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1186

Electromagnetism

A beam of electrons emitted by a point source $\mathrm{P}$ enters the magnetic field $\vec{B}$ of a toroidal coil (toroid) in the direction of the lines of force. The angle of the aperture of the beam $2 \cdot \alpha_{0}$ is assumed to be small $\left(2 \cdot \alpha_{0}<<1\right)$. The injection of the electrons occurs on the mean radius $\mathrm{R}$ of the toroid with acceleration voltage $\mathrm{V}_{0}$. Neglect any interaction between the electrons. The magnitude of $\vec{B}, B$, is assumed to be constant. <img_4363> Data: $\frac{\mathrm{e}}{\mathrm{m}}=1.76 \cdot 10^{11} \mathrm{C} \cdot \mathrm{kg}^{-1} ; \quad \mathrm{V}_{0}=3 \mathrm{kV} ; \quad \mathrm{R}=50 \mathrm{~mm}$ Context question: 1. To guide the electron in the toroidal field a homogeneous magnetic deflection field $\vec{B}_{1}$ is required. Calculate $\vec{B}_{1}$ for an electron moving on a circular orbit of radius $R$ in the torus. Context answer: \boxed{$1.48 \cdot 10^{-2}$} Context question: 2. Determine the value of $\vec{B}$ which gives four focussing points separated by $\pi / 2$ as indicated in the diagram. Note: When considering the electron paths you may disregard the curvature of the magnetic field. Context answer: \boxed{$0.37 \cdot 10^{-2}$} Extra Supplementary Reading Materials: 3. The electron beam cannot stay in the toroid without a deflection field $\vec{B}_{1}$, but will leave it with a systematic motion (drift) perpendicular to the plane of the toroid. Note: The angle of aperture of the electron beam can be neglected. Use the laws of conservation of energy and of angular momentum.

a) Show that the radial deviation of the electrons from the injection radius is finite.

TP_MM_physics_en_COMP

1313

Optics

Thermoacoustic Engine A thermoacoustic engine is a device that converts heat into acoustic power, or sound waves - a form of mechanical work. Like many other heat machines, it can be operated in reverse to become a refrigerator, using sound to pump heat from a cold to a hot reservoir. The high operating frequencies reduce heat conduction and eliminate the need for any working chamber confinement. Unlike many other engine types, the thermoacoustic engine has no moving parts except the working fluid itself. The efficiencies of thermoacoustic machines are typically lower than other engine types, but they have advantages in set up and maintenance costs. This creates opportunities for renewable energy applications, such as solar-thermal power plants and utilization of waste heat. Our analysis will focus on the creation of acoustic energy within the system, ignoring the extraction or conversion for powering external devices. Part A: Sound wave in a closed tube Consider a thermally insulating tube of length $L$ and cross-sectional area $S$, whose axis lies along the $x$ direction. The two ends of the tube are located at $x=0$ and $x=L$. The tube is filled with an ideal gas and is sealed on both ends. At equilibrium, the gas has temperature $T_{0}$, pressure $p_{0}$ and mass density $\rho_{0}$. Assume that viscosity can be ignored and that the gas motion is only in the $x$ direction. The gas properties are uniform in the perpendicular $y$ and $z$ directions. <img_4558> Figure 1 Context question: A.1 When a standing sound wave forms, the gas elements oscillate in the $x$ direction with angular frequency $\omega$. The amplitude of the oscillations depends on each element's equilibrium position $x$ along the tube. The longitudinal displacement of each gas element from its equilibrium position $x$ is given by $$ u(x, t)=a \sin (k x) \cos (\omega t)=u_{1}(x) \cos (\omega t) \tag{1} $$ (please note the $u$ here describes the displacement of a gas element) where $a \ll L$ is a positive constant, $k=2 \pi / \lambda$ is the wavenumber and $\lambda$ is the wavelength. What is the maximum possible wavelength $\lambda_{\max }$ in this system? Context answer: \boxed{$\lambda_{\max }=2 L$} Extra Supplementary Reading Materials: We will assume throughout the question an oscillation mode of $\lambda=\lambda_{\max }$. Now, consider a narrow parcel of gas, located at rest between $x$ and $x+\Delta x(\Delta x \ll L)$. As a result of the displacement wave of Task A.1, the parcel oscillates along the $x$ axis and undergoes a change in volume and other thermodynamic properties. Throughout the following tasks assume all these changes to the thermodynamic properties to be small compared to the unperturbed values. Context question: A.2 The parcel volume $V(x, t)$ oscillates around the equilibrium value of $V_{0}=S \Delta x$ and has the form $$ V(x, t)=V_{0}+V_{1}(x) \cos (\omega t) . \tag{2} $$ Obtain an expression for $V_{1}(x)$ in terms of $V_{0}, a, k$ and $x$. Context answer: \boxed{$V_{1}(x)=a k V_{0} \cos (k x)$} Context question: A.3 Assume that the total pressure of the gas, as a result of the sound wave, takes the approximate form $$ p(x, t)=p_{0}-p_{1}(x) \cos (\omega t) . \tag{3} $$ Considering the forces acting on the parcel of gas, compute the amplitude $p_{1}(x)$ of the pressure oscillation to leading order, in terms of the position $x$, the equilibrium density $\rho_{0}$, the displacement amplitude $a$ and the wave parameters $k$ and $\omega$. Context answer: \boxed{$p_{1}(x)=a \frac{\omega^{2}}{k} \rho_{0} \cos (k x)$} Extra Supplementary Reading Materials: At acoustic frequencies, the thermal conductivity of the gas can be neglected. We will treat the expansion and contraction of gas parcels as purely adiabatic, satisfying the relation $p V^{\gamma}=$ const, where $\gamma$ is the adiabatic constant. Context question: A.4 Use the relation above and the results of the previous tasks to obtain an expression for the speed of sound waves $c=\omega / k$ in the tube, to first order. Express your answer in terms of $p_{0}, \rho_{0}$ and the adiabatic constant $\gamma$. Context answer: \boxed{$c=\sqrt{\frac{\gamma p_{0}}{\rho_{0}}}$} Context question: A.5 The change in the gas temperature due to the adiabatic expansion and contraction, as a result of the sound wave, takes the form: $$ T(x, t)=T_{0}-T_{1}(x) \cos (\omega t) \tag{4} $$ Compute the amplitude $T_{1}(x)$ of the temperature oscillations in terms of $T_{0}, \gamma$, $a, k$ and $x$. Context answer: \boxed{$a k(\gamma-1) T_{0} \cos (k x)$}

A.6 For the purpose of this task only, we assume a weak thermal interaction between the tube and the gas. As a result, the standing sound wave remains almost unchanged, but the gas can exchange a small amount of heat with the tube. The heating due to viscosity can be neglected. For each of the points in Figure 2 (A, C at the edges of the tube, B at the center) state whether the temperature of the tube at that point will increase, decrease or remain the same over a long time. Figure 2

TP_MM_physics_en_COMP

1554

Thermodynamics

The schematic below shows the Hadley circulation in the Earth's tropical atmosphere around the spring equinox. Air rises from the equator and moves poleward in both hemispheres before descending in the subtropics at latitudes $\pm \varphi_{d}$ (where positive and negative latitudes refer to the northern and southern hemisphere respectively). The angular momentum about the Earth's spin axis is conserved for the upper branches of the circulation (enclosed by the dashed oval). Note that the schematic is not drawn to scale. <img_4380> Context question: (a) Assume that there is no wind velocity in the east-west direction around the point $\mathrm{X}$. What is the expression for the east-west wind velocity $u_{Y}$ at the points Y? Convention: positive velocities point from west to east. (The angular velocity of the Earth about its spin axis is $\Omega$, the radius of the Earth is $a$, and the thickness of the atmosphere is much smaller than $a$.) Context answer: \boxed{$u_{Y}=\Omega a\left(\frac{1}{\cos \varphi_{d}}-\cos \varphi_{d}\right)$} Context question: (b) Which of the following explains ultimately why angular momentum is not conserved along the lower branches of the Hadley circulation? Tick the correct answer(s). There can be more than one correct answer. (I) There is friction from the Earth's surface. (II) There is turbulence in the lower atmosphere, where different layers of air are mixed (III) The air is denser lower down and so inertia slows down the motion around the spin axis of the Earth. (IV) The air is moist at the lower levels causing retardation to the wind velocity. Context answer: (I),(II) Extra Supplementary Reading Materials: Around the northern winter solstice, the rising branch of the Hadley circulation is located at the latitude $\varphi_{r}$ and the descending branches are located at $\varphi_{n}$ and $\varphi_{s}$ as shown in the schematic below. Refer to this diagram for parts (c), (d) and (e). <img_4510> Context question: (c) Assume that there is no east-west wind velocity around the point $\mathrm{Z}$. Given that $\varphi_{r}=-8^{\circ}, \varphi_{n}=28^{\circ}$ and $\varphi_{s}=-20^{\circ}$, what are the east-west wind velocities $u_{P}, u_{Q}$ and $u_{R}$ respectively at the points $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ ? (The radius of the Earth is a $=6370 \mathrm{~km}$.) Hence, which hemisphere below has a stronger atmospheric jet stream? (I) Winter Hemisphere (II) Summer Hemisphere (III) Both hemispheres have equally strong jet streams. Context answer: \boxed{(I)} Context question: (d) The near-surface branch of the Hadley circulation blows southward across the equator. Mark by arrows on the figure below the direction of the eastwest component of the Coriolis force acting on the tropical air mass (A) north of the equator; (B) south of the equator. <img_4497> Context answer: <img_4493> Context question: (e) From your answer to part (d) and the fact that surface friction nearly balances the Coriolis forces in the east-west direction, sketch the near-surface wind pattern in the tropics near the equator during northern winter solstice. Context answer: <img_4521> Extra Supplementary Reading Materials: Suppose the Hadley circulation can be simplified as a heat engine shown in the schematic below. Focusing on the Hadley circulation reaching into the winter hemisphere as shown below, the physical transformation of the air mass from A to B and from $\mathrm{D}$ to $\mathrm{E}$ are adiabatic, while that from $\mathrm{B}$ to $\mathrm{C}, \mathrm{C}$ to $\mathrm{D}$ and from $\mathrm{E}$ to $\mathrm{A}$ are isothermal. Air gains heat by contact with the Earth's surface and by condensation of water from the atmosphere, while air loses heat by radiation into space. <img_4349> Context question: (f) Given that atmospheric pressure at a vertical level owes its origin to the weight of the air above that level, order the pressures $p_{A}, p_{B}, p_{C}, p_{D}, p_{E}$, respectively at the points $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}$ by a series of inequalities. (Given that $p_{A}=1000 \mathrm{hPa}$ and $p_{D}=225 \mathrm{hPa}$. Note that $1 \mathrm{hPa}$ is $100 \mathrm{~Pa}$.) Context answer: $p_{E}>p_{A}>p_{D}>p_{B}>p_{C}$ Context question: (g) Let the temperature next to the surface and at the top of the atmosphere be $T_{H}$ and $T_{C}$ respectively. Given that the pressure difference between points $\mathrm{A}$ and $\mathrm{E}$ is $20 \mathrm{hPa}$, calculate $T_{C}$ for $T_{H}=300 \mathrm{~K}$. Note that the ratio of molar gas constant $(R)$ to molar heat capacity at constant pressure $\left(c_{p}\right)$ for air, $\kappa$, is $2 / 7$. Context answer: \boxed{195} Context question: (h) Calculate the pressure $p_{B}$. Context answer: \boxed{220} Context question: (i) For an air mass moving once around the winter Hadley circulation, using the molar gas constant, $R$, and the quantities defined above, obtain expressions for (A) the net work done per unit mole $W_{\text {net }}$ ignoring surface friction; Context answer: \boxed{$R\left(T_{H}-T_{C}\right) \ln \left(\frac{p_{E}}{p_{A}}\right)$} Context question: (B) the heat loss per unit mole $Q_{\text {loss }}$ at the top of the atmosphere. Context answer: \boxed{$R T_{C} \ln \left(\frac{p_{D}}{p_{C}}\right)$} Context question: (j) What is the value of the ideal thermodynamic efficiency $\varepsilon_{i}$ for the winter Hadley circulation? Context answer: \boxed{0.35}

