# Problems of the Week

Contribute a problem

How many ways can you put all the numbers 1 through 9 in this triangular grid, such that numbers along each arrow form an increasing sequence?

The diagram shows one possible configuration.

A ball of mass $$m$$ is moving between two parallel plates of mass much larger than $$m$$, separated by distance $$L$$. The ball moves with velocity $$v$$ in the direction perpendicular to the plates and collides with the plates multiple times. Due to the momentum transfer at the collisions, each plate experiences an average force of $$F$$ that is acting to push the plates apart.

Initially, the velocity of the ball is $$v_0$$ and the distance between the plates is $$L_0$$. As the separation of the planes is slowly changed, the force is changing and it can be expressed as $F= \frac{m v_0^2}{ L} \left(\frac{L_0}{L}\right)^A.$ What is the value of $$A?$$

Assume that the collision between the ball and the plates is perfectly elastic. Neglect the effect of gravity.

A circle of radius $$20$$ is sliced into three congruent sectors, which are then slid apart to create a green equilateral triangle with side length $$9.$$ A larger equilateral triangle is then circumscribed.

The side length of this large triangle can be written as $$\frac{a+b\sqrt{d}}{c}$$ with $$d$$ square-free and $$a, b, c$$ irreducible.

What is the value of $$a+b+c+d?$$

A line is "special" to a curve if it is both a tangent and a normal of the curve.

Only two lines are special to the curve $$x^3+ax^2+bx+c,$$ where $$a,b,c$$ are constants, and the two lines have the same slope.

What is this slope?

An $$8.1\text{ m}$$-long uniform ladder stands on a frictionless floor and leans against a frictionless wall. It is initially held motionless, with its bottom end at a very small distance from the wall.

The ladder is then released from rest.

What is the speed of the ladder's tip when it hits the ground?

Note: Take $$g=10\text{ m}/\text{s}^2.$$

×