Olympiad problems

2009 F = Ma, 22

Tags: 2009 , Problem 22

Determine the period of orbit for the star of mass $3M$. (A) $\pi \sqrt{\frac{d^3}{GM}}$ (B) $\frac{3\pi}{4}\sqrt{\frac{d^3}{GM}}$ (C) $\pi \sqrt{\frac{d^3}{3GM}}$ (D) $2\pi \sqrt{\frac{d^3}{GM}}$ (E) $\frac{\pi}{4} \sqrt{\frac{d^3}{GM}}$

2009 Bosnia Herzegovina Team Selection Test, 1

Tags: induction , combinatorics unsolved , combinatorics , Game Theory , Bosnia , 2009

Given an $1$ x $n$ table ($n\geq 2$), two players alternate the moves in which they write the signs + and - in the cells of the table. The first player always writes +, while the second always writes -. It is not allowed for two equal signs to appear in the adjacent cells. The player who can’t make a move loses the game. Which of the players has a winning strategy?

2009 F = Ma, 20

Tags: 2009 , problem 20

Consider a completely inelastic collision between two lumps of space goo. Lump 1 has mass $m$ and originally moves directly north with a speed $v_\text{0}$. Lump 2 has mass $3m$ and originally moves directly east with speed $v_\text{0}/2$. What is the final speed of the masses after the collision? Ignore gravity, and assume the two lumps stick together after the collision. (A) $7/16 \ v_\text{0}$ (B) $\sqrt{5}/8 \ v_\text{0}$ (C) $\sqrt{13}/8 \ v_\text{0}$ (D) $5/8 \ v_\text{0}$ (E) $\sqrt{13/8} \ v_\text{0}$

2009 Philippine MO, 4

Tags: PMO , 2009 , algebra , Inequality , minimum value

Let $k$ be a positive real number such that $$\frac{1}{k+a} + \frac{1}{k+b} + \frac{1}{k+c} \leq 1$$ for any positive positive real numbers $a$, $b$ and $c$ with $abc = 1$. Find the minimum value of $k$.

2009 Philippine MO, 5

Tags: geometry , PMO , 2009 , concurrency , perpendicular bisector

Segments $AC$ and $BD$ intersect at point $P$ such that $PA = PD$ and $PB = PC$. Let $E$ be the foot of the perpendicular from $P$ to the line $CD$. Prove that the line $PE$ and the perpendicular bisectors of the segments $PA$ and $PB$ are concurrent.

2009 F = Ma, 12

Tags: 2009 , Problem 12

Batman, who has a mass of $\text{M = 100 kg}$, climbs to the roof of a $\text{30 m}$ building and then lowers one end of a massless rope to his sidekick Robin. Batman then pulls Robin, who has a mass of $\text{m = 75 kg}$, up the roof of the building. Approximately how much total work has Batman done after Robin is on the roof? (A) $\text{60 J}$ (B) $\text{7} \times \text{10}^3 \text{J}$ (C) $\text{5} \times \text{10}^4 \text{J}$ (D) $\text{600 J}$ (E) $\text{3} \times \text{10}^4 \text{J}$

2009 F = Ma, 11

Tags: 2009 , Problem 11

A $\text{2.25 kg}$ mass undergoes an acceleration as shown below. How much work is done on the mass? [asy] // Code by riben size(350); // Axes draw((0,0)--(12,0),lightgray); draw((0,-3)--(0,5)); // Tick Marks draw((2,0)--(2,-0.2)); label("2",(2,-0.2),S*2); draw((4,0)--(4,-0.2)); label("4",(4,-0.2),S*2); draw((6,0)--(6,-0.2)); label("6",(6,-0.2),S*2); draw((8,0)--(8,-0.2)); label("8",(8,-0.2),S*2); draw((10,0)--(10,-0.2)); label("10",(10,-0.2),S*2); draw((12,0)--(12,-0.2)); label("12",(12,-0.2),S*2); draw((0,-2)--(-0.2,-2)); label("-2",(-0.2,-2),W); draw((0,0)--(-0.2,0),lightgray); label("0",(-0.2,0),W); draw((0,2)--(-0.2,2),lightgray); label("2",(-0.2,2),W); draw((0,4)--(-0.2,4)); label("4",(-0.2,4),W); // Dashed Lines draw((0,-2)--(12,-2),dashed); draw((0,2)--(12,2),dashed+lightgray); draw((0,4)--(12,4),dashed); draw((2,5)--(2,0.2),dashed); draw((4,5)--(4,0.2),dashed); draw((6,5)--(6,0.2),dashed); draw((8,5)--(8,0.2),dashed); draw((10,5)--(10,0.2),dashed); draw((12,5)--(12,0.2),dashed); draw((2,-1)--(2,-3),dashed); draw((4,-1)--(4,-3),dashed); draw((6,-1)--(6,-3),dashed); draw((8,-1)--(8,-3),dashed); draw((10,-1)--(10,-3),dashed); draw((12,-1)--(12,-3),dashed); // Path path A=(0,0)--(2,4)--(4,4)--(6,2)--(8,0)--(10,-2)--(12,0); draw(A,linewidth(1.5)); // Labels label(scale(0.85)*rotate(90)*"Acceleration (m/s/s)",(0,1),W*7); label(scale(0.75)*"Position (m)",(11,0),N); [/asy] (A) $\text{36 J}$ (B) $\text{22 J}$ (C) $\text{5 J}$ (D)$\text{-17 J}$ (E) $\text{-36 J}$

