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Found: 3

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-60383

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

13–14 3.0

On two parallel lines $a$ and $b$ , the points $A_{1}, A_{2}, \dots, A_{m}$ and $B_{1}, B_{2}, \dots, B_{n}$ are chosen, respectively, and all of the segments of the form $A_{i} B_{j}$ , where $1 \leq i \leq m$ , $1 \leq j \leq n$ . How many intersection points will there be if it is known that no three of these segments intersect at one point?

Problem #PRU-32135

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Combinatorics Systems of points and line segments Systems of points and line segments (other)

12–14 3.0

101 points are marked on a plane; not all of the points lie on the same straight line. A red pencil is used to draw a straight line passing through each possible pair of points. Prove that there will always be a marked point on the plane through which at least 11 red lines pass.