(k) Prove that the actual thermodynamic efficiency $\varepsilon$ for the winter Hadley circulation is always smaller than $\varepsilon_{i}$, showing all mathematical steps.

TP_MM_physics_en_COMP

1581

Modern Physics

Global Positioning System (GPS) is a navigation technology which uses signal from satellites to determine the position of an object (for example an airplane). However, due to the satellites high speed movement in orbit, there should be a special relativistic correction, and due to their high altitude, there should be a general relativistic correction. Both corrections seem to be small but are very important for precise measurement of position. We will explore both corrections in this problem. First we will investigate the special relativistic effect on an accelerated particle. We consider two types of frame, the first one is the rest frame (called $S$ or Earth's frame), where the particle is at rest initially. The other is the proper frame (called $S^{\prime}$ ), a frame that instantaneously moves together with the accelerated particle. Note that this is not an accelerated frame, it is a constant velocity frame that at a particular moment has the same velocity with the accelerated particle. At that short moment, the time rate experienced by the particle is the same as the proper frame's time rate. Of course this proper frame is only good for an infinitesimally short time, and then we need to define a new proper frame afterward. At the beginning we synchronize the particle's clock with the clock in the rest frame by setting them to zero, $t=\tau=0$ ( $t$ is the time in the rest frame, and $\tau$ is the time shown by particle's clock). By applying equivalence principle, we can obtain general relativistic effects from special relavistic results which does not involve complicated metric tensor calculations. By combining the special and general relativistic effects, we can calculate the corrections needed for a GPS (global positioning system) satellite to provide accurate positioning. Some mathematics formulas that might be useful - $\sinh x=\frac{e^{x}-e^{-x}}{2}$ - $\cosh x=\frac{e^{x}+e^{-x}}{2}$ - $\tanh x=\frac{\sinh x}{\cosh x}$ - $1+\sinh ^{2} x=\cosh ^{2} x$ - $\sinh (x-y)=\sinh x \cosh y-\cosh x \sinh y$ - $\int \frac{d x}{\left(1-x^{2}\right)^{\frac{3}{2}}}=\frac{x}{\sqrt{1-x^{2}}}+C$ - $\int \frac{d x}{1-x^{2}}=\ln \sqrt{\frac{1+x}{1-x}}+C$ Part A. Single Accelerated Particle Consider a particle with a rest mass $m$ under a constant and uniform force field $F$ (defined in the rest frame) pointing in the positive $x$ direction. Initially $(t=\tau=0)$ the particle is at rest at the origin $(x=0)$. Context question: 1. When the velocity of the particle is $v$, calculate the acceleration of the particle, $a$ (with respect to the rest frame). Context answer: \boxed{$a=\frac{F}{\gamma^{3} m}$} Context question: 2. Calculate the velocity of the particle $\beta(t)=\frac{v(t)}{c}$ at time $t$ (in rest frame), in terms of $F, m, t$ and $c$. Context answer: \boxed{$\beta=\frac{\frac{F t}{m c}}{\sqrt{1+\left(\frac{F t}{m c}\right)^{2}}}$} Context question: 3. Calculate the position of the particle $x(t)$ at time $t$, in term of $F, m, t$ and $c$. Context answer: \boxed{$x=\frac{m c^{2}}{F}\left(\sqrt{1+\left(\frac{F t}{m c}\right)^{2}}-1\right)$} Context question: 4. Show that the proper acceleration of the particle, $a^{\prime} \equiv g=F / m$, is a constant. The proper acceleration is the acceleration of the particle measured in the instantaneous proper frame. Context answer: \boxed{证明题} Context question: 5. Calculate the velocity of the particle $\beta(\tau)$, when the time as experienced by the particle is $\tau$. Express the answer in $g, \tau$, and $c$. Context answer: \boxed{$\beta=\tanh \frac{g \tau}{c}$} Context question: 6. ( $\mathbf{0 . 4} \mathbf{~ p t s )}$ Also calculate the time $t$ in the rest frame in terms of $g, \tau$, and $c$. Context answer: \boxed{$t=\frac{c}{g} \sinh \frac{g \tau}{c}$} Extra Supplementary Reading Materials: Part B. Flight Time The first part has not taken into account the flight time of the information to arrive to the observer. This part is the only part in the whole problem where the flight time is considered. The particle moves as in part A. Context question: 1. At a certain moment, the time experienced by the particle is $\tau$. What reading $t_{0}$ on a stationary clock located at $x=0$ will be observed by the particle? After a long period of time, does the observed reading $t_{0}$ approach a certain value? If so, what is the value? Context answer: \boxed{证明题} Context question: 2. Now consider the opposite point of view. If an observer at the initial point $(x=0)$ is observing the particle's clock when the observer's time is $t$, what is the reading of the particle's clock $\tau_{0}$ ? After a long period of time, will this reading approach a certain value? If so, what is the value? Context answer: \boxed{证明题} Extra Supplementary Reading Materials: Part C. Minkowski Diagram In many occasion, it is very useful to illustrate relativistic events using a diagram, called as Minkowski Diagram. To make the diagram, we just need to use Lorentz transformation between the rest frame $S$ and the moving frame $S^{\prime}$ that move with velocity $v=\beta c$ with respect to the rest frame. $$ \begin{aligned} & x=\gamma\left(x^{\prime}+\beta c t^{\prime}\right), \\ & c t=\gamma\left(c t^{\prime}+\beta x^{\prime}\right), \\ & x^{\prime}=\gamma(x-\beta c t), \\ & c t^{\prime}=\gamma(c t-\beta x) . \\ & \gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \end{aligned} $$ <img_4526> Let's choose $x$ and $c t$ as the orthogonal axes. A point $\left(x^{\prime}, c t^{\prime}\right)=(1,0)$ in the moving frame $S^{\prime}$ has a coordinate $(x, c t)=(\gamma, \gamma \beta)$ in the rest frame $S$. The line connecting this point and the origin defines the $x^{\prime}$ axis. Another point $\left(x^{\prime}, c t^{\prime}\right)=(0,1)$ in the moving frame $S^{\prime}$ has a coordinate $(x, c t)=(\gamma \beta, \gamma)$ in the rest frame $S$. The line connecting this point and the origin defines the $c t^{\prime}$ axis. The angle between the $x$ and $\mathrm{x}^{\prime}$ axis is $\theta$, where $\tan \theta=\beta$. A unit length in the moving frame $S^{\prime}$ is equal to $\gamma \sqrt{1+\beta^{2}}=\sqrt{\frac{1+\beta^{2}}{1-\beta^{2}}}$ in the rest frame S. To get a better understanding of Minkowski diagram, let us take a look at this example. Consider a stick of proper length $L$ in a moving frame S'. We would like to find the length of the stick in the rest frame S. Consider the figure below. <img_4417> The stick is represented by the segment AC. The length $\mathrm{AC}$ is equal to $\sqrt{\frac{1+\beta^{2}}{1-\beta^{2}}} L$ in the $S$ frame. The stick length in the $S$ frame is represented by the line $A B$. $$ \begin{aligned} \mathrm{AB} & =\mathrm{AD}-\mathrm{BD} \\ & =A C \cos \theta-A C \sin \theta \tan \theta \\ & =L \sqrt{1-\beta^{2}} \end{aligned} $$ Context question: 1. Using a Minkowski diagram, calculate the length of a stick with proper length $L$ in the rest frame, as measured in the moving frame. Context answer: \boxed{$\sqrt{1-\beta^{2}} L$}

2. Now consider the case in part A. Plot the time ct versus the position $x$ of the particle. Draw the $x^{\prime}$ axis and $c t^{\prime}$ axis when $\frac{g t}{c}=1$ in the same graph using length scale $x\left(c^{2} / g\right)$ and $c t\left(c^{2} / g\right)$.