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) $\omega_\text{1}r_\text{1}^4=\omega_\text{2}r_\text{2}^4$

2009 F = Ma, 2

Tags: 2009 , Problem 2

Suppose that all collisions are instantaneous and perfectly elastic. After a long time, which of the following is true? (A) The center block is moving to the left. (B) The center block is moving to the right. (C) The center block is at rest somewhere to the left of its initial position. (D) The center block is at rest at its initial position. (E) The center block is at rest somewhere to the right of its initial position.

2009 F = Ma, 21

Tags: 2009 , problem 21

What is the value of the gravitational potential energy of the two star system? (A) $-\frac{GM^2}{d}$ (B) $\frac{3GM^2}{d}$ (C) $-\frac{GM^2}{d^2}$ (D) $-\frac{3GM^2}{d}$ (E) $-\frac{3GM^2}{d^2}$

2009 F = Ma, 6

Tags: 2009 , Problem 6

An object is thrown with a fixed initial speed $v_\text{0}$ at various angles $\alpha$ relative to the horizon. At some constant height $h$ above the launch point the speed $v$ of the object is measured as a function of the initial angle $\alpha$. Which of the following best describes the dependence of $v$ on $\alpha$? (Assume that the height h is achieved, and assume that there is no air resistance.) (A) $v$ will increase monotonically with $\alpha$. (B) $v$ will increase to some critical value $v_{max}$ and then decrease. (C) $v$ will remain constant, independent of $\alpha$. (D) $v$ will decrease to some critical value $v_{min}$ and then increase. (E) None of the above.

2009 F = Ma, 19

Tags: 2009 , problem 19

A certain football quarterback can throw a football a maximum range of $80$ meters on level ground. What is the highest point reached by the football if thrown this maximum range? Ignore air friction. (A) $\text{10 m}$ (B) $\text{20 m}$ (C) $\text{30 m}$ (D) $\text{40 m}$ (E) $\text{50 m}$

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.

2009 F = Ma, 3

Tags: 2009 , Problem 3

Suppose, instead, that all collisions are instantaneous and perfectly inelastic. After a long time, which of the following is true? (A) The center block is moving to the left. (B) The center block is moving to the right. (C) The center block is at rest somewhere to the left of its initial position. (D) The center block is at rest at its initial position. (E) The center block is at rest somewhere to the right of its initial position.

2009 F = Ma, 1

Tags: 2009 , Problem 1

A $\text{0.3 kg}$ apple falls from rest through a height of $\text{40 cm}$ onto a flat surface. Upon impact, the apple comes to rest in $\text{0.1 s}$, and $\text{4 cm}^2$ of the apple comes into contact with the surface during the impact. What is the average pressure exerted on the apple during the impact? Ignore air resistance. (A) $\text{67,000 Pa}$ (B) $\text{21,000 Pa}$ (C) $\text{6,700 Pa}$ (D) $\text{210 Pa}$ (E) $\text{67 Pa}$

2009 Philippine MO, 1

Tags: algebra , Sequences , PMO , 2009

The sequence ${a_0, a_1, a_2, ...}$ of real numbers satisfies the recursive relation $$n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n$$ for every positive integer $n$, where $a_0 = a_1 = 1$. Calculate the sum $$\frac{a_0}{a_1} + \frac{a_1}{a_2} + ... + \frac{a_{2008}}{a_{2009}}$$.

2009 F = Ma, 15

Tags: 2009 , Problem 15

A $\text{22.0 kg}$ suitcase is dragged in a straight line at a constant speed of $\text{1.10 m/s}$ across a level airport floor by a student on the way to the 40th IPhO in Merida, Mexico. The individual pulls with a $\text{1.00} \times \text{10}^2 \text{N}$ force along a handle with makes an upward angle of $\text{30.0}$ degrees with respect to the horizontal. What is the coefficient of kinetic friction between the suitcase and the floor? (A) $\mu_\text{k} = \text{0.013}$ (B) $\mu_\text{k} = \text{0.394}$ (C) $\mu_\text{k} = \text{0.509}$ (D) $\mu_\text{k} = \text{0.866}$ (E) $\mu_\text{k} = \text{1.055}$

2009 F = Ma, 16

Tags: 2009 , problem 16

Two identical objects of mass $m$ are placed at either end of a spring of spring constant $k$ and the whole system is placed on a horizontal frictionless surface. At what angular frequency $\omega$ does the system oscillate? (A) $\sqrt{k/m}$ (B) $\sqrt{2k/m}$ (C) $\sqrt{k/2m}$ (D) $2\sqrt{k/m}$ (E) $\sqrt{k/m}/2$

