Olympiad problems

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

2009 F = Ma, 13

Tags: 2009 , problem 13

Lucy (mass $\text{33.1 kg}$), Henry (mass $\text{63.7 kg}$), and Mary (mass $\text{24.3 kg}$) sit on a lightweight seesaw at evenly spaced $\text{2.74 m}$ intervals (in the order in which they are listed; Henry is between Lucy and Mary) so that the seesaw balances. Who exerts the most torque (in terms of magnitude) on the seesaw? Ignore the mass of the seesaw. (A) Henry (B) Lucy (C) Mary (D) They all exert the same torque. (E) There is not enough information to answer the question.

2018 AIME Problems, 13

Tags: AMC , AIME , AIME II , Dice , probability , problem 13

Misha rolls a standard, fair six-sided die until she rolls $1$-$2$-$3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2008 F = Ma, 13

Tags: 2008 , problem 13

A mass is attached to the wall by a spring of constant $k$. When the spring is at its natural length, the mass is given a certain initial velocity, resulting in oscillations of amplitude $A$. If the spring is replaced by a spring of constant $2k$, and the mass is given the same initial velocity, what is the amplitude of the resulting oscillation? (a) $\frac{1}{2}A$ (b) $\frac{1}{\sqrt{2}}A$ (c) $\sqrt{2}A$ (d) $2A$ (e) $4A$

2010 F = Ma, 13

Tags: 2010 , problem 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) $\frac{7}{5}h$