We solve problems

Problem #PRU-110072

Problems Discrete Mathematics 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-110103

Problems 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 Discrete Mathematics Algorithm Theory 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?

Problem #PRU-115384

Problems Discrete Mathematics Combinatorics Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 4.0

A game of ’Battleships’ has a fleet consisting of one \(1\times 4\) square, two \(1\times 3\) squares, three \(1\times 2\) squares, and four \(1\times 1\) squares. It is easy to distribute the fleet of ships on a \(10\times 10\) board, see the example below. What is the smallest square board on which this fleet can be placed? Note that by the rules of the game, no two ships can be placed on horizontally, vertically, or diagonally adjacent squares.

Problem #PRU-115449

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

13–16 4.0

On an 8×8 grid (like a chessboard), an L-corner is a shape made of 3 little squares of the board that touch to make an L. You can turn the L any way you like. We place the L-corners so that none overlap. What is the fewest L-corners you must place so that no more L-corners can be added anywhere? Here is an example of how three L-corners may look like:

Problem #PRU-115454

Problemas Métodos Principio del casillero Principio del casillero (otro) Prueba por contradicción

13–15 4.0

An airline flew exactly 10 flights each day over the course of 92 days. Each day, each plane flew no more than one flight. It is known that for any two days in this period there will be exactly one plane which flew on both those days. Prove that there is a plane that flew every day in this period.

Problem #PRU-115467

Problemas Métodos Principio de la casilla Principio de la casilla (número finito de puntos, líneas, etc.)

12–15 4.0

10 children, including Billy, attended Billy’s birthday party. It turns out that any two children picked from those at the party share a grandfather. Prove that 7 of the children share a grandfather.

Problem #PRU-116087

Problems Geometry Plane geometry Triangles Similar triangles Circles Tangent lines and tangent circles Tangent lines to circles

15–16 4.0

Three circles are constructed on a triangle, with the medians of the triangle forming the diameters of the circles. It is known that each pair of circles intersects. Let \(C_{1}\) be the point of intersection, further from the vertex \(C\), of the circles constructed from the medians \(AM_{1}\) and \(BM_{2}\). Points \(A_{1}\) and \(B_{1}\) are defined similarly. Prove that the lines \(AA_{1}\), \(BB_{1}\) and \(CC_{1}\) intersect at the same point.

Problem #PRU-116278

Problemas Métodos Ejemplos y contraejemplos. Pruebas constructivas Principio del casillero Principio del casillero (otro)

13–14 4.0

Two ants crawled along their own closed route on a \(7\times7\) board. Each ant crawled only on the sides of the cells of the board and visited each of the 64 vertices of the cells exactly once. What is the smallest possible number of cell edges, along which both the first and second ants crawled?

Problem #PRU-116301

Problemas Geometría Geometría plana Desigualdades geométricas Desigualdades de área Métodos Principio de la casilla Principio de la casilla (número finito de puntos, líneas, etc.) Área

13–14 4.0

101 random points are chosen inside a unit square, including on the edges of the square, so that no three points lie on the same straight line. Prove that there exist some triangles with vertices on these points, whose area does not exceed 0.01.