We solve problems

Problems Set theory and logic Mathematical logic Puzzles

Can I replace the letters with numbers in the puzzle \(RE \times CTS + 1 = CE \times MS\) so that the correct equality is obtained (different letters need to be replaced by different numbers, and the same letters must correspond to the same digits)?

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.

Problem #PRU-116208

Problems Set theory and logic Theory of algotithms Balance puzzles

11–13 2.0

At the vertices of the hexagon \(ABCDEF\) (see Fig.) There were 6 identical balls: at \(A\) – one with mass 1 g, at \(B\) – 2 g, ..., at \(F\) – 6 g. Callum changed the places of two balls in opposite vertices. A set of weighing scales with 2 plates is available, which let you know which plate contains the balls of greater mass. How, in one weighing, can it be determined which balls were rearranged?

Problem #PRU-116661

Problems Algebra and arithmetic Word problems Solving problems by inequalities. Going through the cases. Sequences Progressions Arithmetic progression Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

10–12 3.0

Can 100 weights of masses 1, 2, 3, ..., 99, 100 be arranged into 10 piles of different masses so that the following condition is fulfilled: the heavier the pile, the fewer weights in it?

Problem #PRU-116842

Problems Set theory and logic Mathematical logic Mathematical logic (other)

11–12 2.0

Going to school, Michael found everything he needed under the pillow, under the sofa, on the table or under the table. The items he needed to find were a notebook, a cheat sheet, an mp3 player and sneakers. Under the table, he did not find a notebook or an mp3 player. His cheat sheet never lies on the floor. The mp3 player was neither on the table nor under the sofa. What was lying where, if there was only one object in each of the places?

Problem #PRU-116978

Problems Methods Pigeonhole principle Pigeonhole principle (other)

10–12 3.0

When cleaning her children’s room, a mother found \(9\) socks. In a group of any \(4\) of the socks at least two belonged to the same child. In a group of any \(5\) of the socks no more than \(3\) had the same owner. How many children are there in the room and how many socks belong to each child?

Problem #PRU-21970

Problems Methods Pigeonhole principle Pigeonhole principle (other)

10–12 2.0

A bag contains balls of two different colours – black and white. What is the minimum number of balls you need to remove, without looking, to guarantee that within the removed balls at least two are the same colour.

Problem #PRU-21971

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–12 2.0

A forest contains a million fir trees. It is known that any given tree has at most 600,000 needles. Prove that there will be two trees with the same number of needles.

Problem #PRU-21972

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

11–13 2.0

You are given 12 different whole numbers. Prove that it is possible to choose two of these whose difference is divisible by 11.

Problem #PRU-21974

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–13 2.0

A supermarket received a delivery of 25 crates of apples of 3 different types; each crate contains only one type of apple. Prove that there are at least 9 crates of apples of the same sort in the delivery.