TP_MM_physics_en_COMP

1600

Modern Physics

All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual classical formalism. In this problem, we are investigating the influence of magnetic field on spin using its classical analogue. The classical torque equation of spin is given by $$ \boldsymbol{\tau}=\frac{d \boldsymbol{L}}{d t}=\boldsymbol{\mu} \times \boldsymbol{B} $$ In this case, the angular momentum $\boldsymbol{L}$ represents the "intrinsic" spin of the particles, $\boldsymbol{\mu}$ is the magnetic moment of the particles, and $\boldsymbol{B}$ is magnetic field. The spin of a particle is associated with a magnetic moment via the equation $$ \boldsymbol{\mu}=-\gamma \boldsymbol{L} $$ where $\gamma$ is the gyromagnetic ratio. In this problem, the term "frequency" means angular frequency (rad/s), which is a scalar quantity. All bold letters represent vectors; otherwise they represent scalars. Part A. Larmor precession Context question: 1. Prove that the magnitude of magnetic moment $\mu$ is always constant under the influence of a magnetic field $\boldsymbol{B}$. For a special case of stationary (constant) magnetic field, also show that the angle between $\boldsymbol{\mu}$ and $\boldsymbol{B}$ is constant. (Hint: You can use properties of vector products.) Context answer: \boxed{证明题} Context question: 2. A uniform magnetic field $\boldsymbol{B}$ exists and it makes an angle $\phi$ with a particle's magnetic moment $\boldsymbol{\mu}$. Due to the torque by the magnetic field, the magnetic moment $\boldsymbol{\mu}$ rotates around the field $\boldsymbol{B}$, which is also known as Larmor precession. Determine the Larmor precession frequency $\omega_{0}$ of the magnetic moment with respect to $\boldsymbol{B}=B_{0} \boldsymbol{k}$. Context answer: \boxed{$\omega_{0}=\gamma B_{0}$} Extra Supplementary Reading Materials: Part B. Rotating frame In this section, we choose a rotating frame $S^{\prime}$ as our frame of reference. The rotating frame $S^{\prime}=\left(x^{\prime}, y^{\prime}, z^{\prime}\right)$ rotates with an angular velocity $\omega \boldsymbol{k}$ as seen by an observer in the laboratory frame $S=(x, y, z)$, where the axes $x^{\prime}, y^{\prime}, z^{\prime}$ intersect with $x, y, z$ at time $t=0$. Any vector $\boldsymbol{A}=A_{x} \boldsymbol{i}+A_{y} \boldsymbol{j}+A_{z} \boldsymbol{k}$ in a lab frame can be written as $\boldsymbol{A}=A_{x}{ }^{\prime} \boldsymbol{i}^{\prime}+A_{y}{ }^{\prime} \boldsymbol{j}^{\prime}+A_{z}{ }^{\prime} \boldsymbol{k}^{\prime}$ in the rotating frame $S^{\prime}$. The time derivative of the vector becomes $$ \frac{d \boldsymbol{A}}{d t}=\left(\frac{d A_{x}{ }^{\prime}}{d t} \boldsymbol{i}^{\prime}+\frac{d A_{y}{ }^{\prime}}{d t} \boldsymbol{j}^{\prime}+\frac{d A_{z}{ }^{\prime}}{d t} \boldsymbol{k}^{\prime}\right)+\left(A_{x}{ }^{\prime} \frac{d \boldsymbol{i}^{\prime}}{d t}+A_{y}{ }^{\prime} \frac{d \boldsymbol{j}^{\prime}}{d t}+A_{z}{ }^{\prime} \frac{d \boldsymbol{k}^{\prime}}{d t}\right) $$ $$ \left(\frac{d \boldsymbol{A}}{d t}\right)_{l a b}=\left(\frac{d \boldsymbol{A}}{d t}\right)_{r o t}+(\omega \mathbf{k} \times \boldsymbol{A}) $$ where $\left(\frac{d \boldsymbol{A}}{d t}\right)_{l a b}$ is the time derivative of vector $\boldsymbol{A}$ seen by an observer in the lab frame, and $\left(\frac{d A}{d t}\right)_{\text {rot }}$ is the time derivative seen by an observer in the rotating frame. For all the following problems in this part, the answers are referred to the rotating frame $S^{\prime}$. Context question: 1. Show that the time evolution of the magnetic moment follows the equation $$ \left(\frac{d \boldsymbol{\mu}}{d t}\right)_{r o t}=-\gamma \boldsymbol{\mu} \times \boldsymbol{B}_{e f f} $$ where $\boldsymbol{B}_{\text {eff }}=\boldsymbol{B}-\frac{\omega}{\gamma} \boldsymbol{k}^{\prime}$ is the effective magnetic field. Context answer: \boxed{证明题} Context question: 2. For $\boldsymbol{B}=B_{0} \boldsymbol{k}$, what is the new precession frequency $\Delta$ in terms of $\omega_{0}$ and $\omega$ ? Context answer: \boxed{$\Delta =\gamma B_{0}-\omega$} Context question: 3. Now, let us consider the case of a time-varying magnetic field. Besides a constant magnetic field, we also apply a rotating magnetic field $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}+\sin \omega t \boldsymbol{j})$, so $\boldsymbol{B}=B_{0} \boldsymbol{k}+\boldsymbol{b}(t)$. Show that the new Larmor precession frequency of the magnetic moment is $$ \Omega=\gamma \sqrt{\left(B_{0}-\frac{\omega}{\gamma}\right)^{2}+b^{2}} $$ Context answer: \boxed{证明题} Context question: 4. Instead of applying the field $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}+\sin \omega t \boldsymbol{j})$, now we apply $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}-\sin \omega t \boldsymbol{j})$, which rotates in the opposite direction and hence $\boldsymbol{B}=B_{0} \boldsymbol{k}+b(\cos \omega t \boldsymbol{i}-\sin \omega t \boldsymbol{j})$. What is the effective magnetic field $\boldsymbol{B}_{\text {eff }}$ for this case (in terms of the unit vectors $\boldsymbol{i}^{\prime}, \boldsymbol{j}^{\prime}, \boldsymbol{k}^{\prime}$ )? What is its time average, $\overline{\boldsymbol{B}_{\text {eff }}}$ (recall that $\overline{\cos 2 \pi t / T}=\overline{\sin 2 \pi t / T}=0$ )? Context answer: \boxed{$\mathbf{B}_{\mathrm{eff}}=\left(B_{0}-\frac{\omega}{\gamma}\right) \mathbf{k}^{\prime}+b\left(\cos 2 \omega t \mathbf{i}^{\prime}-\sin 2 \omega t \mathbf{j}^{\prime}\right)$ , $\overline{\mathbf{B}_{\mathrm{eff}}}=\left(B_{0}-\frac{\omega}{\gamma}\right) \mathbf{k}^{\prime}$} Extra Supplementary Reading Materials: Part C. Rabi oscillation For an ensemble of $N$ particles under the influence of a large magnetic field, the spin can have two quantum states: "up" and "down". Consequently, the total population of spin up $N_{\uparrow}$ and down $N_{\downarrow}$ obeys the equation $$ N_{\uparrow}+N_{\downarrow}=N $$ The difference of spin up population and spin down population yields the macroscopic magnetization along the $z$ axis: $$ M=\left(N_{\uparrow}-N_{\downarrow}\right) \mu=N \mu_{z} . $$ In a real experiment, two magnetic fields are usually applied, a large bias field $B_{0} \boldsymbol{k}$ and an oscillating field with amplitude $2 b$ perpendicular to the bias field $\left(b \ll B_{0}\right)$. Initially, only the large bias is applied, causing all the particles lie in the spin up states ( $\boldsymbol{\mu}$ is oriented in the $z$-direction at $t=0$ ). Then, the oscillating field is turned on, where its frequency $\omega$ is chosen to be in resonance with the Larmor precession frequency $\omega_{0}$, i.e. $\omega=\omega_{0}$. In other words, the total field after time $t=0$ is given by $$ \boldsymbol{B}(t)=B_{0} \boldsymbol{k}+2 b \cos \omega_{0} t \boldsymbol{i} . $$ Context question: 1. In the rotating frame $S^{\prime}$, show that the effective field can be approximated by $$ \boldsymbol{B}_{\text {eff }} \approx b \boldsymbol{i}^{\prime}, $$ which is commonly known as rotating wave approximation. What is the precession frequency $\Omega$ in frame $S^{\prime}$ ? Context answer: \boxed{$\Omega=\gamma b$} Context question: 2. Determine the angle $\alpha$ that $\boldsymbol{\mu}$ makes with $\boldsymbol{B}_{\text {eff }}$. Also, prove that the magnetization varies with time as $$ M(t)=N \mu(\cos \Omega t) . $$ Context answer: \boxed{证明题} Context question: 3. Under the application of magnetic field described above, determine the fractional population of each spin up $P_{\uparrow}=N_{\uparrow} / N$ and spin down $P_{\downarrow}=N_{\downarrow} / N$ as a function of time. Plot $P_{\uparrow}(t)$ and $P_{\downarrow}(t)$ on the same graph vs. time $t$. The alternating spin up and spin down population as a function of time is called Rabi oscillation. Context answer: \boxed{$P_{\downarrow}=\sin ^{2} \frac{\Omega t}{2}$ , $P_{\uparrow}=\cos ^{2} \frac{\Omega t}{2}$} Extra Supplementary Reading Materials: Part D. Measurement incompatibility Spin is in fact a vector quantity; but due to its quantum properties, we cannot measure each of its components simultaneously (i.e. we can know both $|\boldsymbol{\mu}|$ and $\mu_{z}$ as in above problems; but not all $|\boldsymbol{\mu}|, \mu_{x}, \mu_{y}$, and $\mu_{z}$ simultaneously). In this problem, we will do a calculation based on the Heisenberg uncertainty principle (using the relation $\Delta p_{q} \Delta q \geq \hbar$ ) to show how these measurements are incompatible with each other.

