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Found: 5

Problem #PRU-57750

Problems Combinatorics Systems of points and line segments Center of mass Center of mass formula

14–14 2.0

Prove that the medians of the triangle \(ABC\) intersect at one point and that point divides the medians in a ratio of \(2: 1\), counting from the vertex.

Problem #PRU-60381

Problems Combinatorics Systems of points and line segments Systems of points and line segments (other)

12–13 2.0

There are \(n\) points on the plane. How many lines are there with endpoints at these points?

Problem #PRU-60382

Problems Combinatorics Systems of points and line segments Systems of points, lines, and line segments

13–15 2.0

On a plane \(n\) randomly placed lines are given. What is the number of triangles formed by them?

Problem #PRU-32992

Problems Geometry Solid geometry Regular polyhedra Main features and definitions of regular polyhedra Space transformations Projection Projection helps solve the problem Combinatorics Systems of points and line segments Systems of points and line segments (counterexamples)

13–13 2.0

a) Can 4 points be placed on a plane so that each of them is connected by segments with three points (without intersections)?

b) Can 6 points be placed on a plane and connected by non-intersecting segments so that exactly 4 segments emerge from each point?

Problem #PRU-34962

Problems Methods Pigeonhole principle Pigeonhole principle (other) Combinatorics Systems of points and line segments Systems of points, lines, and line segments

12–14 2.0

Is it possible to arrange 1000 line segments in a plane so that both ends of each line segment rest strictly inside another line segment?