ball); add(shift(2*up)*shift(60*right)*ball); add(shift(10*up)*shift(60*right)*ball); add(shift(28*up)*shift(60*right)*ball); label(scale(0.75)*"(E)",(59,0),2*S); [/asy]

2011 India Regional Mathematical Olympiad, 6

Tags: inequalities , inequalities proposed , india , 2011

Find the largest real constant $\lambda$ such that \[\frac{\lambda abc}{a+b+c}\leq (a+b)^2+(a+b+4c)^2\] For all positive real numbers $a,b,c.$

2011 F = Ma, 25

Tags: 2011 , problem 25

A hollow cylinder with a very thin wall (like a toilet paper tube) and a block are placed at rest at the top of a plane with inclination $\theta$ above the horizontal. The cylinder rolls down the plane without slipping and the block slides down the plane; it is found that both objects reach the bottom of the plane simultaneously. What is the coefficient of kinetic friction between the block and the plane? (A) $0$ (B) $\frac{1}{3}\tan \theta$ (C) $\frac{1}{2}\tan \theta$ (D) $\frac{2}{3}\tan \theta$ (E) $\tan \theta$

2011 F = Ma, 6

Tags: 2011 , Problem 6

A child is sliding out of control with velocity $v_\text{c}$ across a frozen lake. He runs head-on into another child, initially at rest, with $3$ times the mass of the first child, who holds on so that the two now slide together. What is the velocity of the couple after the collision? (A) $2v_\text{c}$ (B) $v_\text{c}$ (C) $v_\text{c}/2$ (D) $v_\text{c}/3$ (E) $v_\text{c}/4$

2011 F = Ma, 12

Tags: 2011 , Problem 12

You are given a large collection of identical heavy balls and lightweight rods. When two balls are placed at the ends of one rod and interact through their mutual gravitational attraction (as is shown on the left), the compressive force in the rod is $F$. Next, three balls and three rods are placed at the vertexes and edges of an equilateral triangle (as is shown on the right). What is the compressive force in each rod in the latter case? [asy] size(300); real x=-25; draw((x,-8)--(x,8),linewidth(6)); filldraw(Circle((x,8),2.5),grey); filldraw(Circle((x,-8),2.5),grey); draw((0,-8)--(0,8)--(8*sqrt(3),0)--cycle,linewidth(6)); filldraw(Circle((0,8),2.5),grey); filldraw(Circle((0,-8),2.5),grey); filldraw(Circle((8*sqrt(3),0),2.5),grey); [/asy] (A) $\frac{1}{\sqrt{3}}F$ (B) $\frac{\sqrt{3}}{2}F$ (C) $F$ (D) $\sqrt{3}F$ (E) $2F$

2011 F = Ma, 15

Tags: 2011 , Problem 15

A vertical mass-spring oscillator is displaced $\text{2.0 cm}$ from equilibrium. The $\text{100 g}$ mass passes through the equilibrium point with a speed of $\text{0.75 m/s}$. What is the spring constant of the spring? (A) $\text{90 N/m}$ (B) $\text{100 N/m}$ (C) $\text{110 N/m}$ (D) $\text{140 N/m}$ (E) $\text{160 N/m}$

2011 ELMO Shortlist, 3

Tags: Pascal's Triangle , combinatorics proposed , combinatorics , Elmo , 2011 , #2 , Lucas' Theorem

Wanda the Worm likes to eat Pascal's triangle. One day, she starts at the top of the triangle and eats $\textstyle\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a,b,c$ such that $a+b=c$, prove that it is possible for her to eat 100,000 numbers in the first 2011 rows given that she is not restricted to traveling only in the first 2011 rows. (Here, the $n+1$st row of Pascal's triangle consists of entries of the form $\textstyle\binom{n}{k}$ for integers $0\le k\le n$. Thus, the entry $\textstyle\binom{n}{k}$ is considered adjacent to the entries $\textstyle\binom{n-1}{k-1}$, $\textstyle\binom{n-1}{k}$, $\textstyle\binom{n}{k-1}$, $\textstyle\binom{n}{k+1}$, $\textstyle\binom{n+1}{k}$, $\textstyle\binom{n+1}{k+1}$.) [i]Linus Hamilton.[/i]

2011 ELMO Shortlist, 3

Tags: Pascal's Triangle , combinatorics proposed , combinatorics , Elmo , 2011 , #2 , Lucas' Theorem

Wanda the Worm likes to eat Pascal's triangle. One day, she starts at the top of the triangle and eats $\textstyle\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a,b,c$ such that $a+b=c$, prove that it is possible for her to eat 100,000 numbers in the first 2011 rows given that she is not restricted to traveling only in the first 2011 rows. (Here, the $n+1$st row of Pascal's triangle consists of entries of the form $\textstyle\binom{n}{k}$ for integers $0\le k\le n$. Thus, the entry $\textstyle\binom{n}{k}$ is considered adjacent to the entries $\textstyle\binom{n-1}{k-1}$, $\textstyle\binom{n-1}{k}$, $\textstyle\binom{n}{k-1}$, $\textstyle\binom{n}{k+1}$, $\textstyle\binom{n+1}{k}$, $\textstyle\binom{n+1}{k+1}$.) [i]Linus Hamilton.[/i]

2011 F = Ma, 3

Tags: 2011 , Problem 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) $\text{I = II = III}$

