We solve problems

Problem #PRU-111328

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

12–14 3.0

A cinema contains 7 rows each with 10 seats. A group of 50 children went to see the morning screening of a film, and returned for the evening screening. Prove that there will be two children who sat in the same row for both the morning and the evening screening.

Problem #PRU-111682

Problems Algebra and arithmetic Word problems Tables and tournaments Chessboards and chess pieces Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Proof by contradiction

12–14 3.0

100 queens, that cannot capture each other, are placed on a \(100 \times 100\) chessboard. Prove that at least one queen is in each \(50 \times 50\) corner square.

Problem #PRU-111770

Problems Methods Pigeonhole principle Pigeonhole principle (other)

13–15 3.0

In 25 boxes there are spheres of different colours. It is known that for any \(k\) where \(1 \leq k \leq 25\) in any \(k\) of the boxes there are spheres of exactly \(k+1\) different colours. Prove that a sphere of one particular colour lies in every single box.

Problem #PRU-111923

Problems Calculus Derivative Derivative and extrema Derivative and multiple roots

15–16 3.0

When water is drained from a pool, the water level \(h\) in it varies depending on the time \(t\) according to the function \(h (t) = at^2 + bt + c\), and at the time \(t_0\) of when the draining is ending, the equalities \(h (t_0) = h' (t_0) = 0\) are satisfied. For how many hours does the pool drain completely, if in the first hour the water level in it is reduced by half?

Problem #PRU-115384

Problems 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.) Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

13–16 4.0

What is the smallest number of ‘L’ shaped ‘corners’ out of 3 squares that can be marked on an \(8\times 8\) square grid, so that no more ’corners’ would fit?

Problem #PRU-115454

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

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

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines 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-116145

Problems Algebra and arithmetic Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Set theory and logic Logic and set theory

12–14 2.0

Of the four inequalities \(2x > 70\), \(x < 100\), \(4x > 25\) and \(x > 5\), two are true and two are false. Find the value of \(x\) if it is known that it is an integer.