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Problem #PRU-109766

Problems Algebra and arithmetic Word problems Tables and tournaments Number tables and its properties Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Is it possible to arrange natural numbers from 1 to \(2002^2\) in the cells of a \(2002\times2002\) table so that for each cell of this table one could choose a triplet of numbers, from a row or column, where one of the numbers is equal to the product of the other two?

Problem #PRU-109808

Problems Combinatorics Geometry on grid paper Integer lattices Integer lattices (other) Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

All of the points with whole number co-ordinates in a plane are plotted in one of three colours; all three colours are present. Prove that there will always be possible to form a right-angle triangle from these points so that its vertices are of three different colours.

Problem #PRU-109877

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

11–13 3.0

A regular hexagon with sides of length \(5\) is divided by straight lines, that are parallel to its sides, to form regular triangles with sides of length 1. We call the vertices of all such triangles nodes. It is known that more than half of the nodes are marked. Prove that there are five marked nodes lying on one circle.

Problem #PRU-109937

Problems Methods Invariants and semi-invariants Invariants Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Algebraic methods Processes and operations Proof by exhaustion

14–16 4.0

We are given a table of size \(n \times n\). \(n-1\) of the cells in the table contain the number \(1\). The remainder contain the number \(0\). We are allowed to carry out the following operation on the table:

1. Pick a cell.

2. Subtract 1 from the number in that cell.

3. Add 1 to every other cell in the same row or column as the chosen cell.

Is it possible, using only this operation, to create a table in which all the cells contain the same number?

Problem #PRU-109946

Problems Algebra and arithmetic Algebraic equations and systems of equations Algebraic equations and systems of equations (other) Calculus Real numbers Integer and fractional parts. Archimedean property Polynomials Quadratic polynomials Investigating the quadratic polynomial

14–15 3.0

Solve the equation \((x + 1)^3 = x^3\).

Problem #PRU-110061

Problems Methods Pigeonhole principle Pigeonhole principle (other)

13–16 4.0

Prove that in any set of 117 unique three-digit numbers it is possible to pick 4 non-overlapping subsets, so that the sum of the numbers in each subset is the same.

Problem #PRU-110072

Problems Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 4.0

A target consists of a triangle divided by three families of parallel lines into 100 equilateral unit triangles. A sniper shoots at the target. He aims at a particular equilateral triangle and either hits it or hits one of the adjacent triangles that share a side with the one he was aiming for. He can see the results of his shots and can choose when to stop shooting. What is the largest number of triangles that the sniper can guarantee he can hit exactly 5 times?

Problem #PRU-110083

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Algebra and arithmetic Word problems Tables and tournaments Tables and tournaments (other)

12–14 3.0

Can the cells of a \(5 \times 5\) board be painted in 4 colours so that the cells located at the intersection of any two rows and any two columns are painted in at least three colours?

Problem #PRU-110103

Problems Algebra and arithmetic Number theory. Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Is it possible to arrange the numbers 1, 2, ..., 60 in a circle in such an order that the sum of every two numbers, between which lies one number, is divisible by 2, the sum of every two numbers between which lie two numbers, is divisible by 3, the sum of every two numbers between which lie six numbers, is divisible by 7?

Problem #PRU-110166

Problems Set theory and logic Theory of algotithms Balance puzzles Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

A set of weights has the following properties: It contains \(5\) weights, which are all different in weight. For any two weights, there are two other weights of the same total weight. What is the smallest number of weights that can be in this set?