2011 ELMO Problems, 2

Tags: Pascal's Triangle , combinatorics proposed , combinatorics , Elmo , 2011 , #2 , Lucas' Theorem

Wanda the Worm likes to eat Pascal's triangle. One day, she starts at the top of the triangle and eats $\textstyle\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a,b,c$ such that $a+b=c$, prove that it is possible for her to eat 100,000 numbers in the first 2011 rows given that she is not restricted to traveling only in the first 2011 rows. (Here, the $n+1$st row of Pascal's triangle consists of entries of the form $\textstyle\binom{n}{k}$ for integers $0\le k\le n$. Thus, the entry $\textstyle\binom{n}{k}$ is considered adjacent to the entries $\textstyle\binom{n-1}{k-1}$, $\textstyle\binom{n-1}{k}$, $\textstyle\binom{n}{k-1}$, $\textstyle\binom{n}{k+1}$, $\textstyle\binom{n+1}{k}$, $\textstyle\binom{n+1}{k+1}$.) [i]Linus Hamilton.[/i]

2011 F = Ma, 11

Tags: 2011 , Problem 11

A large metal cylindrical cup floats in a rectangular tub half-filled with water. The tap is placed over the cup and turned on, releasing water at a constant rate. Eventually the cup sinks to the bottom and is completely submerged. Which of the following five graphs could represent the water level in the sink as a function of time? [asy] size(450); picture pic; draw(pic,(0,0)--(10,0)--(10,7)--(0,7)--cycle); for (int i=1;i<10;++i) { draw(pic,(i,0)--(i,7),dashed+linewidth(0.4)); } for (int j=1;j<7;++j) { draw(pic,(0,j)--(10,j),dashed+linewidth(0.4)); } label(pic,scale(1.2)*"time",(5.5,-0.5),S); label(pic,rotate(90)*scale(1.2)*"water level",(-0.5,2.5),W); add(pic); path A=(0,1)--(10,6); draw(A,linewidth(2)); label("(A)",(4.5,-1.5),1.5*S); picture pic2=shift(13*right)*pic; add(pic2); path B=(0,1)--(4,4)--(10,6); draw(shift(13*right)*B,linewidth(2)); label("(B)",(17.5,-1.5),1.5*S); picture pic3=shift(26*right)*pic; add(pic3); path C=(0,1)--(4,3)--(4,2)--(10,5); draw(shift(26*right)*C,linewidth(2)); label("(C)",(30.5,-1.5),1.5*S); picture pic4=shift(13*down)*pic; add(pic4); path D=(0,1)--(4,3)--(4,4)--(10,7); draw(shift(13*down)*D,linewidth(2)); label("(D)",(4.5,-14.5),1.5*S); picture pic5=shift(13*down)*shift(13*right)*pic; add(pic5); path E=(0,1)--(4,3)--(4,2)--(10,4); draw(shift(13*down)*shift(13*right)*E,linewidth(2)); label("(E)",(17.5,-14.5),1.5*S); [/asy]

2011 F = Ma, 19

Tags: 2011 , problem 19

After how much time will the particle first return to the origin? (A) $\text{0.785 s}$ (B) $\text{1.26 s}$ (C) $\text{1.57 s}$ (D) $\text{2.00 s}$ (E) $\text{3.14 s}$

2011 F = Ma, 8

Tags: 2011 , Problem 8

When a block of wood with a weight of $\text{30 N}$ is completely submerged under water the buoyant force on the block of wood from the water is $\text{50 N}$. When the block is released it floats at the surface. What fraction of the block will then be visible above the surface of the water when the block is floating? (A) $1/15$ (B) $1/5$ (C) $1/3$ (D) $2/5$ (E) $3/5$

2011 F = Ma, 24

Tags: 2011 , problem 24

A turntable is supported on a Teflon ring of inner radius $R$ and outer radius $R+\sigma$ ($\sigma<<R$), as shown in the diagram. To rotate the turntable at a constant rate, power must be supplied to overcome friction. The manufacturer of the turntable wishes to reduce the power required without changing the rotation rate, the weight of the turntable, or the coefficient of friction of the Teflon surface. Engineers propose two solutions: increasing the width of the bearing (increasing $\sigma$), or increasing the radius (increasing $R$). What are the effects of these proposed changes? [asy] size(200); draw(circle((0,0),5.5),linewidth(2)); draw(circle((0,0),7),linewidth(2)); path arrow1 = (0,0)--5*dir(50); draw(arrow1,EndArrow); label("R",arrow1,NW); draw((3,0)--(5.5,0),EndArrow); path arrow2 = ((10,0)--(7,0)); draw(arrow2,EndArrow); label("$\delta$",arrow2,N); [/asy] (A) Increasing $\sigma$ has no significant effect on the required power; increasing $R$ increases the required power. (B) Increasing $\sigma$ has no significant effect on the required power; increasing $R$ decreases the required power. (C) Increasing $\sigma$ increases the required power; increasing $R$ has no significant effect on the required power. (D) Increasing $\sigma$ decreases the required power; increasing $R$ has no significant effect on the required power. (E) Neither change has a significant effect on the required power.

2011 F = Ma, 16

Tags: 2011 , problem 16

What magnitude force does Jonathan need to exert on the physics book to keep the rope from slipping? (A) $Mg$ (B) $\mu_k Mg$ (C) $\mu_k Mg/\mu_s$ (D) $(\mu_s + \mu_k)Mg$ (E) $Mg/\mu_s$