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Problem #PRU-35558

Problems Algebra and arithmetic 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

Let $M$ be a finite set of numbers. It is known that among any three of its elements there are two, the sum of which belongs to $M$ .

What is the largest number of elements in $M$ ?

Problem #PRU-58313

Problems Algebra and arithmetic 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

Ten circles are marked on the circle. How many non-closed non-self-intersecting nine-point broken lines exist with vertices at these points?

Problem #PRU-60347

Problems Algebra and arithmetic 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

How many nine-digit numbers exist, the sum of the digits of which is even?

Problem #PRU-60388

Problems Algebra and arithmetic Algebraic equations and systems of equations Binomial theorem Higher order equations. Palindromic polynomial equations Equations of higher order (other)

12–14 2.0

Prove the validity of the following formula of Newton’s binom $(x + y)^{n} = (\binom{n}{0}) x^{n} + (\binom{n}{1}) x^{n - 1} y + \dots + (\binom{n}{n}) y^{n} .$

Problem #PRU-60412

Problems Algebra and arithmetic 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

Calculate the following sums:

a) $(\binom{5}{0}) + 2 (\binom{5}{1}) + 2^{2} (\binom{5}{2}) + \dots + 2^{5} (\binom{5}{5})$ ;

b) $(\binom{n}{0}) - (\binom{n}{1}) + \dots + (- 1)^{n} (\binom{n}{n})$ ;

c) $(\binom{n}{0}) + (\binom{n}{1}) + \dots + (\binom{n}{n})$ .

Problem #PRU-60416

Problems Algebra and arithmetic 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

In the expansion of $(x + y)^{n}$ , using the Newton binomial formula, the second term was 240, the third – 720, and the fourth – 1080. Find $x$ , $y$ and $n$ .

Problem #PRU-60424

Problems Algebra and arithmetic 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

Here is a fragment of the table, which is called the Leibniz triangle. Its properties are “analogous in the sense of the opposite” to the properties of Pascal’s triangle. The numbers on the boundary of the triangle are the inverses of consecutive natural numbers. Each number is equal to the sum of two numbers below it. Find the formula that connects the numbers from Pascal’s and Leibniz triangles.

Problem #PRU-61421

Problems Algebra and arithmetic 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

Draw all of the stairs made from four bricks in descending order, starting with the steepest $(4, 0, 0, 0)$ and ending with the shallowest $(1, 1, 1, 1)$ .

Problem #PRU-61473

Problems Algebra and arithmetic 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

A frog jumps over the vertices of the triangle $A B C$ , moving each time to one of the neighbouring vertices.

How many ways can it get from $A$ to $A$ in $n$ jumps?

Problem #PRU-78020

Problems Algebra and arithmetic 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 $O$ ).