We solve problems

Problem #PRU-103787

Problems Methods Algebraic methods Counting in two ways Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

One term a school ran 20 sessions of an after-school Astronomy Club. Exactly five pupils attended each session and no two students encountered one another over all of the sessions more than once. Prove that no fewer than 20 pupils attended the Astronomy Club at some point during the term.

Problem #PRU-107762

Problems Methods Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

Each of the 1994 deputies in parliament slapped exactly one of his colleagues. Prove that it is possible to draw up a parliamentary commission of 665 people whose members did not clarify the relationship between themselves in the manner indicated above.

Problem #PRU-32132

Problems Methods Extremal principle Extremal principle (other) Set theory and logic Order relations Algebraic methods Proof by exhaustion

13–15 3.0

The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from \(n\) books of different heights, if: a) \(n = 4\); b) \(n = 5\)?

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 \(P (x) = (qx - p) Q (x)\), where the polynomial \(Q (x)\) also has integer coefficients.

Problem #PRU-34991

Problems Methods Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (other) Combinatorics Systems of points and line segments Systems of points, lines, and line segments

13–15 4.0

On a line, there are 50 segments. Prove that either it is possible to find some 8 segments all of which have a shared intersection, or there can be found 8 segments, no two of which intersect.

Problem #PRU-35020

Problems Algebra and arithmetic Arithmetics. Mental maths Methods Pigeonhole principle Pigeonhole principle (other) Extremal principle Sorting in increasing (decreasing) order

11–13 3.0

10 people collected a total of 46 mushrooms in a forest. It is known that no two people collected the same number of mushrooms. How many mushrooms did each person collect?

Problem #PRU-35578

Problems Methods Extremal principle Extremal principle (other)

13–14 2.0

In how many ways can you rearrange the numbers from 1 to 100 so that the neighbouring numbers differ by no more than 1?

Problem #PRU-35791

Problems Algebra and arithmetic Word problems Solving problems by inequalities. Going through the cases. Methods Extremal principle Algebraic methods Processes and operations

13–14 3.0

Sam and Lena have several chocolates, each weighing not more than 100 grams. No matter how they share these chocolates, one of them will have a total weight of chocolate that does not exceed 100 grams. What is the maximum total weight of all of the chocolates?

Problem #PRU-60489

Problems Algebra and arithmetic Number theory. Divisibility Equations in integer numbers Methods Extremal principle Extremal principle (other) Methods from geometry Geometric interpretations in algebra Pigeonhole principle Pigeonhole principle (other) The greatest common divisor (GCD) and the least common multiplier (LCM). Mutually prime numbers

13–15 4.0

Let \(a\), \(b\), \(c\) be integers; where \(a\) and \(b\) are not equal to zero.

Prove that the equation \(ax + by = c\) has integer solutions if and only if \(c\) is divisible by \(d = \mathrm{GCD} (a, b)\).

Problem #PRU-61399

Problems Algebra and arithmetic Number theory. Divisibility Equations in integer numbers Methods Extremal principle Sorting in increasing (decreasing) order

13–15 2.0

Prove that the equation \(\frac {x}{y} + \frac {y}{z} + \frac {z}{x} = 1\) is unsolvable using positive integers.