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?
Four children said the following about each other.
Mary: Sarah, Nathan and George solved the problem.
Sarah: Mary, Nathan and George didn’t solve the problem.
Nathan: Mary and Sarah lied.
George: Mary, Sarah and Nathan told the truth.
How many of the children actually told the truth?
On a board there are written four three-digit numbers, totaling 2012. To write them all, only two different digits were used.
Give an example of such numbers.
The teacher wrote on the board in alphabetical order all possible \(2^n\) words consisting of \(n\) letters A or B. Then he replaced each word with a product of \(n\) factors, correcting each letter A by \(x\), and each letter B by \((1 - x)\), and added several of the first of these polynomials in \(x\). Prove that the resulting polynomial is either a constant or increasing function in \(x\) on the interval \([0, 1]\).
The graph of the function \(y=kx+b\) is shown on the diagram below. Compare \(|k|\) and \(|b|\).
We are given a polynomial \(P(x)\) and numbers \(a_1\), \(a_2\), \(a_3\), \(b_1\), \(b_2\), \(b_3\) such that \(a_1a_2a_3 \ne 0\). It turned out that \(P (a_1x + b_1) + P (a_2x + b_2) = P (a_3x + b_3)\) for any real \(x\). Prove that \(P (x)\) has at least one real root.
A monkey, donkey and goat decided to play a game. They sat in a row, with the monkey on the right. They started to play the violin, but very poorly. They changed places and then the donkey was in the middle. However the violin trio still didn’t sound as they wanted it to. They changed places once more. After changing places 3 times, each of the three “musicians” had a chance to sit in the left, middle and right of the row. Who sat where after the third change of seats?
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?
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?
2012 pine cones lay under the fir-tree. Winnie the Pooh and the donkey Eeyore play a game: they take turns picking up these pine cones. Winnie-the-Pooh takes either one or four cones in each of his turns, and Eeyore – either one or three. Winnie the Pooh goes first. The player who cannot make a move loses. Which of the players can be guaranteed to win, no matter how their opponent plays?