Found problems: 259
On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$ and replace by $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.
In a room there are several children and a pile of 1000 sweets. The children come to the pile one after another in some order. Upon reaching the pile each of them divides the current number of sweets in the pile by the number of children in the room, rounds the result if it is not integer, takes the resulting number of sweets from the pile and leaves the room. All the boys round upwards and all the girls round downwards. The process continues until everyone leaves the room. Prove that the total number of sweets received by the boys does not depend on the order in which the children reach the pile.
[i]Maxim Didin[/i]
Consider the following transformation of the Cartesian plane: choose a lattice point and rotate the plane $90^\circ$ counterclockwise about that lattice point. Is it possible, through a sequence of such transformations, to take the triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$ to the triangle with vertices $(0,0)$, $(1,0)$ and $(1,1)$?
Let $ R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $ R_1 \equal{} (1),$ and if $ R_{n - 1} \equal{} (x_1, \ldots, x_s),$ then
\[ R_n \equal{} (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).\]
For example, $ R_2 \equal{} (1, 2),$ $ R_3 \equal{} (1, 1, 2, 3),$ $ R_4 \equal{} (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $ n > 1,$ then the $ k$th term from the left in $ R_n$ is equal to 1 if and only if the $ k$th term from the right in $ R_n$ is different from 1.
Let $a,b$ be integers with $0<a<b$. A set $\{x,y,z\}$ of non-negative integers is [i]olympic[/i] if $x<y<z$ and if $\{z-y,y-x\}=\{a,b\}$. Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets.
Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.
Given is $2022\times 2022$ cells table. We can select $4$ cells, such that they make the figure $L$ (rotations, symmetric still count) (left one) and put a ball in each of them, or select $4$ cell which makes up the right figure (rotations, symmetric still count) and get one ball from each of them.
For which $k$ is it possible in a given moment to be exactly $k$ points in each of the cells
As the graph, a pond is divided into 2n (n $\geq$ 5) parts. Two parts are called neighborhood if they have a common side or arc. Thus every part has three neighborhoods. Now there are 4n+1 frogs at the pond. If there are three or more frogs at one part, then three of the frogs of the part will jump to the three neighborhoods repsectively. Prove that for some time later, the frogs at the pond will uniformily distribute. That is, for any part either there are frogs at the part or there are frogs at the each of its neighborhoods.
[img]http://www.mathlinks.ro/Forum/files/china2005_2_214.gif[/img]
The cirumcentre of the cyclic quadrilateral $ABCD$ is $O$. The second intersection point of the circles $ABO$ and $CDO$, other than $O$, is $P$, which lies in the interior of the triangle $DAO$. Choose a point $Q$ on the extension of $OP$ beyond $P$, and a point $R$ on the extension of $OP$ beyond $O$. Prove that $\angle QAP=\angle OBR$ if and only if $\angle PDQ=\angle RCO$.
The three roots of $P(x) = x^3 - 2x^2 - x + 1$ are $a>b>c \in \mathbb{R}$. Find the value of $a^2b+b^2c+c^2a$. :D
Let $f_n\ (n=1,\ 2,\ \cdots)$ be a linear transformation expressed by a matrix $\left(
\begin{array}{cc}
1-n & 1 \\
-n(n+1) & n+2
\end{array}
\right)$ on the $xy$ plane. Answer the following questions:
(1) Prove that there exists 2 lines passing through the origin $O(0,\ 0)$ such that all points of the lines are mapped to the same lines, then find the equation of the lines.
(2) Find the area $S_n$ of the figure enclosed by the lines obtained in (1) and the curve $y=x^2$.
(3) Find $\sum_{n=1}^{\infty} \frac{1}{S_n-\frac 16}.$
[i]2011 Tokyo Institute of Technlogy entrance exam, Problem 1[/i]
On the table lay a pencil, sharpened at one end. The student can rotate the pencil around one of its ends at $45^{\circ}$ clockwise or counterclockwise. Can the student, after a few turns of the pencil, go back to the starting position so that the sharpened end and the not sharpened are reversed?
There are $n+1$ containers arranged in a circle. One container has $n$ stones, the others are empty. A move is to choose two containers $A$ and $B$, take a stone from $A$ and put it in one of the containers adjacent to $B$, and to take a stone from $B$ and put it in one of the containers adjacent to $A$. We can take $A = B$. For which $n$ is it possible by series of moves to end up with one stone in each container except that which originally held $n$ stones.
George plays the following game: At every step he can replace a triple of integers $(x,y,z)$ which is written on the blackboard, with any of the following triples:
(i) $(x,z,y)$
(ii) $(-x,y,z)$
(iii) $(x+y,y,2x+y+z)$
(iv) $(x-y,y,y+z-2x)$
Initially, the triple $(1,1,1)$ is written on the blackboard. Determine whether George can, with a sequence of allowed steps, end up at the triple $(2021,2019,2023)$, fully justifying your answer.
Around a circular table an even number of persons have a discussion. After a break they sit again around the circular table in a different order. Prove that there are at least two people such that the number of participants sitting between them before and after a break is the same.
Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations.
[i]Proposed by Warut Suksompong, Thailand[/i]
On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$ and replace by $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.
Three circles $k_1,k_2$ and $k_3$ go through the points $A$ and $B$. A secant through $A$ intersects the circles $k_1,k_2$ and $k_3$ again in the points $C,D$ resp. $E$. Prove that the ratio $|CD|:|DE|$ does not depend on the choice of the secant.
The numbers $1$ through $4^{n}$ are written on a board. In each step, Pedro erases two numbers $a$ and $b$ from the board, and writes instead the number $\frac{ab}{\sqrt{2a^2+2b^2}}$. Pedro repeats this procedure until only one number remains. Prove that this number is less than $\frac{1}{n}$, no matter what numbers Pedro chose in each step.
Prove that for any positive integer $n$, \[\left\lfloor \frac{n}{3}\right\rfloor+\left\lfloor \frac{n+2}{6}\right\rfloor+\left\lfloor \frac{n+4}{6}\right\rfloor = \left\lfloor \frac{n}{2}\right\rfloor+\left\lfloor \frac{n+3}{6}\right\rfloor .\]
Find all pair of positive integers $(x, y)$ satisfying the equation
\[x^2 + y^2 - 5 \cdot x \cdot y + 5 = 0.\]
In each cell of an $n\times n$ board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off).
Show that if after a certain number of moves, at least one light is on, then at this moment at least $n$ lights are on.
A needle (a segment) lies on a plane. One can rotate it $45^{\circ}$ round any of its endpoints. Is it possible that after several rotations the needle returns to initial position with the endpoints interchanged?
Let $ABCD$ be a quadrilateral $AB = BC = CD = DA$. Let $MN$ and $PQ$ be two segments perpendicular to the diagonal $BD$ and such that the distance between them is $d > \frac{BD}{2}$, with $M \in AD$, $N \in DC$, $P \in AB$, and $Q \in BC$. Show that the perimeter of hexagon $AMNCQP$ does not depend on the position of $MN$ and $PQ$ so long as the distance between them remains constant.
Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow?
[i]Proposed by Gerhard Woeginger, Netherlands[/i]