1. Let us consider an oven source of silver atoms, which has a small opening. The atoms stream out of the opening along $-y$ direction (see Figure below) and experience a spatial varying field $\boldsymbol{B}_{1}$. The field $\boldsymbol{B}_{1}$ has strong bias field component in the $z$ direction, where the atoms with different magnetic moment $\mu_{z}= \pm \gamma \hbar$ are split in the $z$ direction. At a distance $D$ from the oven source, a screen $S C_{1}$ is put to allow only spin up atoms to pass (blocking spin down atoms). Thus, at the instant after passing the screen, the atoms are prepared in spin up states. After the screen, the atoms enter a region of nonhomogenous field $\boldsymbol{B}_{2}$ where the atoms feel a force $$ F_{x}=\mu_{x} C $$ The field $\boldsymbol{B}_{2}$ has strong bias field component in the $x$ direction, where the atoms have magnetic moment $\mu_{x}= \pm \gamma \hbar$. In order to determine $\mu_{x}$ by observing the splitting in $x$ direction, show that the following condition must be fulfilled: $$ \frac{1}{\hbar}\left|\mu_{x}\right| \Delta x C t \gg 1 $$ where $t$ is the duration after leaving the screen $S C_{1}$ and $\Delta x$ is the opening width on $S C_{1}$.

TP_MM_physics_en_COMP

1601

Modern Physics

All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual classical formalism. In this problem, we are investigating the influence of magnetic field on spin using its classical analogue. The classical torque equation of spin is given by $$ \boldsymbol{\tau}=\frac{d \boldsymbol{L}}{d t}=\boldsymbol{\mu} \times \boldsymbol{B} $$ In this case, the angular momentum $\boldsymbol{L}$ represents the "intrinsic" spin of the particles, $\boldsymbol{\mu}$ is the magnetic moment of the particles, and $\boldsymbol{B}$ is magnetic field. The spin of a particle is associated with a magnetic moment via the equation $$ \boldsymbol{\mu}=-\gamma \boldsymbol{L} $$ where $\gamma$ is the gyromagnetic ratio. In this problem, the term "frequency" means angular frequency (rad/s), which is a scalar quantity. All bold letters represent vectors; otherwise they represent scalars. Part A. Larmor precession Context question: 1. Prove that the magnitude of magnetic moment $\mu$ is always constant under the influence of a magnetic field $\boldsymbol{B}$. For a special case of stationary (constant) magnetic field, also show that the angle between $\boldsymbol{\mu}$ and $\boldsymbol{B}$ is constant. (Hint: You can use properties of vector products.) Context answer: \boxed{证明题} Context question: 2. A uniform magnetic field $\boldsymbol{B}$ exists and it makes an angle $\phi$ with a particle's magnetic moment $\boldsymbol{\mu}$. Due to the torque by the magnetic field, the magnetic moment $\boldsymbol{\mu}$ rotates around the field $\boldsymbol{B}$, which is also known as Larmor precession. Determine the Larmor precession frequency $\omega_{0}$ of the magnetic moment with respect to $\boldsymbol{B}=B_{0} \boldsymbol{k}$. Context answer: \boxed{$\omega_{0}=\gamma B_{0}$} Extra Supplementary Reading Materials: Part B. Rotating frame In this section, we choose a rotating frame $S^{\prime}$ as our frame of reference. The rotating frame $S^{\prime}=\left(x^{\prime}, y^{\prime}, z^{\prime}\right)$ rotates with an angular velocity $\omega \boldsymbol{k}$ as seen by an observer in the laboratory frame $S=(x, y, z)$, where the axes $x^{\prime}, y^{\prime}, z^{\prime}$ intersect with $x, y, z$ at time $t=0$. Any vector $\boldsymbol{A}=A_{x} \boldsymbol{i}+A_{y} \boldsymbol{j}+A_{z} \boldsymbol{k}$ in a lab frame can be written as $\boldsymbol{A}=A_{x}{ }^{\prime} \boldsymbol{i}^{\prime}+A_{y}{ }^{\prime} \boldsymbol{j}^{\prime}+A_{z}{ }^{\prime} \boldsymbol{k}^{\prime}$ in the rotating frame $S^{\prime}$. The time derivative of the vector becomes $$ \frac{d \boldsymbol{A}}{d t}=\left(\frac{d A_{x}{ }^{\prime}}{d t} \boldsymbol{i}^{\prime}+\frac{d A_{y}{ }^{\prime}}{d t} \boldsymbol{j}^{\prime}+\frac{d A_{z}{ }^{\prime}}{d t} \boldsymbol{k}^{\prime}\right)+\left(A_{x}{ }^{\prime} \frac{d \boldsymbol{i}^{\prime}}{d t}+A_{y}{ }^{\prime} \frac{d \boldsymbol{j}^{\prime}}{d t}+A_{z}{ }^{\prime} \frac{d \boldsymbol{k}^{\prime}}{d t}\right) $$ $$ \left(\frac{d \boldsymbol{A}}{d t}\right)_{l a b}=\left(\frac{d \boldsymbol{A}}{d t}\right)_{r o t}+(\omega \mathbf{k} \times \boldsymbol{A}) $$ where $\left(\frac{d \boldsymbol{A}}{d t}\right)_{l a b}$ is the time derivative of vector $\boldsymbol{A}$ seen by an observer in the lab frame, and $\left(\frac{d A}{d t}\right)_{\text {rot }}$ is the time derivative seen by an observer in the rotating frame. For all the following problems in this part, the answers are referred to the rotating frame $S^{\prime}$. Context question: 1. Show that the time evolution of the magnetic moment follows the equation $$ \left(\frac{d \boldsymbol{\mu}}{d t}\right)_{r o t}=-\gamma \boldsymbol{\mu} \times \boldsymbol{B}_{e f f} $$ where $\boldsymbol{B}_{\text {eff }}=\boldsymbol{B}-\frac{\omega}{\gamma} \boldsymbol{k}^{\prime}$ is the effective magnetic field. Context answer: \boxed{证明题} Context question: 2. For $\boldsymbol{B}=B_{0} \boldsymbol{k}$, what is the new precession frequency $\Delta$ in terms of $\omega_{0}$ and $\omega$ ? Context answer: \boxed{$\Delta =\gamma B_{0}-\omega$} Context question: 3. Now, let us consider the case of a time-varying magnetic field. Besides a constant magnetic field, we also apply a rotating magnetic field $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}+\sin \omega t \boldsymbol{j})$, so $\boldsymbol{B}=B_{0} \boldsymbol{k}+\boldsymbol{b}(t)$. Show that the new Larmor precession frequency of the magnetic moment is $$ \Omega=\gamma \sqrt{\left(B_{0}-\frac{\omega}{\gamma}\right)^{2}+b^{2}} $$ Context answer: \boxed{证明题} Context question: 4. Instead of applying the field $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}+\sin \omega t \boldsymbol{j})$, now we apply $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}-\sin \omega t \boldsymbol{j})$, which rotates in the opposite direction and hence $\boldsymbol{B}=B_{0} \boldsymbol{k}+b(\cos \omega t \boldsymbol{i}-\sin \omega t \boldsymbol{j})$. What is the effective magnetic field $\boldsymbol{B}_{\text {eff }}$ for this case (in terms of the unit vectors $\boldsymbol{i}^{\prime}, \boldsymbol{j}^{\prime}, \boldsymbol{k}^{\prime}$ )? What is its time average, $\overline{\boldsymbol{B}_{\text {eff }}}$ (recall that $\overline{\cos 2 \pi t / T}=\overline{\sin 2 \pi t / T}=0$ )? Context answer: \boxed{$\mathbf{B}_{\mathrm{eff}}=\left(B_{0}-\frac{\omega}{\gamma}\right) \mathbf{k}^{\prime}+b\left(\cos 2 \omega t \mathbf{i}^{\prime}-\sin 2 \omega t \mathbf{j}^{\prime}\right)$ , $\overline{\mathbf{B}_{\mathrm{eff}}}=\left(B_{0}-\frac{\omega}{\gamma}\right) \mathbf{k}^{\prime}$} Extra Supplementary Reading Materials: Part C. Rabi oscillation For an ensemble of $N$ particles under the influence of a large magnetic field, the spin can have two quantum states: "up" and "down". Consequently, the total population of spin up $N_{\uparrow}$ and down $N_{\downarrow}$ obeys the equation $$ N_{\uparrow}+N_{\downarrow}=N $$ The difference of spin up population and spin down population yields the macroscopic magnetization along the $z$ axis: $$ M=\left(N_{\uparrow}-N_{\downarrow}\right) \mu=N \mu_{z} . $$ In a real experiment, two magnetic fields are usually applied, a large bias field $B_{0} \boldsymbol{k}$ and an oscillating field with amplitude $2 b$ perpendicular to the bias field $\left(b \ll B_{0}\right)$. Initially, only the large bias is applied, causing all the particles lie in the spin up states ( $\boldsymbol{\mu}$ is oriented in the $z$-direction at $t=0$ ). Then, the oscillating field is turned on, where its frequency $\omega$ is chosen to be in resonance with the Larmor precession frequency $\omega_{0}$, i.e. $\omega=\omega_{0}$. In other words, the total field after time $t=0$ is given by $$ \boldsymbol{B}(t)=B_{0} \boldsymbol{k}+2 b \cos \omega_{0} t \boldsymbol{i} . $$ Context question: 1. In the rotating frame $S^{\prime}$, show that the effective field can be approximated by $$ \boldsymbol{B}_{\text {eff }} \approx b \boldsymbol{i}^{\prime}, $$ which is commonly known as rotating wave approximation. What is the precession frequency $\Omega$ in frame $S^{\prime}$ ? Context answer: \boxed{$\Omega=\gamma b$} Context question: 2. Determine the angle $\alpha$ that $\boldsymbol{\mu}$ makes with $\boldsymbol{B}_{\text {eff }}$. Also, prove that the magnetization varies with time as $$ M(t)=N \mu(\cos \Omega t) . $$ Context answer: \boxed{证明题} Context question: 3. Under the application of magnetic field described above, determine the fractional population of each spin up $P_{\uparrow}=N_{\uparrow} / N$ and spin down $P_{\downarrow}=N_{\downarrow} / N$ as a function of time. Plot $P_{\uparrow}(t)$ and $P_{\downarrow}(t)$ on the same graph vs. time $t$. The alternating spin up and spin down population as a function of time is called Rabi oscillation. Context answer: \boxed{$P_{\downarrow}=\sin ^{2} \frac{\Omega t}{2}$ , $P_{\uparrow}=\cos ^{2} \frac{\Omega t}{2}$} Extra Supplementary Reading Materials: Part D. Measurement incompatibility Spin is in fact a vector quantity; but due to its quantum properties, we cannot measure each of its components simultaneously (i.e. we can know both $|\boldsymbol{\mu}|$ and $\mu_{z}$ as in above problems; but not all $|\boldsymbol{\mu}|, \mu_{x}, \mu_{y}$, and $\mu_{z}$ simultaneously). In this problem, we will do a calculation based on the Heisenberg uncertainty principle (using the relation $\Delta p_{q} \Delta q \geq \hbar$ ) to show how these measurements are incompatible with each other. Context question: 1. Let us consider an oven source of silver atoms, which has a small opening. The atoms stream out of the opening along $-y$ direction (see Figure below) and experience a spatial varying field $\boldsymbol{B}_{1}$. The field $\boldsymbol{B}_{1}$ has strong bias field component in the $z$ direction, where the atoms with different magnetic moment $\mu_{z}= \pm \gamma \hbar$ are split in the $z$ direction. At a distance $D$ from the oven source, a screen $S C_{1}$ is put to allow only spin up atoms to pass (blocking spin down atoms). Thus, at the instant after passing the screen, the atoms are prepared in spin up states. After the screen, the atoms enter a region of nonhomogenous field $\boldsymbol{B}_{2}$ where the atoms feel a force $$ F_{x}=\mu_{x} C $$ The field $\boldsymbol{B}_{2}$ has strong bias field component in the $x$ direction, where the atoms have magnetic moment $\mu_{x}= \pm \gamma \hbar$. <img_4533> In order to determine $\mu_{x}$ by observing the splitting in $x$ direction, show that the following condition must be fulfilled: $$ \frac{1}{\hbar}\left|\mu_{x}\right| \Delta x C t \gg 1 $$ where $t$ is the duration after leaving the screen $S C_{1}$ and $\Delta x$ is the opening width on $S C_{1}$. Context answer: \boxed{证明题}

