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Problems Methods Algebraic methods Counting in two ways

An \(8 \times 8\) chessboard has 30 diagonals total (15 in each direction). Is it possible to place several chess pieces on this chessboard in such a way that the total number of pieces on each diagonal would be odd?

Problem #PRU-100636

Problems Methods Algebraic methods Counting in two ways

Anna’s garden is a grid of \(n \times m\) squares. She wants to have trees in some of these squares, but she wants the total number of trees in each column and in each row to be an odd number (not necessarily the same, they just all need to be odd). Show that it is possible only if \(m\) and \(n\) are both even or both odd and calculate in how many different ways she can place the trees in the grid.

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-109704

Problems Algebra Algebraic inequalities and systems of inequalities Algebraic inequalities (other) Methods Algebraic methods Counting in two ways Calculus Real numbers Integer and fractional parts. Archimedean property

13–15 4.0

Prove that for any positive integer \(n\) the inequality

is true.

Problem #PRU-116817

Problems Methods Algebraic methods Counting in two ways Set theory and logic Mathematical logic Mathematical logic (other)

13–14 3.0

There is a group of 5 people: Alex, Beatrice, Victor, Gregory and Deborah. Each of them has one of the following codenames: V, W, X, Y, Z. We know that:

Alex is 1 year older than V,

Beatrice is 2 years older than W,

Victor is 3 years older than X,

Gregory is 4 years older than Y.

Who is older and by how much: Deborah or Z?

Problem #PRU-21985

Problems Methods Algebraic methods Counting in two ways Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

12–13 3.0

The total age of a group of 7 people is 332 years. Prove that it is possible to choose three members of this group so that the sum of their ages is no less than 142 years.

Problem #PRU-32046

Problems Methods Algebraic methods Counting in two ways Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

11–13 3.0

You are given 25 numbers. The sum of any 4 of these numbers is positive. Prove that the sum of all 25 numbers is also positive.

Problem #PRU-32092

Problems Methods Algebraic methods Counting in two ways Algebra Word Problems Tables and tournaments Number tables and its properties

10–12 2.0

In each square of a rectangular table of size \(M \times K\), a number is written. The sum of the numbers in each row and in each column, is 1. Prove that \(M = K\).

Problem #PRU-78206

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

13–14 3.0

30 pupils in years 7 to 11 took part in the creation of 40 maths problems. Every possible pair of pupils in the same year created the same number of problems. Every possible pair of pupils in different years created a different number of problems. How many pupils created exactly one problem?

Problem #PRU-88120

Problems Methods Algebraic methods Counting in two ways Algebra Word Problems Tables and tournaments Number tables and its properties

10–12 2.0

Is it possible to fill a \(5 \times 5\) table with numbers so that the sum of the numbers in each row is positive and the sum of the numbers in each column is negative?