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2018-02-12 Advanced

         

I've placed 2018 points in a plane such that it is not possible to draw a straight line that passes through exactly two points.

If a line passes through more than two of the points, then what is the minimum number of points that it passes through?

If 3 points are placed on a plane such that it is not possible to draw a line through exactly two points, then the 3 points must be collinear.

If 3 points are placed on a plane such that it is not possible to draw a line through exactly two points, then the 3 points must be collinear.

A simple bridge of three triangular elements spans a river and is carried by the foundation at \(A\) and \(C\) in the diagram. The connecting lines between the points correspond to steel girders, each of which can bear a maximum tensile and compressive load of 10 tons.

What is the minimum angle \(\alpha\) needed so that a truck with a weight of 10 tons can safely cross the bridge?


Hints:

  • Assume that the weight load of the truck is concentrated at point \(B \), so here a force of \(\vec F_B = (10 \,\text{tons})\,\cdot \vec g \) acts on the bridge.
  • The bridge is held by the opposing forces \(\vec F_A = \vec F_C = - \frac{1}{2} \vec F_B \) on points \(A \) and \(C \). Calculate the forces acting along the steel girders.

Note: for every point of the polygon, the sum of all forces must be zero.

A function \(f(n)\) is defined over positive integers as follows:

\[f(n) = \begin{cases} 0 & \text{if }n\text{ is a perfect square}; \\ 1 & \text{if }n\text{ is closer to the perfect square before it than to the one after it}; \\ -1 & \text{otherwise}. \end{cases}\]

For example, \(f(1) = 0\) because 1 is a perfect square; \(f(2) = 1\) because 2 is closer to 1 than it is to 4; \(f(7) = -1\) because 7 is closer to 9 than it is to 4.

What can be said about the series \(\displaystyle \sum_{n=1}^{\infty} \frac{f(n)}{n}?\)

In the diagram below, two semicircles and one quarter-circle are inscribed in a unit square.

Inside one of the partitioned sections, infinitely many circles are drawn that are tangent to each other and to the boundaries. Let \(r_n\) denote the radius of the \(n^\text{th} \) largest circle.

Express \(r_n \) in terms of \(n.\)

Imagine a pyramid made out of blocks, where the first layer has 1 block, the second layer is a \(2\times 2\) square made up of 4 blocks, the third layer is a \(3\times 3\) square made up of 9 blocks, and so on, so that each \(n^\text{th}\) layer is a square made from \(n^2\) blocks.

Each block has a number written on it. The top block has the number 1 written on it, but every other block has the sum of all the blocks that touch its top side written on it. The first five layers are as follows:

What number would be written on the block that is in the \(3^\text{rd}\) column and \(4^\text{th}\) row of the \(100^\text{th}\) layer?

Note: This problem does not require a programming solution, although you may want to use a calculator for the last step!

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