Let $a>1.$ A uniform wire is bent into a form coinciding with the portion of the curve $y=e^x$ for $x\in [0,a]$, and the line segment $y=e^a$ for $x\in [a-1,a].$ The wire is then suspended from the point $(a-1, e^a)$ and a horizontal force $F$ is applied to the point $(0,1)$ to hold the wire in coincidence with the curve and segment. Show that the force $F$ is directed to the right.

A particle of unit mass moves on a straight line under the action of a force which is a function $f(v)$ of the velocity $v$ of the particle, but the form of the function is not known. A motion is observed, and the distance $x$ covered in time $t$ satisfies the formula $x= at^2 + bt+c$, where $a,b,c$ have numerical values determined by observation of the motion. Find the function $f(v)$ for the range of $v$ covered by the experiment.

$n$ cockroaches are sitting on the plane at the vertices of the regular $ n $ -gon. They simultaneously begin to move at a speed of $ v $ on the sides of the polygon in the direction of the clockwise adjacent cockroach, then they continue moving in the initial direction with the initial speed. Vasya a entomologist moves on a straight line in the plane at a speed of $u$. After some time, it turned out that Vasya has crushed three cockroaches. Prove that $ v = u $.

A spider built a web on the unit circle. The web is a planar graph with straight edges inside the circle, bounded by the circumference of the circle. Each vertex of the graph lying on the circle belongs to a unique edge, which goes perpendicularly inward to the circle. For each vertex of the graph inside the circle, the sum of the unit outgoing vectors along the edges of the graph is zero. Prove that the total length of the web is equal to the number of its vertices on the circle.

A particle moves under a central force inversely proportional to the $k$-th power of the distance. If the particle describes a circle ( the central force proceeding from a point on the circumference of the circle ), find $k$.

A particle falls in a vertical plane from rest under the influence of gravity and a force perpendicular to and proportional to its velocity. Obtain the equations of the trajectory and identify the curve.

A projectile, thrown with initial velocity $v_0$ in a direction making angle $\alpha$ with the horizontal, is acted on by no force except gravity. Find the lenght of its path until it strikes a horizontal plane through the starting point. Show that the flight is longest when $$\sin \alpha \log(\sec \alpha+ \tan \alpha)=1.$$

Two uniform solid spheres of equal radii are so placed that one is directly above the other. The bottom sphere is fixed, and the top sphere, initially at rest, rolls off. At what point will contact between the two spheres be "lost"? Assume the coefficient of friction is such that no slipping occurs.

A particle moves in a straight line with monotonically decreasing acceleration. It starts from rest and has velocity $v$ a distance $d$ from the start. What is the maximum time it could have taken to travel the distance $d$?

A projectile moves in a resisting medium. The resisting force is a function of the velocity and is directed along the velocity vector. The equation $x=f(t)$ (where $f(t)$ is not constant) gives the horizontal distance in terms of the time $t$. Show that the vertical distance $y$ is given by $$y=-gf(t) \int \frac{dt}{f'(t)} + g \int \frac{f(t)}{f'(t)} \, dt +Af(t)+B$$ where $A$ and $B$ are constants and $g$ is the acceleration due to gravity.

The driver starts driving every morning at the same time from office to the house of his boss, picks up the boss and then drives back to the office. He always drives with the same speed on the same road. Because the time of arrival of the car to the boss's house is predetermined, the boss always leaves the house on time, and thus the driver does not spend any time waiting for his boss. Once the driver started driving from the office $42$ minutes later, than usual. The boss saw that the car didn't come and started walking in the direction of office. When he met the car on the road, the driver picked him up and started driving back to the office. The speed of the boss is 20 times lower than the speed of the car, and the time usually spent on the route from office to the house is at least an hour. Determine did the car come earlier or later to the office and by how many minutes.

There are $n\ge 3$ particles on a circle situated at the vertices of a regular $n$-gon. All these particles move on the circle with the same constant speed. One of the particles moves in the clockwise direction while all others move in the anti-clockwise direction. When particles collide, that is, they are all at the same point, they all reverse the direction of their motion and continue with the same speed as before. Let $s$ be the smallest number of collisions after which all particles return to their original positions. Find $s$. [i]Proposed by N.V. Tejaswi[/i]

A uniform rod of length $2k$ and weight $w$ rests with the end $A$ against a vertical wall, while the lower end $B$ is fastened by a string $BC$ of length $2b$ coming from a point $C$ in the wall above $A.$ If the system is in equilibrium, determine the angle $ABC.$

A particle moves in $3$-space according to the equations: $$ \frac{dx}{dt} =yz,\; \frac{dy}{dt} =xz,\; \frac{dz}{dt}= xy.$$ Show that: (a) If two of $x(0), y(0), z(0)$ equal $0,$ then the particle never moves. (b) If $x(0)=y(0)=1, z(0)=0,$ then the solution is $$ x(t)= \sec t ,\; y(t) =\sec t ,\; z(t)= \tan t;$$ whereas if $x(0)=y(0)=1, z(0)=-1,$ then $$ x(t) =\frac{1}{t+1} ,\; y(t)=\frac{1}{t+1}, z(t)=- \frac{1}{t+1}.$$ (c) If at least two of the values $x(0), y(0), z(0)$ are different from zero, then either the particle moves to infinity at some finite time in the future, or it came from infinity at some finite time in the past (a point $(x, y, z)$ in $3$-space "moves to infinity" if its distance from the origin approaches infinity).

In a solid sphere of radius $R$ the density $\rho$ is a function of $r$, the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is $k r^2$, where $k$ is a constant, find $\rho$ as a function of $r.$ Find also the magnitude of the force of attraction at a point outside the sphere at a distance $r$ from the center.

On a river with a current speed of \(3 \, \text{km/h}\), there are two harbors. Every Saturday, a cruise ship departs from Harbor 1 to Harbor 2, stays overnight, and returns to Harbor 1 on Sunday. On the ship live two snails, Romeo and Juliet. One Saturday, immediately after the ship departs, both snails start moving to meet each other and do so exactly when the ship arrives at Harbor 2. On the following Sunday, as the ship departs from Harbor 2, Romeo starts moving towards Juliet's house and reaches there exactly when the ship arrives back at Harbor 1. Given that Juliet moves half as fast as Romeo, determine the speed of the ship in still water. [i]Proposed by Demetre Gelashvili, Georgia [/i]

Astrophysicists have discovered a minor planet of radius $30$ kilometers whose surface is completely covered in water. A spherical meteor hits this planet and is submerged in the water. This incidence causes an increase of $1$ centimeters to the height of the water on this planet. What is the radius of the meteor in meters?

The motion of the particles of a fluid in the plane is specified by the following components of velocity $$\frac{dx}{dt}=y+2x(1-x^2 -y^2),\;\; \frac{dy}{dt}=-x.$$ Sketch the shape of the trajectories near the origin. Discuss what happens to an individual particle as $t\to \infty$, and justify your conclusion.

A coast artillery gun can fire at every angle of elevation between $0^{\circ}$ and $90^{\circ}$ in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant ($=v_0 $), determine the set $H$ of points in the plane and above the horizontal which can be hit.

The distance between the city and the house of Borya is 2km. Once Borya went from the city to his house with speed 4km/h. Simultaneously with that a dog Sharik started running out of house in the direction to city, and whenever Sharik meets Borya or the house, it starts running back (so the dog runs between Borya and the house), and when the dog runs to the house, its speed is 8km/h, and when it runs from the house, its speed is 12km/h. What distance will Sharik run until Borya comes to the house? [i]Yauheni Barabanau[/i]