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Problem #PRU-35635

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

13–15 3.0

Two players play the following game. They take turns. One names two numbers that are at the ends of a line segment. The next then names two other numbers, which are at the ends of a segment nested in the previous one. The game goes on indefinitely. The first aims to have at least one rational number within the intersection of all of these segments, and the second aims to prevent such occurring. Who wins in this game?

Problem #PRU-35708

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

13–15 3.0

The positive irrational numbers $a$ and $b$ are such that $1 / a + 1 / b = 1$ . Prove that among the numbers $⌊ m a ⌋, ⌊ n b ⌋$ each natural number occurs exactly once.

Problem #PRU-35714

Problems Methods Mathematical induction Mathematical induction (other) Set theory and logic Theory of algotithms Theory of algorithms (other)

12–14 3.0

There are 8 glasses of water on the table. You are allowed to take any two of the glasses and make them have equal volumes of water (by pouring some water from one glass into the other). Prove that, by using such operations, you can eventually get all the glasses to contain equal volumes of water.

Problem #PRU-35733

Problems Algebra and arithmetic Rational functions Identical transformations Set theory and logic Theory of algotithms Theory of algorithms (other)

12–14 3.0

A broken calculator carries out only one operation “asterisk”: $a * b = 1 - a / b$ . Prove that using this calculator it is possible to carry out all four arithmetic operations (addition, subtraction, multiplication, division).

Problem #PRU-55621

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

13–15 3.0

A rectangular billiard with sides 1 and $\sqrt{2}$ is given. From its angle at an angle of $45^{\circ}$ to the side a ball is released. Will it ever get into one of the pockets? (The pockets are in the corners of the billiard table).

Problem #PRU-58299

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

13–15 3.0

Suppose that $n \geq 3$ . Are there n points that do not lie on one line, whose pairwise distances are irrational, and the areas of all of the triangles with vertices in them are rational?

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 $(x + y)^{n} = (\binom{n}{0}) x^{n} + (\binom{n}{1}) x^{n - 1} y + \dots + (\binom{n}{n}) y^{n} .$