2. The atoms are initially prepared in the spin up states right after leaving the screen, where $\mu_{z}=\gamma \hbar=\left|\mu_{x}\right|$. This means the atoms will precess at rates covering a range of values $\Delta \omega$ with respect to the $x$ component of $\boldsymbol{B}_{2}$, specifically $B_{2 x}=B_{0}+C x$. Prove that the spread in the precession angle $\Delta \omega t$ is so large and hence we cannot measure both $\mu_{x}$ and $\mu_{z}$ simultaneously. In other words, the measurement of $\mu_{x}$ destroys the information on $\mu_{z}$.

TP_MM_physics_en_COMP

2231

Geometry

Turbo the snail sits on a point on a circle with circumference 1. Given an infinite sequence of positive real numbers $c_{1}, c_{2}, c_{3}, \ldots$. Turbo successively crawls distances $c_{1}, c_{2}, c_{3}, \ldots$ around the circle, each time choosing to crawl either clockwise or counterclockwise. For example, if the sequence $c_{1}, c_{2}, c_{3}, \ldots$ is $0.4,0.6,0.3, \ldots$, then Turbo may start crawling as follows: Determine the largest constant $C>0$ with the following property: for every sequence of positive real numbers $c_{1}, c_{2}, c_{3}, \ldots$ with $c_{i}<C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.

Numerical

OE_MM_maths_en_COMP

2237

Geometry

In the diagram, $\angle A B F=41^{\circ}, \angle C B F=59^{\circ}, D E$ is parallel to $B F$, and $E F=25$. If $A E=E C$, determine the length of $A E$, to 2 decimal places.

Numerical

1e-1

OE_MM_maths_en_COMP

2240

Geometry

In triangle $A B C, A B=B C=25$ and $A C=30$. The circle with diameter $B C$ intersects $A B$ at $X$ and $A C$ at $Y$. Determine the length of $X Y$.

Numerical

OE_MM_maths_en_COMP

2245

Geometry

Points $P$ and $Q$ are located inside the square $A B C D$ such that $D P$ is parallel to $Q B$ and $D P=Q B=P Q$. Determine the minimum possible value of $\angle A D P$.

Numerical

OE_MM_maths_en_COMP

2246

Geometry

In the diagram, $\angle E A D=90^{\circ}, \angle A C D=90^{\circ}$, and $\angle A B C=90^{\circ}$. Also, $E D=13, E A=12$, $D C=4$, and $C B=2$. Determine the length of $A B$.

Numerical

OE_MM_maths_en_COMP

2250

Geometry

In the diagram, $A B C D$ is a quadrilateral with $A B=B C=C D=6, \angle A B C=90^{\circ}$, and $\angle B C D=60^{\circ}$. Determine the length of $A D$.

Numerical

OE_MM_maths_en_COMP

2252

Geometry

A triangle has vertices $A(0,3), B(4,0)$, $C(k, 5)$, where $0<k<4$. If the area of the triangle is 8 , determine the value of $k$.

Numerical

OE_MM_maths_en_COMP

2264

Geometry

A helicopter hovers at point $H$, directly above point $P$ on level ground. Lloyd sits on the ground at a point $L$ where $\angle H L P=60^{\circ}$. A ball is droppped from the helicopter. When the ball is at point $B, 400 \mathrm{~m}$ directly below the helicopter, $\angle B L P=30^{\circ}$. What is the distance between $L$ and $P$ ?

Numerical

OE_MM_maths_en_COMP

2267

Geometry

In the diagram, $A B C D$ is a quadrilateral in which $\angle A+\angle C=180^{\circ}$. What is the length of $C D$ ?

Numerical

OE_MM_maths_en_COMP

2269

Geometry

In the diagram, the parabola $$ y=-\frac{1}{4}(x-r)(x-s) $$ intersects the axes at three points. The vertex of this parabola is the point $V$. Determine the value of $k$ and the coordinates of $V$.

Numerical,Tuple

OE_MM_maths_en_COMP

2273

Combinatorics

A school has a row of $n$ open lockers, numbered 1 through $n$. After arriving at school one day, Josephine starts at the beginning of the row and closes every second locker until reaching the end of the row, as shown in the example below. Then on her way back, she closes every second locker that is still open. She continues in this manner along the row, until only one locker remains open. Define $f(n)$ to be the number of the last open locker. For example, if there are 15 lockers, then $f(15)=11$ as shown below: Determine $f(50)$.

Numerical

OE_MM_maths_en_COMP

2278

Geometry

In the diagram, $P Q R S$ is a quadrilateral. What is its perimeter?

Numerical

OE_MM_maths_en_COMP

2279

Geometry

In the diagram, $A$ has coordinates $(0,8)$. Also, the midpoint of $A B$ is $M(3,9)$ and the midpoint of $B C$ is $N(7,6)$. What is the slope of $A C$ ?

Numerical

OE_MM_maths_en_COMP

2284

Geometry

In the diagram, $A B D E$ is a rectangle, $\triangle B C D$ is equilateral, and $A D$ is parallel to $B C$. Also, $A E=2 x$ for some real number $x$. Determine the length of $A B$ in terms of $x$.

Expression

OE_MM_maths_en_COMP

2285

Geometry

In the diagram, $A B D E$ is a rectangle, $\triangle B C D$ is equilateral, and $A D$ is parallel to $B C$. Also, $A E=2 x$ for some real number $x$. Determine positive integers $r$ and $s$ for which $$ \frac{A C}{A D}=\sqrt{\frac{r}{s}} $$

Numerical

OE_MM_maths_en_COMP

2290

Number Theory

Five distinct integers are to be chosen from the set $\{1,2,3,4,5,6,7,8\}$ and placed in some order in the top row of boxes in the diagram. Each box that is not in the top row then contains the product of the integers in the two boxes connected to it in the row directly above. Determine the number of ways in which the integers can be chosen and placed in the top row so that the integer in the bottom box is 9953280000 .

Numerical

OE_MM_maths_en_COMP

2295

Geometry

In the diagram, eleven circles of four different radius 1, each circle labelled $X$ has radius 2, the circle labelled $Y$ has radius 4 , and the circle labelled $Z$ has radius $r$. Each of the circles labelled $W$ or $X$ is tangent to three other circles. The circle labelled $Y$ is tangent to all ten of the other circles. The circle labelled $Z$ is tangent to three other circles. Determine positive integers $s$ and $t$ for which $r=\frac{s}{t}$.

Numerical

OE_MM_maths_en_COMP

2297

Algebra

A circular disc is divided into 36 sectors. A number is written in each sector. When three consecutive sectors contain $a, b$ and $c$ in that order, then $b=a c$. If the number 2 is placed in one of the sectors and the number 3 is placed in one of the adjacent sectors, as shown, what is the sum of the 36 numbers on the disc?

Numerical

OE_MM_maths_en_COMP

2299

Geometry

In the diagram, $A C D F$ is a rectangle with $A C=200$ and $C D=50$. Also, $\triangle F B D$ and $\triangle A E C$ are congruent triangles which are right-angled at $B$ and $E$, respectively. What is the area of the shaded region?

Numerical

OE_MM_maths_en_COMP

2302

Geometry

In the diagram, $\triangle X Y Z$ is isosceles with $X Y=X Z=a$ and $Y Z=b$ where $b<2 a$. A larger circle of radius $R$ is inscribed in the triangle (that is, the circle is drawn so that it touches all three sides of the triangle). A smaller circle of radius $r$ is drawn so that it touches $X Y, X Z$ and the larger circle. Determine an expression for $\frac{R}{r}$ in terms of $a$ and $b$.

