# Problems of the Week

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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.

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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