Filter Problems

Showing 41 to 60 of 2116 entries

A student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number. The result was three times greater.

Find these numbers.

An after school club was attended by 60 pupils. It turns out that in any group of 10 there will always be 3 classmates. Prove that within the group of 60 who attended there will always be at least 15 pupils from the same class.

Initially, a natural number A is written on a board. You are allowed to add to it one of its divisors, distinct from itself and one. With the resulting number you are permitted to perform a similar operation, and so on.

Prove that from the number A = 4 one can, with the help of such operations, come to any given in advance composite number.

Ben noticed that all 25 of his classmates have a different number of friends in this class. How many friends does Ben have?

Sage stated the sum of some three natural numbers, and the Patricia named their product.

“If I knew,” said Sage, “that your number is greater than mine, then I would immediately name the three numbers that are needed.”

“My number is smaller than yours,” Patricia answered, “and the numbers you want are …, … and ….”

What numbers did Patricia name?

A numerical sequence is defined by the following conditions:

$\\$

$\\$

How many complete squares are found among the first members of this sequence, not exceeding 1,000,000?

The function f $(x)$ on the interval [a, b] is equal to the maximum of several functions of the form $y = C \times 10^{- | x-d |}$ $($where d and C are different, and all C are positive$)$. It is given that f $(a)$ = f $(b)$. Prove that the sum of the lengths of the sections on which the function increases is equal to the sum of the lengths of the sections on which the function decreases.

A numerical sequence is defined by the following conditions:

$\\$

Prove that among the terms of this sequence there are an infinite number of complete squares.

A gang contains 101 bandits. The whole gang has never taken part in a raid together, but every possible pair of bandits have taken part in a raid together exactly once. Prove that one of the bandits has taken part in no less than 11 different raids.

In a certain kingdom there were 32 knights. Some of them were vassals of others $($ a vassal can have only one suzerain, and the suzerain is always richer than his vassal $)$. A knight with at least four vassals is given the title of Baron. What is the largest number of barons that can exist under these conditions?

$($ In the kingdom the following law is enacted: ” the vassal of my vassal is not my vassal”$)$.

10 friends sent one another greetings cards; each sent 5 cards. Prove that there will be two friends who sent cards to one another.

PFind the number of solutions in natural numbers of the equation [x / 10] = [x / 11] + 1.

A staircase has 100 steps. Vivian wants to go down the stairs, starting from the top, and she can only do so by jumping down and then up, down and then up, and so on. The jumps can be of three types – six steps $($jumping over five to land on the sixth$)$, seven steps or eight steps. Note that Vivian does not jump onto the same step twice. Will she be able to go down the stairs?

We are given 101 rectangles with integer-length sides that do not exceed 100.

Prove that amongst them there will be three rectangles $A, B, C$, which will fit completely inside one another so that $A \subset B \subset C$.

What figure should I put in place of the “?” in the number 888 … 88? 99 … 999 $($ eights and nines are written 50 times each $)$ so that it is divisible by 7?

What is the minimum number of squares that need to be marked on a chessboard, so that:

1) There are no horizontally, vertically, or diagonally adjacent marked squares.

2) Adding any single new marked square breaks rule 1.

a) We are given two cogs, each with 14 teeth. They are placed on top of one another, so that their teeth are in line with one another and their projection looks like a single cog. After this 4 teeth are removed from each cog, the same 4 teeth on each one. Is it always then possible to rotate one of the cogs with respect to the other so that the projection of the two partially toothless cogs appears as a single complete cog? The cogs can be rotated in the same plane, but cannot be flipped over.

b) The same question, but this time two cogs of 13 teeth each from which 4 are again removed?

Prove that in any group of 7 natural numbers – not necessarily consecutive – it is possible to choose three numbers such that their sum is divisible by 3.

We consider a sequence of words consisting of the letters “A” and “B”. The first word in the sequence is “A”, the k-th word is obtained from the $(k-1)$-th by the following operation: each “A” is replaced by “AAB” and each “B” by “A”. It is easy to see that each word is the beginning of the next, thus obtaining an infinite sequence of letters: AABAABAAABAABAAAB …

$\\$ a) Where in this sequence will the 1000th letter “A” be?

$\\$ b) Prove that this sequence is non-periodic.

a) The vertices (corners) in a regular polygon with 10 sides are coloured black and white in an alternating fashion (i.e. one vertice is black, the next is white, etc). Two people play the following game. Each player in turn draws a line connecting two vertices of the same colour. These lines must not have common vertices (i.e. must not begin or end on the same dot as another line) with the lines already drawn. The winner of the game is the player who made the final move. Which player, the first or the second, would win if the right strategy is used?

b) The same problem, but for a regular polygon with 12 sides.