We solve problems

Problems Methods Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (other)

Each of the 1994 deputies in parliament slapped exactly one of his colleagues. Prove that it is possible to draw up a parliamentary commission of 665 people whose members did not clarify the relationship between themselves in the manner indicated above.

Problem #PRU-116560

Problems Methods Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

14–15 4.0

2011 numbers are written on a blackboard. It turns out that the sum of any of these written numbers is also one of the written numbers. What is the minimum number of zeroes within this set of 2011 numbers?

Problem #PRU-116873

Problems Methods Extremal principle Extremal principle (other) Mathematical induction Mathematical induction (other) Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

14–15 4.0

We are given \(n+1\) different natural numbers, which are less than \(2n\) (\(n>1\)). Prove that among them there will always be three numbers, where the sum of two of them is equal to the third.

Problem #PRU-34991

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

13–15 4.0

On a line, there are 50 segments. Prove that either it is possible to find some 8 segments all of which have a shared intersection, or there can be found 8 segments, no two of which intersect.

Problem #PRU-60489

Problems Number Theory Divisibility Equations in integer numbers Methods Extremal principle Extremal principle (other) Methods from geometry Geometric interpretations in algebra Pigeonhole principle Pigeonhole principle (other) The greatest common divisor (GCD) and the least common multiplier (LCM). Mutually prime numbers

13–15 4.0

Let \(a\), \(b\), \(c\) be integers; where \(a\) and \(b\) are not equal to zero.

Prove that the equation \(ax + by = c\) has integer solutions if and only if \(c\) is divisible by \(d = \mathrm{GCD} (a, b)\).

Problem #PRU-65204

Problems Geometry Plane geometry Triangles Similar triangles Auxiliary similar triangles Methods Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (area and volume)

14–16 4.0

A unit square is divided into \(n\) triangles. Prove that one of the triangles can be used to completely cover a square with side length \(\frac{1}{n}\).

Problem #PRU-78557

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Covers Methods Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

Several pieces of carpet are laid along a corridor. Pieces cover the entire corridor from end to end without omissions and even overlap one another, so that over some parts of the floor lie several layers of carpet. Prove that you can remove a few pieces, perhaps by taking them out from under others and leaving the rest exactly in the same places they used to be, so that the corridor will still be completely covered and the total length of the pieces left will be less than twice the length corridor.

Problem #PRU-79322

Problems Geometry Solid geometry Spheres Circle on a sphere Methods Pigeonhole principle Pigeonhole principle (area and volume) Extremal principle Smallest or largest distance (length)

14–16 4.0

A spherical sun is observed to have a finite number of circular sunspots, each of which covers less than half of the sun’s surface. These sunspots are said to be enclosed, that is no two sunspots can touch, and they do not overlap with one another. Prove that the sun will have two diametrically opposite points that are not covered by sunspots.

Problem #PRU-98170

Problems Methods Extremal principle Extremal principle (other) Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

10–14 4.0

Ben noticed that all 25 of his classmates have a different number of friends in this class. How many friends does Ben have?