We solve problems

Problem #PRU-78020

Problems Algebra Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

The segment $O A$ is given. From the end of the segment $A$ there are 5 segments $A B_{1}, A B_{2}, A B_{3}, A B_{4}, A B_{5}$ . From each point $B_{i}$ there can be five more new segments or not a single new segment, etc. Can the number of free ends of the constructed segments be 1001? By the free end of a segment we mean a point belonging to only one segment (except point $O$ ).

Problem #PRU-78043

Problems Methods Methods of calculus Monotonicity and boundedness Pigeonhole principle Pigeonhole principle (other) Algebra Polynomials Quadratic polynomials

13–15 4.0

The expression $a x^{2} + b x + c$ is an exact fourth power for all integers $x$ . Prove that $a = b = 0$ .

Problem #PRU-78051

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

The numbers $⌊ a ⌋, ⌊ 2 a ⌋, \dots, ⌊ N a ⌋$ are all different, and the numbers $⌊ 1 / a ⌋, ⌊ 2 / a ⌋, \dots, ⌊ M / a ⌋$ are also all different. Find all such $a$ .

Problem #PRU-78086

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–15 4.0

2022 points are selected from a cube, whose edge is equal to 13 units. Is it possible to place a cube with edge of 1 unit in this cube so that there is not one selected point inside it?

Problem #PRU-78102

Problems Calculus Real numbers Integer and fractional parts. Archimedean property

15–16 3.0

Solve the equation $x^{3} - ⌊ x ⌋ = 3$ .

Problem #PRU-78130

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 each created at least one maths problem, making 40 maths problems altogether. 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 $2 A$ is presented in the form of a sum of some natural numbers smaller than 10, $2 A = a_{1} + a_{2} + \dots + a_{k}$ , 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 $A$ .