Olympiad problems

2009 F = Ma, 25

Tags: 2009 , problem 25

Two discs are mounted on thin, lightweight rods oriented through their centers and normal to the discs. These axles are constrained to be vertical at all times, and the discs can pivot frictionlessly on the rods. The discs have identical thickness and are made of the same material, but have differing radii $r_\text{1}$ and $r_\text{2}$. The discs are given angular velocities of magnitudes $\omega_\text{1}$ and $\omega_\text{2}$, respectively, and brought into contact at their edges. After the discs interact via friction it is found that both discs come exactly to a halt. Which of the following must hold? Ignore effects associated with the vertical rods. [asy] //Code by riben, Improved by CalTech_2023 // Solids import solids; //bigger cylinder draw(shift(0,0,-1)*scale(0.1,0.1,0.59)*unitcylinder,surfacepen=white,black); draw(shift(0,0,-0.1)*unitdisk, surfacepen=black); draw(unitdisk, surfacepen=white,black); draw(scale(0.1,0.1,1)*unitcylinder,surfacepen=white,black); //smaller cylinder draw(rotate(5,X)*shift(-2,3.2,-1)*scale(0.1,0.1,0.6)*unitcylinder,surfacepen=white,black); draw(rotate(4,X)*scale(0.5,0.5,1)*shift(1,8,0.55)*unitdisk, surfacepen=black); draw(rotate(4,X)*scale(0.5,0.5,1)*shift(1,8,0.6)*unitdisk, surfacepen=white,black); draw(rotate(5,X)*shift(-2,3.2,-0.2)*scale(0.1,0.1,1)*unitcylinder,surfacepen=white,black); // Lines draw((0,-2)--(1,-2),Arrows(size=5)); draw((4,-2)--(4.7,-2),Arrows(size=5)); // Labels label("r1",(0.5,-2),S); label("r2",(4.35,-2),S); // Curved Lines path A=(-0.694, 0.897)-- (-0.711, 0.890)-- (-0.742, 0.886)-- (-0.764, 0.882)-- (-0.790, 0.873)-- (-0.815, 0.869)-- (-0.849, 0.867)-- (-0.852, 0.851)-- (-0.884, 0.844)-- (-0.895, 0.837)-- (-0.904, 0.824)-- (-0.879, 0.800)-- (-0.841, 0.784)-- (-0.805, 0.772)-- (-0.762, 0.762)-- (-0.720, 0.747)-- (-0.671, 0.737)-- (-0.626, 0.728)-- (-0.591, 0.720)-- (-0.556, 0.715)-- (-0.504, 0.705)-- (-0.464, 0.700)-- (-0.433, 0.688)-- (-0.407, 0.683)-- (-0.371, 0.685)-- (-0.316, 0.673)-- (-0.271, 0.672)-- (-0.234, 0.667)-- (-0.192, 0.664)-- (-0.156, 0.663)-- (-0.114, 0.663)-- (-0.070, 0.660)-- (-0.033, 0.662)-- (0.000, 0.663)-- (0.036, 0.663)-- (0.067, 0.665)-- (0.095, 0.667)-- (0.125, 0.666)-- (0.150, 0.673)-- (0.187, 0.675)-- (0.223, 0.676)-- (0.245, 0.681)-- (0.274, 0.687)-- (0.300, 0.696)-- (0.327, 0.707)-- (0.357, 0.709)-- (0.381, 0.718)-- (0.408, 0.731)-- (0.443, 0.740)-- (0.455, 0.754)-- (0.458, 0.765)-- (0.453, 0.781)-- (0.438, 0.795)-- (0.411, 0.809)-- (0.383, 0.817)-- (0.344, 0.829)-- (0.292, 0.839)-- (0.254, 0.846)-- (0.216, 0.851)-- (0.182, 0.857)-- (0.153, 0.862)-- (0.124, 0.867); draw(shift(0.2,0)*A,EndArrow(size=5)); path B=(2.804, 0.844)-- (2.790, 0.838)-- (2.775, 0.838)-- (2.758, 0.831)-- (2.740, 0.831)-- (2.709, 0.827)-- (2.688, 0.825)-- (2.680, 0.818)-- (2.660, 0.810)-- (2.639, 0.810)-- (2.628, 0.803)-- (2.618, 0.799)-- (2.604, 0.790)-- (2.598, 0.778)-- (2.596, 0.769)-- (2.606, 0.757)-- (2.630, 0.748)-- (2.666, 0.733)-- (2.696, 0.721)-- (2.744, 0.707)-- (2.773, 0.702)-- (2.808, 0.697)-- (2.841, 0.683)-- (2.867, 0.680)-- (2.912, 0.668)-- (2.945, 0.665)-- (2.973, 0.655)-- (3.010, 0.648)-- (3.040, 0.647)-- (3.069, 0.642)-- (3.102, 0.640)-- (3.136, 0.632)-- (3.168, 0.629)-- (3.189, 0.627)-- (3.232, 0.619)-- (3.254, 0.624)-- (3.281, 0.621)-- (3.328, 0.618)-- (3.355, 0.618)-- (3.397, 0.617)-- (3.442, 0.616)-- (3.468, 0.611)-- (3.528, 0.611)-- (3.575, 0.617)-- (3.611, 0.619)-- (3.634, 0.625)-- (3.666, 0.622)-- (3.706, 0.626)-- (3.742, 0.635)-- (3.772, 0.635)-- (3.794, 0.641)-- (3.813, 0.646)-- (3.837, 0.654)-- (3.868, 0.659)-- (3.886, 0.672)-- (3.903, 0.681)-- (3.917, 0.688)-- (3.931, 0.697)-- (3.943, 0.711)-- (3.951, 0.720)-- (3.948, 0.731)-- (3.924, 0.745)-- (3.900, 0.757)-- (3.874, 0.774)-- (3.851, 0.779)-- (3.821, 0.779)-- (3.786, 0.786)-- (3.754, 0.792)-- (3.726, 0.797)-- (3.677, 0.806)-- (3.642, 0.812); draw(shift(0.7,0)*B,EndArrow(size=5)); [/asy] (A) $\omega_\text{1}^2r_\text{1}=\omega_\text{2}^2r_\text{2}$ (B) $\omega_\text{1}r_\text{1}=\omega_\text{2}r_\text{2}$ (C) $\omega_\text{1}r_\text{1}^2=\omega_\text{2}r_\text{2}^2$ (D) $\omega_\text{1}r_\text{1}^3=\omega_\text{2}r_\text{2}^3$ (E) $\omega_\text{1}r_\text{1}^4=\omega_\text{2}r_\text{2}^4$

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) $\tan \theta$

2008 F = Ma, 25

Tags: 2008 , problem 25

Two satellites are launched at a distance $R$ from a planet of negligible radius. Both satellites are launched in the tangential direction. The first satellite launches correctly at a speed $v_\text{0}$ and enters a circular orbit. The second satellite, however, is launched at a speed $\frac{1}{2}v_\text{0}$. What is the minimum distance between the second satellite and the planet over the course of its orbit? (a) $\frac{1}{\sqrt{2}}R$ (b) $\frac{1}{2}R$ (c) $\frac{1}{3}R$ (d) $\frac{1}{4}R$ (e) $\frac{1}{7}R$

2019 AMC 8, 25

Tags: 2019 AMC 8 , problem 25

Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the people has at least 2 apples? $\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380$

2010 F = Ma, 25

Tags: 2010 , problem 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) $2v_\text{0}$