We solve problems

Problems Calculus Real numbers Integer and fractional parts. Archimedean property

Solve the equation \(x^3 - \lfloor x\rfloor = 3\).

Problems Discrete Mathematics Algorithm Theory Game theory Game theory (other) Algebra Algebraic equations and systems of equations Systems of linear equations

12–14 3.0

There is a system of equations \[\begin{aligned} * x + * y + * z &= 0,\\ * x + * y + * z &= 0,\\ * x + * y + * z &= 0. \end{aligned}\] Two people alternately enter a number instead of a star. Prove that the player that goes first can always ensure that the system has a non-zero solution.

Problem #PRU-78150

Problems Discrete Mathematics Combinatorics Painting problems Algorithm Theory

12–14 3.0

There are two sets of numbers made up of 1s and \(-1\)s, and in each there are 2022 numbers. Prove that in some number of steps it is possible to turn the first set into the second one if for each step you are allowed to simultaneously change the sign of any 11 numbers of the starting set. (Two sets are considered the same if they have the same numbers in the same places.)

Problem #PRU-78175

Problems Number Theory Numeral systems Decimal number system

12–15 2.0

Note that if you turn over a sheet on which numbers are written, then the digits 0, 1, 8 will not change and the digits 6 and 9 will switch places, whilst the others will lose their meaning. How many nine-digit numbers exist that do not change when a sheet is turned over?

Problem #PRU-78206

Problems Methods Algebraic methods Counting in two ways Pigeonhole principle Pigeonhole principle (other)

13–14 3.0

30 pupils in years 7 to 11 took part in the creation of 40 maths problems. Every possible pair of pupils in the same year created the same number of problems. Every possible pair of pupils in different years created a different number of problems. How many pupils created exactly one problem?

Problem #PRU-78234

Problems Methods Pigeonhole principle Algebraic methods Processes and operations

13–15 4.0

The number \(A\) is divisible by \(1, 2, 3, \dots , 9\). Prove that if \(2A\) is presented in the form of a sum of some natural numbers smaller than 10, \(2A= a_1 +a_2 +\dots +a_k\), then we can always choose some of the numbers \(a_1, a_2, \dots , a_k\) so that the sum of the chosen numbers is equal to \(A\).

Problem #PRU-78259

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other)

14–16 4.0

A table of \(4\times4\) cells is given, in some cells of which a star is placed. Show that you can arrange seven stars so that when you remove any two rows and any two columns of this table, there will always be at least one star in the remaining cells. Prove that if there are fewer than seven stars, you can always remove two rows and two columns so that all the remaining cells are empty.

Problem #PRU-78266

Graph theory

12–14 2.0

Given \(n\) points that are connected by segments so that each point is connected to some other and there are no two points that would be connected in two different ways. Prove that the total number of segments is \(n - 1\).

Problem #PRU-78270

Problems Geometry Plane geometry Area Area of curvilinear shapes Methods Pigeonhole principle Pigeonhole principle (area and volume)

13–15 4.0

120 unit squares are placed inside a \(20 \times 25\) rectangle. Prove that it will always be possible to place a circle with diameter 1 inside the rectangle, without it overlapping with any of the unit squares.

Problem #PRU-78475

Problems Geometry Plane geometry Lines, segments, rays, and angles

13–14 3.0

You are given \(7\) straight lines on a plane, no two of which are parallel. Prove that there will be two lines such that the angle between them is less than \(26^{\circ}\).