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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 AA and CC 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 α\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 BB , so here a force of FB=(10tons)g\vec F_B = (10 \,\text{tons})\,\cdot \vec g acts on the bridge.
  • The bridge is held by the opposing forces FA=FC=12FB\vec F_A = \vec F_C = - \frac{1}{2} \vec F_B on points AA and CC . 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)f(n) is defined over positive integers as follows:

f(n)={0if n is a perfect square;1if n is closer to the perfect square before it than to the one after it;1otherwise.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)=0f(1) = 0 because 1 is a perfect square; f(2)=1f(2) = 1 because 2 is closer to 1 than it is to 4; f(7)=1f(7) = -1 because 7 is closer to 9 than it is to 4.

What can be said about the series n=1f(n)n?\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 rnr_n denote the radius of the nthn^\text{th} largest circle.

Express rnr_n in terms of n.n.

Imagine a pyramid made out of blocks, where the first layer has 1 block, the second layer is a 2×22\times 2 square made up of 4 blocks, the third layer is a 3×33\times 3 square made up of 9 blocks, and so on, so that each nthn^\text{th} layer is a square made from n2n^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 3rd3^\text{rd} column and 4th4^\text{th} row of the 100th100^\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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