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2018-10-01 Advanced

         

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 \(g=10\text{ m}/\text{s}^2.\)

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