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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-32132

Problems Methods Extremal principle Extremal principle (other) Set theory and logic Order relations Algebraic methods Proof by exhaustion

13–15 3.0

The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from \(n\) books of different heights, if: a) \(n = 4\); b) \(n = 5\)?

Problem #PRU-34991

Problems Methods Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (other) 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 Algebra 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-61399

Problems Algebra Number theory. Divisibility Equations in integer numbers Methods Extremal principle Sorting in increasing (decreasing) order

13–15 2.0

Prove that the equation \(\frac {x}{y} + \frac {y}{z} + \frac {z}{x} = 1\) is unsolvable using positive integers.

Problem #PRU-65204

Problems Geometry Plane geometry Triangles Similar triangles Auxiliary simialr 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-65266

Problems Probability and statistics Probability theory Discrete distribution Methods Extremal principle Sorting in increasing (decreasing) order

13–16 3.0

On each of four cards there is written a natural number. Take two cards at random and add the numbers on them. With equal probability, this amount can be less than 9, equal to 9 or more 9. What numbers can be written on the cards?

Problem #PRU-65269

Problems Algebra Mean values Methods Algebraic methods Proof by exhaustion Extremal principle Sorting in increasing (decreasing) order Probability and statistics Statistics

13–16 3.0

In the first term of the year Daniel received five grades in mathematics with each of them being on a scale of 1 to 5, and the most common grade among them was a 5 . In this case it turned out that the median of all his grades was 4, and the arithmetic mean was 3.8. What grades could Daniel have?