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Problem #PRU-58300

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

13–15 3.0

Do there exist three points $A$ , $B$ and $C$ on the plane such that for any point $X$ the length of at least one of the segments $X A$ , $X B$ and $X C$ is irrational?

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-60389

Problems Algebra Algebraic equations and systems of equations Binomial theorem

14–16 3.0

How many rational terms are contained in the expansion of

a) $(\sqrt{2} + \sqrt[4]{3})^{100}$ ;

b) $(\sqrt{2} + \sqrt[3]{3})^{300}$ ?

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-60420

Problems Calculus Number sequences Number sequences (other)

15–16 3.0

Which term in the expansion $(1 + \sqrt{3})^{100}$ will be the largest by the Newton binomial formula?

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-60427

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

13–15 3.0

Find the sums of the following series:

a) $\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \frac{1}{4 \times 5} + \dots$ ;

b) $\frac{1}{1 \times 2 \times 3} + \frac{1}{2 \times 3 \times 4} + \frac{1}{3 \times 4 \times 5} + \frac{1}{4 \times 5 \times 6} + \dots$ ;

c) $\frac{0!}{r!} + \frac{1!}{(r - 1)!} + \frac{2!}{(r - 2)!} + \frac{3!}{(r - 3)!} + \dots$ for $r \geq 2$ .