We solve problems

Problems Set theory and logic Theory of algotithms Game theory Game theory (other)

Prove that in a game of noughts and crosses on a $3 \times 3$ grid, if the first player uses the right strategy then the second player cannot win.

Problem #PRU-32817

Problems Set theory and logic Mathematical logic Mathematical logic (other)

11–12 2.0

a) There are 10 coins. It is known that one of them is fake (by weight, it is heavier than the real ones). How can you determine the counterfeit coin with three weighings on scales without weights?

b) How can you determine the counterfeit coin with three weighings, if there are 27 coins?

Problem #PRU-32819

Problems Set theory and logic Theory of algotithms Balance puzzles

12–13 3.0

Vincent makes small weights. He made 4 weights which should have masses (in grams) of 1, 3, 4 and 7, respectively. However, he made a mistake and one of these weights has the wrong mass. By weighing them twice using balance scales (without the use of weights other than those mentioned) can he find which weight has the wrong mass?

Problem #PRU-32820

Problems Set theory and logic Theory of algotithms Balance puzzles

12–14 3.0

There are some coins on a table. One of these coins is fake (has a different weight than a real coin). By weighing them twice using balance scales, determine whether the fake coin is lighter or heavier than a real coin (you don’t need to find the fake coin) if the number of coins is: a) 100; b) 99; c) 98?

Problem #PRU-32884

Problems Methods Extremal principle Extremal principle (other) Mathematical induction Mathematical induction (other) Algebra and arithmetic Polynomials Polynomial remainder theorem (little Bezout's theorem). Factorisation. Polynomials with integer coefficients and integer values Proof by contradiction Calculus Real numbers Rational and irrational numbers

12–12 3.0

Prove that if the irreducible rational fraction $p / q$ is a root of the polynomial $P (x)$ with integer coefficients, then $P (x) = (q x - p) Q (x)$ , where the polynomial $Q (x)$ also has integer coefficients.

Problem #PRU-32901

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Find a natural number greater than one that occurs in the Pascal triangle a) more than three times; b) more than four times.

Problem #PRU-32904

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

Can you find

a) in the 100th line of Pascal’s triangle, the number $1 + 2 + 3 + \dots + 98 + 99$ ?

b) in the 200th line the sum of the squares of the numbers in the 100th line?

Problem #PRU-32987

Problems Algebra and arithmetic Algebraic equations and systems of equations

11–13 2.0

Prove there are no integer solutions for the equation $3 x^{2} + 2 = y^{2}$ .

Problem #PRU-34931

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

On the dining room table, there is a choice of six dishes. Every day Valentina takes a certain set of dishes (perhaps, she does not take a single dish), and this set of dishes should be different from all of the sets that she took in the previous days. What is the maximum number of days that Valentina will be able to eat according to such rules and how many meals will she eat on average during the day?

Problem #PRU-34972

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

For which $n > 3$ , can a set of weights with masses of $1, 2, 3, . . ., n$ grams be divided into three groups of equal mass?