Expression

OE_MM_maths_en_COMP

2307

Geometry

In the diagram, what is the area of figure $A B C D E F$ ?

Numerical

OE_MM_maths_en_COMP

2308

Geometry

In the diagram, $A B C D$ is a rectangle with $A E=15, E B=20$ and $D F=24$. What is the length of $C F$ ?

Numerical

OE_MM_maths_en_COMP

2309

Geometry

In the diagram, $A B C D$ is a square of side length 6. Points $E, F, G$, and $H$ are on $A B, B C, C D$, and $D A$, respectively, so that the ratios $A E: E B, B F: F C$, $C G: G D$, and $D H: H A$ are all equal to $1: 2$. What is the area of $E F G H$ ?

Numerical

OE_MM_maths_en_COMP

2310

Geometry

In the diagram, line $A$ has equation $y=2 x$. Line $B$ is obtained by reflecting line $A$ in the $y$-axis. Line $C$ is perpendicular to line $B$. What is the slope of line $C$ ?

Numerical

OE_MM_maths_en_COMP

2311

Geometry

Three squares, each of side length 1 , are drawn side by side in the first quadrant, as shown. Lines are drawn from the origin to $P$ and $Q$. Determine, with explanation, the length of $A B$.

Numerical

OE_MM_maths_en_COMP

2314

Geometry

In the diagram, the parabola with equation $y=x^{2}+t x-2$ intersects the $x$-axis at points $P$ and $Q$. Also, the line with equation $y=3 x+3$ intersects the parabola at points $P$ and $R$. Determine the value of $t$ and the area of triangle $P Q R$.

Numerical

OE_MM_maths_en_COMP

2316

Geometry

In the diagram, $A C=B C, A D=7, D C=8$, and $\angle A D C=120^{\circ}$. What is the value of $x$ ?

Numerical

OE_MM_maths_en_COMP

2324

Geometry

Donna has a laser at $C$. She points the laser beam at the point $E$. The beam reflects off of $D F$ at $E$ and then off of $F H$ at $G$, as shown, arriving at point $B$ on $A D$. If $D E=E F=1 \mathrm{~m}$, what is the length of $B D$, in metres?

Numerical

OE_MM_maths_en_COMP

2335

Number Theory

An L shape is made by adjoining three congruent squares. The L is subdivided into four smaller L shapes, as shown. Each of the resulting L's is subdivided in this same way. After the third round of subdivisions, how many L's of the smallest size are there?

Numerical

OE_MM_maths_en_COMP

2337

Geometry

Jimmy is baking two large identical triangular cookies, $\triangle A B C$ and $\triangle D E F$. Each cookie is in the shape of an isosceles right-angled triangle. The length of the shorter sides of each of these triangles is $20 \mathrm{~cm}$. He puts the cookies on a rectangular baking tray so that $A, B, D$, and $E$ are at the vertices of the rectangle, as shown. If the distance between parallel sides $A C$ and $D F$ is $4 \mathrm{~cm}$, what is the width $B D$ of the tray?

cm

Numerical

OE_MM_maths_en_COMP

2347

Geometry

In the diagram, $\angle A C B=\angle A D E=90^{\circ}$. If $A B=75, B C=21, A D=20$, and $C E=47$, determine the exact length of $B D$.

Numerical

OE_MM_maths_en_COMP

2353

Geometry

A circle, with diameter $A B$ as shown, intersects the positive $y$-axis at point $D(0, d)$. Determine $d$.

Numerical

OE_MM_maths_en_COMP

2354

Geometry

A square $P Q R S$ with side of length $x$ is subdivided into four triangular regions as shown so that area (A) + area $(B)=\text{area}(C)$. If $P T=3$ and $R U=5$, determine the value of $x$.

Numerical

OE_MM_maths_en_COMP

2357

Geometry

In the diagram, $A D=D C, \sin \angle D B C=0.6$ and $\angle A C B=90^{\circ}$. What is the value of $\tan \angle A B C$ ?

Numerical

OE_MM_maths_en_COMP

2358

Geometry

On a cross-sectional diagram of the Earth, the $x$ and $y$-axes are placed so that $O(0,0)$ is the centre of the Earth and $C(6.40,0.00)$ is the location of Cape Canaveral. A space shuttle is forced to land on an island at $A(5.43,3.39)$, as shown. Each unit represents $1000 \mathrm{~km}$. Determine the distance from Cape Canaveral to the island, measured on the surface of the earth, to the nearest $10 \mathrm{~km}$.

km

Numerical

OE_MM_maths_en_COMP

2360

Geometry

The parabola $y=-x^{2}+4$ has vertex $P$ and intersects the $x$-axis at $A$ and $B$. The parabola is translated from its original position so that its vertex moves along the line $y=x+4$ to the point $Q$. In this position, the parabola intersects the $x$-axis at $B$ and $C$. Determine the coordinates of $C$.

Tuple

OE_MM_maths_en_COMP

2362

Geometry

In the isosceles trapezoid $A B C D$, $A B=C D=x$. The area of the trapezoid is 80 and the circle with centre $O$ and radius 4 is tangent to the four sides of the trapezoid. Determine the value of $x$.

Numerical

OE_MM_maths_en_COMP

2368

Geometry

In the diagram, points $P(p, 4), B(10,0)$, and $O(0,0)$ are shown. If $\triangle O P B$ is right-angled at $P$, determine all possible values of $p$.

Numerical

OE_MM_maths_en_COMP

2373

Geometry

A snail's shell is formed from six triangular sections, as shown. Each triangle has interior angles of $30^{\circ}, 60^{\circ}$ and $90^{\circ}$. If $A B$ has a length of $1 \mathrm{~cm}$, what is the length of $A H$, in $\mathrm{cm}$ ?

Numerical

OE_MM_maths_en_COMP

2374

Geometry

In rectangle $A B C D$, point $E$ is on side $D C$. Line segments $A E$ and $B D$ are perpendicular and intersect at $F$. If $A F=4$ and $D F=2$, determine the area of quadrilateral $B C E F$.

Numerical

OE_MM_maths_en_COMP

2378

Geometry

In the diagram, points $B, P, Q$, and $C$ lie on line segment $A D$. The semi-circle with diameter $A C$ has centre $P$ and the semi-circle with diameter $B D$ has centre $Q$. The two semi-circles intersect at $R$. If $\angle P R Q=40^{\circ}$, determine the measure of $\angle A R D$.

Numerical

OE_MM_maths_en_COMP

2385

Geometry

In the diagram, a line is drawn through points $P, Q$ and $R$. If $P Q=Q R$, what are the coordinates of $R$ ?

Tuple

OE_MM_maths_en_COMP

2386

Geometry

In the diagram, $O A=15, O P=9$ and $P B=4$. Determine the equation of the line through $A$ and $B$. Explain how you got your answer.

Expression

OE_MM_maths_en_COMP

2387

Geometry

In the diagram, $\triangle A B C$ is right-angled at $B$ and $A B=10$. If $\cos (\angle B A C)=\frac{5}{13}$, what is the value of $\tan (\angle A C B)$ ?

Numerical

OE_MM_maths_en_COMP

2389

Geometry

In the diagram, $A B=B C=2 \sqrt{2}, C D=D E$, $\angle C D E=60^{\circ}$, and $\angle E A B=75^{\circ}$. Determine the perimeter of figure $A B C D E$. Explain how you got your answer.

Numerical

OE_MM_maths_en_COMP

2393

Geometry

In the diagram, the parabola intersects the $x$-axis at $A(-3,0)$ and $B(3,0)$ and has its vertex at $C$ below the $x$-axis. The area of $\triangle A B C$ is 54 . Determine the equation of the parabola. Explain how you got your answer.

Expression

OE_MM_maths_en_COMP

2394

Geometry

In the diagram, $A(0, a)$ lies on the $y$-axis above $D$. If the triangles $A O B$ and $B C D$ have the same area, determine the value of $a$. Explain how you got your answer.

Numerical

OE_MM_maths_en_COMP

2395

Geometry

The Little Prince lives on a spherical planet which has a radius of $24 \mathrm{~km}$ and centre $O$. He hovers in a helicopter $(H)$ at a height of $2 \mathrm{~km}$ above the surface of the planet. From his position in the helicopter, what is the distance, in kilometres, to the furthest point on the surface of the planet that he can see?

km

Numerical

OE_MM_maths_en_COMP

2396

Geometry

In the diagram, points $A$ and $B$ are located on islands in a river full of rabid aquatic goats. Determine the distance from $A$ to $B$, to the nearest metre. (Luckily, someone has measured the angles shown in the diagram as well as the distances $C D$ and $D E$.)

m

Numerical

OE_MM_maths_en_COMP

2399

Geometry

In the $4 \times 4$ grid shown, three coins are randomly placed in different squares. Determine the probability that no two coins lie in the same row or column.

Numerical

OE_MM_maths_en_COMP

2400

Geometry

In the diagram, the area of $\triangle A B C$ is 1 . Trapezoid $D E F G$ is constructed so that $G$ is to the left of $F, D E$ is parallel to $B C$, $E F$ is parallel to $A B$ and $D G$ is parallel to $A C$. Determine the maximum possible area of trapezoid $D E F G$.

Numerical

OE_MM_maths_en_COMP

2404

Geometry

In the diagram, $\triangle P Q R$ has $P Q=a, Q R=b, P R=21$, and $\angle P Q R=60^{\circ}$. Also, $\triangle S T U$ has $S T=a, T U=b, \angle T S U=30^{\circ}$, and $\sin (\angle T U S)=\frac{4}{5}$. Determine the values of $a$ and $b$.

Numerical

OE_MM_maths_en_COMP

2405

Geometry

A triangle of area $770 \mathrm{~cm}^{2}$ is divided into 11 regions of equal height by 10 lines that are all parallel to the base of the triangle. Starting from the top of the triangle, every other region is shaded, as shown. What is the total area of the shaded regions?

