Problems

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Found: 2370

There are 16 cities in the kingdom. Prove that it is not possible to build a system of roads in such a way that one can get from any city to any other without passing through more than one city on the way, and with at most four roads coming out of each city.

Recall that a line is tangent to a circle if they have only one point of intersection, a circle is called inscribed in a polygon if it is tangent to every side as a segment of that polygon.
In the triangle CDE the angle CDE=90 and the line DH is the median. A circle with center A is inscribed in the triangle CDH and is tangent to the segment DH in its middle, let’s denote it as G, so GH=DG. Find the angles of the triangle CDE.

A circle with center A is inscribed into a square CDFE. A line GH intersects the sides CD and CE of the square and is tangent to the circle at the point I. Find the perimeter of the triangle CHG (the sum of lengths of all the sides) if the side of the square is 10cm.

Recall that a line is tangent to a circle if they have only one point of intersection, a circle is called inscribed in a polygon if it is tangent to every side as a segment of that polygon.
In the triangle EFG the line EH is the median. Two circles with centres A and C are inscribed into triangles EFH and EGH respectively, they are tangent to the median EH at the points B and D. Find the length of BD if EFEG=2.

Is it possible to cover a 6×6 board with the L-tetraminos without overlapping? The pieces can be flipped and turned.

Is it possible to cover a (4n+2)×(4n+2) board with the L-tetraminos without overlapping for any n? The pieces can be flipped and turned.

Is it possible to cover a 4n×4n board with the L-tetraminos without overlapping for any n? The pieces can be flipped and turned.

Each number denotes the area of a rectangle it is written into. What is the area of the last rectangle?

Divide the trapezium into two parts such that they can be reassembled to make a triangle