2009 F = Ma, 24

Tags: 2009 , problem 24

A uniform rectangular wood block of mass $M$, with length $b$ and height $a$, rests on an incline as shown. The incline and the wood block have a coefficient of static friction, $\mu_s$. The incline is moved upwards from an angle of zero through an angle $\theta$. At some critical angle the block will either tip over or slip down the plane. Determine the relationship between $a$, $b$, and $\mu_s$ such that the block will tip over (and not slip) at the critical angle. The box is rectangular, and $a \neq b$. [asy] draw((-10,0)--(0,0)--20/sqrt(3)*dir(150)); label("$\theta$",(0,0),dir(165)*6); real x = 3; fill((0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x)--(3,3)*dir(60)+(-x*sqrt(3),x)--(0,3)*dir(60)+(-x*sqrt(3),x)--cycle,grey); draw((0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x)--(3,3)*dir(60)+(-x*sqrt(3),x)--(0,3)*dir(60)+(-x*sqrt(3),x)--cycle); label("$a$",(0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x)); label("$b$",(3,3)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x),dir(60)); [/asy] (A) $\mu_s > a/b$ (B) $\mu_s > 1-a/b$ (C) $\mu_s >b/a$ (D) $\mu_s < a/b$ (E) $\mu_s < b/a-1$

2009 Middle European Mathematical Olympiad, 1

Tags: function , algebra proposed , algebra , functional equation , Reals , 2009 , memo

Find all functions $ f: \mathbb{R} \to \mathbb{R}$, such that \[ f(xf(y)) \plus{} f(f(x) \plus{} f(y)) \equal{} yf(x) \plus{} f(x \plus{} f(y))\] holds for all $ x$, $ y \in \mathbb{R}$, where $ \mathbb{R}$ denotes the set of real numbers.

2009 F = Ma, 8

Tags: 2009 , Problem 8

Determine the angular acceleration of the disk when $t=\text{2.0 s}$. (A) $\text{-12 rad/s}^2$. (B) $\text{-8 rad/s}^2$. (C) $\text{-4 rad/s}^2$. (D) $\text{-2 rad/s}^2$. (E) $\text{0 rad/s}^2$.

2009 F = Ma, 10

Tags: 2009 , problem 10

A person standing on the edge of a fire escape simultaneously launches two apples, one straight up with a speed of $\text{7 m/s}$ and the other straight down at the same speed. How far apart are the two apples $2$ seconds after they were thrown, assuming that neither has hit the ground? (A) $\text{14 m}$ (B) $\text{20 m}$ (C) $\text{28 m}$ (D) $\text{34 m}$ (E) $\text{56 m}$

2009 F = Ma, 23

Tags: 2009 , problem 23

A mass is attached to an ideal spring. At time $t = \text{0}$ the spring is at its natural length and the mass is given an initial velocity; the period of the ensuing (one-dimensional) simple harmonic motion is $T$. At what time is the power delivered [i]to[/i] the mass by the spring first a maximum? (A) $t = \text{0}$ (B) $t = T/\text{8}$ (C) $t = T/\text{4}$ (D) $t = \text{3}T/\text{8}$ (E) $t = T/\text{2}$

2009 F = Ma, 14

Tags: 2009 , Problem 14

A wooden block (mass $M$) is hung from a peg by a massless rope. A speeding bullet (with mass $m$ and initial speed $v_\text{0}$) collides with the block at time $t = \text{0}$ and embeds in it. Let $S$ be the system consisting of the block and bullet. Which quantities are conserved between $t = -\text{10 s}$ and $ t = \text{+10 s}$? [asy] // Code by riben draw(circle((0,0),0.3),linewidth(2)); filldraw(circle((0,0),0.3),gray); draw((0,-0.8)--(0,-15.5),linewidth(2)); draw((5,-15.5)--(-5,-15.5)--(-5,-20.5)--(5,-20.5)--cycle,linewidth(2)); filldraw((5,-15.5)--(-5,-15.5)--(-5,-20.5)--(5,-20.5)--cycle,gray); draw((-15,-18)--(-16,-17)--(-18,-17)--(-18,-19)--(-16,-19)--cycle,linewidth(2)); filldraw((-15,-18)--(-16,-17)--(-18,-17)--(-18,-19)--(-16,-19)--cycle,gray); [/asy] (A) The total linear momentum of $S$. (B) The horizontal component of the linear momentum of $S$. (C) The mechanical energy of $S$. (D) The angular momentum of $S$ as measured about a perpendicular axis through the peg. (E) None of the above are conserved.

2009 F = Ma, 4

Tags: 2009 , Problem 4

A spaceman of mass $\text{80 kg}$ is sitting in a spacecraft near the surface of the Earth. The spacecraft is accelerating upward at five times the acceleration due to gravity. What is the force of the spaceman on the spacecraft? (A) $\text{4800 N}$ (B) $\text{4000 N}$ (C) $\text{3200 N}$ (D) $\text{800 N}$ (E) $\text{400 N}$