Numerical

OE_MM_maths_en_COMP

2406

Geometry

A square lattice of 16 points is constructed such that the horizontal and vertical distances between adjacent points are all exactly 1 unit. Each of four pairs of points are connected by a line segment, as shown. The intersections of these line segments are the vertices of square $A B C D$. Determine the area of square $A B C D$.

Numerical

OE_MM_maths_en_COMP

2412

Combinatorics

At the Canadian Eatery with Multiple Configurations, there are round tables, around which chairs are placed. When a table has $n$ chairs around it for some integer $n \geq 3$, the chairs are labelled $1,2,3, \ldots, n-1, n$ in order around the table. A table is considered full if no more people can be seated without having two people sit in neighbouring chairs. For example, when $n=6$, full tables occur when people are seated in chairs labelled $\{1,4\}$ or $\{2,5\}$ or $\{3,6\}$ or $\{1,3,5\}$ or $\{2,4,6\}$. Thus, there are 5 different full tables when $n=6$. A full table with $6 k+5$ chairs, for some positive integer $k$, has $t$ people seated in its chairs. Determine, in terms of $k$, the number of possible values of $t$.

Expression

OE_MM_maths_en_COMP

2413

Combinatorics

At the Canadian Eatery with Multiple Configurations, there are round tables, around which chairs are placed. When a table has $n$ chairs around it for some integer $n \geq 3$, the chairs are labelled $1,2,3, \ldots, n-1, n$ in order around the table. A table is considered full if no more people can be seated without having two people sit in neighbouring chairs. For example, when $n=6$, full tables occur when people are seated in chairs labelled $\{1,4\}$ or $\{2,5\}$ or $\{3,6\}$ or $\{1,3,5\}$ or $\{2,4,6\}$. Thus, there are 5 different full tables when $n=6$. Determine the number of different full tables when $n=19$.

Numerical

OE_MM_maths_en_COMP

2417

Geometry

In the diagram, $\triangle A B C$ is right-angled at $B$ and $\triangle A C D$ is right-angled at $A$. Also, $A B=3, B C=4$, and $C D=13$. What is the area of quadrilateral $A B C D$ ?

Numerical

OE_MM_maths_en_COMP

2418

Geometry

Three identical rectangles $P Q R S$, WTUV and $X W V Y$ are arranged, as shown, so that $R S$ lies along $T X$. The perimeter of each of the three rectangles is $21 \mathrm{~cm}$. What is the perimeter of the whole shape?

cm

Numerical

OE_MM_maths_en_COMP

2424

Geometry

The diagram shows two hills that meet at $O$. One hill makes a $30^{\circ}$ angle with the horizontal and the other hill makes a $45^{\circ}$ angle with the horizontal. Points $A$ and $B$ are on the hills so that $O A=O B=20 \mathrm{~m}$. Vertical poles $B D$ and $A C$ are connected by a straight cable $C D$. If $A C=6 \mathrm{~m}$, what is the length of $B D$ for which $C D$ is as short as possible?

m

Numerical

OE_MM_maths_en_COMP

2428

Geometry

In the diagram, line segments $A C$ and $D F$ are tangent to the circle at $B$ and $E$, respectively. Also, $A F$ intersects the circle at $P$ and $R$, and intersects $B E$ at $Q$, as shown. If $\angle C A F=35^{\circ}, \angle D F A=30^{\circ}$, and $\angle F P E=25^{\circ}$, determine the measure of $\angle P E Q$.

Numerical

OE_MM_maths_en_COMP

2429

Geometry

In the diagram, $A B C D$ and $P N C D$ are squares of side length 2, and $P N C D$ is perpendicular to $A B C D$. Point $M$ is chosen on the same side of $P N C D$ as $A B$ so that $\triangle P M N$ is parallel to $A B C D$, so that $\angle P M N=90^{\circ}$, and so that $P M=M N$. Determine the volume of the convex solid $A B C D P M N$.

Numerical

OE_MM_maths_en_COMP

2436

Geometry

In the diagram, $\triangle A B C$ is right-angled at $B$ and $A C=20$. If $\sin C=\frac{3}{5}$, what is the length of side $B C$ ?

Numerical

OE_MM_maths_en_COMP

2437

Geometry

A helicopter is flying due west over level ground at a constant altitude of $222 \mathrm{~m}$ and at a constant speed. A lazy, stationary goat, which is due west of the helicopter, takes two measurements of the angle between the ground and the helicopter. The first measurement the goat makes is $6^{\circ}$ and the second measurement, which he makes 1 minute later, is $75^{\circ}$. If the helicopter has not yet passed over the goat, as shown, how fast is the helicopter travelling to the nearest kilometre per hour?

km/h

Numerical

OE_MM_maths_en_COMP

2446

Geometry

A regular hexagon is a six-sided figure which has all of its angles equal and all of its side lengths equal. In the diagram, $A B C D E F$ is a regular hexagon with an area of 36. The region common to the equilateral triangles $A C E$ and $B D F$ is a hexagon, which is shaded as shown. What is the area of the shaded hexagon?

Numerical

OE_MM_maths_en_COMP

2447

Geometry

At the Big Top Circus, Herc the Human Cannonball is fired out of the cannon at ground level. (For the safety of the spectators, the cannon is partially buried in the sand floor.) Herc's trajectory is a parabola until he catches the vertical safety net, on his way down, at point $B$. Point $B$ is $64 \mathrm{~m}$ directly above point $C$ on the floor of the tent. If Herc reaches a maximum height of $100 \mathrm{~m}$, directly above a point $30 \mathrm{~m}$ from the cannon, determine the horizontal distance from the cannon to the net.

m

Numerical

OE_MM_maths_en_COMP

2455

Geometry

In the diagram, $V$ is the vertex of the parabola with equation $y=-x^{2}+4 x+1$. Also, $A$ and $B$ are the points of intersection of the parabola and the line with equation $y=-x+1$. Determine the value of $A V^{2}+B V^{2}-A B^{2}$.

Numerical

OE_MM_maths_en_COMP

2456

Geometry

In the diagram, $A B C$ is a quarter of a circular pizza with centre $A$ and radius $20 \mathrm{~cm}$. The piece of pizza is placed on a circular pan with $A, B$ and $C$ touching the circumference of the pan, as shown. What fraction of the pan is covered by the piece of pizza?

Numerical

OE_MM_maths_en_COMP

2457

Geometry

The deck $A B$ of a sailboat is $8 \mathrm{~m}$ long. Rope extends at an angle of $60^{\circ}$ from $A$ to the top $(M)$ of the mast of the boat. More rope extends at an angle of $\theta$ from $B$ to a point $P$ that is $2 \mathrm{~m}$ below $M$, as shown. Determine the height $M F$ of the mast, in terms of $\theta$.

\mathrm{~m}

Expression

OE_MM_maths_en_COMP

2465

Geometry

In the diagram, triangle ABC is right-angled at B. MT is the perpendicular bisector of $B C$ with $M$ on $B C$ and $T$ on $A C$. If $A T=A B$, what is the size of $\angle A C B$ ?

Numerical

OE_MM_maths_en_COMP

2468

Geometry

In the diagram, $A B C D E F$ is a regular hexagon with a side length of 10 . If $X, Y$ and $Z$ are the midpoints of $A B, C D$ and $E F$, respectively, what is the length of $X Z$ ?

Numerical

OE_MM_maths_en_COMP

2470

Geometry

In the diagram, $A C=2 x, B C=2 x+1$ and $\angle A C B=30^{\circ}$. If the area of $\triangle A B C$ is 18 , what is the value of $x$ ?

Numerical

OE_MM_maths_en_COMP

2471

Geometry

A ladder, $A B$, is positioned so that its bottom sits on horizontal ground and its top rests against a vertical wall, as shown. In this initial position, the ladder makes an angle of $70^{\circ}$ with the horizontal. The bottom of the ladder is then pushed $0.5 \mathrm{~m}$ away from the wall, moving the ladder to position $A^{\prime} B^{\prime}$. In this new position, the ladder makes an angle of $55^{\circ}$ with the horizontal. Calculate, to the nearest centimetre, the distance that the ladder slides down the wall (that is, the length of $B B^{\prime}$ ).

cm

Numerical

OE_MM_maths_en_COMP

2480

Geometry

In the diagram, $P Q R S$ is an isosceles trapezoid with $P Q=7, P S=Q R=8$, and $S R=15$. Determine the length of the diagonal $P R$.

Numerical

OE_MM_maths_en_COMP

2483

Geometry

In the diagram, $\triangle A B C$ has $A B=A C$ and $\angle B A C<60^{\circ}$. Point $D$ is on $A C$ with $B C=B D$. Point $E$ is on $A B$ with $B E=E D$. If $\angle B A C=\theta$, determine $\angle B E D$ in terms of $\theta$.

Expression

OE_MM_maths_en_COMP

2484

Geometry

In the diagram, the ferris wheel has a diameter of $18 \mathrm{~m}$ and rotates at a constant rate. When Kolapo rides the ferris wheel and is at its lowest point, he is $1 \mathrm{~m}$ above the ground. When Kolapo is at point $P$ that is $16 \mathrm{~m}$ above the ground and is rising, it takes him 4 seconds to reach the highest point, $T$. He continues to travel for another 8 seconds reaching point $Q$. Determine Kolapo's height above the ground when he reaches point $Q$.

m

Numerical

OE_MM_maths_en_COMP

2485

Algebra

On Saturday, Jimmy started painting his toy helicopter between 9:00 a.m. and 10:00 a.m. When he finished between 10:00 a.m. and 11:00 a.m. on the same morning, the hour hand was exactly where the minute hand had been when he started, and the minute hand was exactly where the hour hand had been when he started. Jimmy spent $t$ hours painting. Determine the value of $t$.

Numerical

OE_MM_maths_en_COMP

2491

Geometry

In the diagram, the circle with centre $C(1,1)$ passes through the point $O(0,0)$, intersects the $y$-axis at $A$, and intersects the $x$-axis at $B(2,0)$. Determine, with justification, the coordinates of $A$ and the area of the part of the circle that lies in the first quadrant.

Tuple,Numerical

OE_MM_maths_en_COMP

2495

Geometry

Survivors on a desert island find a piece of plywood $(A B C)$ in the shape of an equilateral triangle with sides of length $2 \mathrm{~m}$. To shelter their goat from the sun, they place edge $B C$ on the ground, lift corner $A$, and put in a vertical post $P A$ which is $h \mathrm{~m}$ long above ground. When the sun is directly overhead, the shaded region $(\triangle P B C)$ on the ground directly underneath the plywood is an isosceles triangle with largest angle $(\angle B P C)$ equal to $120^{\circ}$. Determine the value of $h$, to the nearest centimetre.

cm

Numerical

OE_MM_maths_en_COMP

2503

Geometry

Points $A_{1}, A_{2}, \ldots, A_{N}$ are equally spaced around the circumference of a circle and $N \geq 3$. Three of these points are selected at random and a triangle is formed using these points as its vertices. Through this solution, we will use the following facts: When an acute triangle is inscribed in a circle: - each of the three angles of the triangle is the angle inscribed in the major arc defined by the side of the triangle by which it is subtended, - each of the three arcs into which the circle is divided by the vertices of the triangles is less than half of the circumference of the circle, and - it contains the centre of the circle. Why are these facts true? - Consider a chord of a circle which is not a diameter. Then the angle subtended in the major arc of this circle is an acute angle and the angle subtended in the minor arc is an obtuse angle. Now consider an acute triangle inscribed in a circle. Since each angle of the triangle is acute, then each of the three angles is inscribed in the major arc defined by the side of the triangle by which it is subtended. - It follows that each arc of the circle that is outside the triangle must be a minor arc, thus less than the circumference of the circle. - Lastly, if the centre was outside the triangle, then we would be able to draw a diameter of the circle with the triangle entirely on one side of the diameter. In this case, one of the arcs of the circle cut off by one of the sides of the triangle would have to be a major arc, which cannot happen, because of the above. Therefore, the centre is contained inside the triangle. If $N=7$, what is the probability that the triangle is acute? (A triangle is acute if each of its three interior angles is less than $90^{\circ}$.)

Numerical

OE_MM_maths_en_COMP

2505

Geometry

In the diagram, $\triangle P Q S$ is right-angled at $P$ and $\triangle Q R S$ is right-angled at $Q$. Also, $P Q=x, Q R=8, R S=x+8$, and $S P=x+3$ for some real number $x$. Determine all possible values of the perimeter of quadrilateral $P Q R S$.

Numerical

OE_MM_maths_en_COMP

2511

Geometry

In the diagram, $\triangle A B D$ has $C$ on $B D$. Also, $B C=2, C D=1, \frac{A C}{A D}=\frac{3}{4}$, and $\cos (\angle A C D)=-\frac{3}{5}$. Determine the length of $A B$.

Numerical

OE_MM_maths_en_COMP

2512

Algebra

Suppose that $a>\frac{1}{2}$ and that the parabola with equation $y=a x^{2}+2$ has vertex $V$. The parabola intersects the line with equation $y=-x+4 a$ at points $B$ and $C$, as shown. If the area of $\triangle V B C$ is $\frac{72}{5}$, determine the value of $a$.

Numerical

OE_MM_maths_en_COMP

2514

Number Theory

Suppose that $m$ and $n$ are positive integers with $m \geq 2$. The $(m, n)$-sawtooth sequence is a sequence of consecutive integers that starts with 1 and has $n$ teeth, where each tooth starts with 2, goes up to $m$ and back down to 1 . For example, the $(3,4)$-sawtooth sequence is The $(3,4)$-sawtooth sequence includes 17 terms and the average of these terms is $\frac{33}{17}$. Determine the sum of the terms in the $(4,2)$-sawtooth sequence.

Numerical

OE_MM_maths_en_COMP

2515

Number Theory

Suppose that $m$ and $n$ are positive integers with $m \geq 2$. The $(m, n)$-sawtooth sequence is a sequence of consecutive integers that starts with 1 and has $n$ teeth, where each tooth starts with 2, goes up to $m$ and back down to 1 . For example, the $(3,4)$-sawtooth sequence is The $(3,4)$-sawtooth sequence includes 17 terms and the average of these terms is $\frac{33}{17}$. For each positive integer $m \geq 2$, determine a simplified expression for the sum of the terms in the $(m, 3)$-sawtooth sequence.

Expression

OE_MM_maths_en_COMP

2517

Combinatorics

At Pizza by Alex, toppings are put on circular pizzas in a random way. Every topping is placed on a randomly chosen semicircular half of the pizza and each topping's semi-circle is chosen independently. For each topping, Alex starts by drawing a diameter whose angle with the horizonal is selected uniformly at random. This divides the pizza into two semi-circles. One of the two halves is then chosen at random to be covered by the topping. For a 2-topping pizza, determine the probability that at least $\frac{1}{4}$ of the pizza is covered by both toppings.

Numerical

OE_MM_maths_en_COMP

2518

Combinatorics

At Pizza by Alex, toppings are put on circular pizzas in a random way. Every topping is placed on a randomly chosen semicircular half of the pizza and each topping's semi-circle is chosen independently. For each topping, Alex starts by drawing a diameter whose angle with the horizonal is selected uniformly at random. This divides the pizza into two semi-circles. One of the two halves is then chosen at random to be covered by the topping. For a 3-topping pizza, determine the probability that some region of the pizza with non-zero area is covered by all 3 toppings. (The diagram above shows an example where no region is covered by all 3 toppings.)

Numerical

OE_MM_maths_en_COMP

2519

Combinatorics

At Pizza by Alex, toppings are put on circular pizzas in a random way. Every topping is placed on a randomly chosen semicircular half of the pizza and each topping's semi-circle is chosen independently. For each topping, Alex starts by drawing a diameter whose angle with the horizonal is selected uniformly at random. This divides the pizza into two semi-circles. One of the two halves is then chosen at random to be covered by the topping. Suppose that $N$ is a positive integer. For an $N$-topping pizza, determine the probability, in terms of $N$, that some region of the pizza with non-zero area is covered by all $N$ toppings.

Expression

OE_MM_maths_en_COMP

2530

Geometry

In rectangle $A B C D, F$ is on diagonal $B D$ so that $A F$ is perpendicular to $B D$. Also, $B C=30, C D=40$ and $A F=x$. Determine the value of $x$.

Numerical

OE_MM_maths_en_COMP

2533

Geometry

In the diagram, $\triangle A B C$ is right-angled at $C$. Also, $2 \sin B=3 \tan A$. Determine the measure of angle $A$.

Numerical

OE_MM_maths_en_COMP

2535

Geometry

Alice drove from town $E$ to town $F$ at a constant speed of $60 \mathrm{~km} / \mathrm{h}$. Bob drove from $F$ to $E$ along the same road also at a constant speed. They started their journeys at the same time and passed each other at point $G$. Alice drove from $G$ to $F$ in 45 minutes. Bob drove from $G$ to $E$ in 20 minutes. Determine Bob's constant speed.

km/h

Numerical

OE_MM_maths_en_COMP

2537

Geometry

In the diagram, $D$ is the vertex of a parabola. The parabola cuts the $x$-axis at $A$ and at $C(4,0)$. The parabola cuts the $y$-axis at $B(0,-4)$. The area of $\triangle A B C$ is 4. Determine the area of $\triangle D B C$.

Numerical

OE_MM_maths_en_COMP

2540

Geometry

In the diagram, $P Q R S$ is a square with sides of length 4. Points $T$ and $U$ are on sides $Q R$ and $R S$ respectively such that $\angle U P T=45^{\circ}$. Determine the maximum possible perimeter of $\triangle R U T$.

Numerical

OE_MM_maths_en_COMP

2541

Algebra

Suppose there are $n$ plates equally spaced around a circular table. Ross wishes to place an identical gift on each of $k$ plates, so that no two neighbouring plates have gifts. Let $f(n, k)$ represent the number of ways in which he can place the gifts. For example $f(6,3)=2$, as shown below. Throughout this problem, we represent the states of the $n$ plates as a string of 0's and 1's (called a binary string) of length $n$ of the form $p_{1} p_{2} \cdots p_{n}$, with the $r$ th digit from the left (namely $p_{r}$ ) equal to 1 if plate $r$ contains a gift and equal to 0 if plate $r$ does not. We call a binary string of length $n$ allowable if it satisfies the requirements - that is, if no two adjacent digits both equal 1. Note that digit $p_{n}$ is also "adjacent" to digit $p_{1}$, so we cannot have $p_{1}=p_{n}=1$. Determine the value of $f(7,3)$.

Numerical

OE_MM_maths_en_COMP

2543

Algebra

Suppose there are $n$ plates equally spaced around a circular table. Ross wishes to place an identical gift on each of $k$ plates, so that no two neighbouring plates have gifts. Let $f(n, k)$ represent the number of ways in which he can place the gifts. For example $f(6,3)=2$, as shown below. Throughout this problem, we represent the states of the $n$ plates as a string of 0's and 1's (called a binary string) of length $n$ of the form $p_{1} p_{2} \cdots p_{n}$, with the $r$ th digit from the left (namely $p_{r}$ ) equal to 1 if plate $r$ contains a gift and equal to 0 if plate $r$ does not. We call a binary string of length $n$ allowable if it satisfies the requirements - that is, if no two adjacent digits both equal 1. Note that digit $p_{n}$ is also "adjacent" to digit $p_{1}$, so we cannot have $p_{1}=p_{n}=1$. Determine the smallest possible value of $n+k$ among all possible ordered pairs of integers $(n, k)$ for which $f(n, k)$ is a positive multiple of 2009 , where $n \geq 3$ and $k \geq 2$.

Numerical

OE_MM_maths_en_COMP