Found problems: 815
For two real numbers $ a$, $ b$, with $ ab\neq 1$, define the $ \ast$ operation by
\[ a\ast b=\frac{a+b-2ab}{1-ab}.\] Start with a list of $ n\geq 2$ real numbers whose entries $ x$ all satisfy $ 0<x<1$. Select any two numbers $ a$ and $ b$ in the list; remove them and put the number $ a\ast b$ at the end of the list, thereby reducing its length by one. Repeat this procedure until a single number remains.
$ a.$ Prove that this single number is the same regardless of the choice of pair at each stage.
$ b.$ Suppose that the condition on the numbers $ x$ is weakened to $ 0<x\leq 1$. What happens if the list contains exactly one $ 1$?
Suppose that $X$ is a compact metric space and $T: X\rightarrow X$ is a continous function. Prove that $T$ has a returning point. It means there is a strictly increasing sequence $n_i$ such that $\lim_{k\rightarrow \infty} T^{n_k}(x_0)=x_0$ for some $x_0$.
Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$. We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$. After each contact of light ray with a horizontal side, we put $0$, and after each contact with a vertical side, we put $1$. For each $n\geq 1$, let $B_{n}$ be set of all blocks of length $n$, in this sequence.
a) Prove that $B_{n}$ does not depend on location of $P$.
b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$.
Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform [i]stone moves[/i], defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\leq i, j, k, l\leq n$, such that $i<j$ and $k<l$. A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively, or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively.
Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.
How many different non-equivalent ways can Steve pile the stones on the grid?
Starting with the triple $(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14})$, define a sequence of triples $(x_{n},y_{n},z_{n})$ by
$x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})}$
$y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})}$
$ z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})}$
for $n\geq 0$.Show that each of the sequences $\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0}$ converges to a limit and find these limits.
Consider functions $f$ that satisfy $|f(x)-f(y)|\leq \frac{1}{2}|x-y|$ for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300) = f(900)$, what is the greatest possible value of
$$f(f(800))-f(f(400))?$$
$ \textbf{(A)}\ 25 \qquad
\textbf{(B)}\ 50 \qquad
\textbf{(C)}\ 100 \qquad
\textbf{(D)}\ 150 \qquad
\textbf{(E)}\ 200$
We are given a polyhedron with at least $5$ vertices, such that exactly $3$ edges meet in each of the vertices. Prove that we can assign a rational number to every vertex of the given polyhedron such that the following conditions are met:
$(i)$ At least one of the numbers assigned to the vertices is equal to $2020$.
$(ii)$ For every polygonal face, the product of the numbers assigned to the vertices of that face is equal to $1$.
We have \( n \) chips that are initially placed on the number line at position 0. On each move, we select a position \( x \in \mathbb{Z} \) where there are at least two chips; we take two of these chips, then place one at \( x-1 \) and the other at \( x+1 \).
a) Prove that after a finite number of moves, regardless of how the moves are chosen, we will reach a final position where no two chips occupy the same number on the number line.
b) For every possible final position, let \( \Delta \) represent the difference between the numbers where the rightmost and the leftmost chips are located. Find all possible values of \( \Delta \) in terms of \( n \).
The points $(1,1),(2,3),(4,5)$ and $(999,111)$ are marked in the coordinate system. We continue to mark points in the following way :
[list]
[*]If points $(a,b)$ are marked then $(b,a)$ and $(a-b,a+b)$ can be marked
[*]If points $(a,b)$ and $(c,d)$ are marked then so can be $(ad+bc, 4ac-4bd)$.
[/list]
Can we, after some finite number of these steps, mark a point belonging to the line $y=2x$.
Let $n>1$ be an integer. Suppose we are given $2n$ points in the plane such that no three of them are collinear. The points are to be labelled $A_1, A_2, \dots , A_{2n}$ in some order. We then consider the $2n$ angles $\angle A_1A_2A_3, \angle A_2A_3A_4, \dots , \angle A_{2n-2}A_{2n-1}A_{2n}, \angle A_{2n-1}A_{2n}A_1, \angle A_{2n}A_1A_2$. We measure each angle in the way that gives the smallest positive value (i.e. between $0^{\circ}$ and $180^{\circ}$). Prove that there exists an ordering of the given points such that the resulting $2n$ angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group.
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.
$101$ numbers are written on a blackboard: $1^2, 2^2, 3^2, \cdots, 101^2$. Alex choses any two numbers and replaces them by their positive difference. He repeats this operation until one number is left on the blackboard. Determine the smallest possible value of this number.
Let $n\ge 2$ be an integer. Elwyn is given an $n\times n$ table filled with real numbers (each cell of the table contains exactly one number). We define a [i]rook set[/i] as a set of $n$ cells of the table situated in $n$ distinct rows as well as in n distinct columns. Assume that, for every rook set, the sum of $n$ numbers in the cells forming the set is nonnegative.\\
\\ By a move, Elwyn chooses a row, a column, and a real number $a,$ and then he adds $a$ to each number in the chosen row, and subtracts $a$ from each number in the chosen column (thus, the number at the intersection of the chosen row and column does not change). Prove that Elwyn can perform a sequence of moves so that all numbers in the table become nonnegative.
The numbers $\frac{50}{1},\frac{50}{2},...\frac{50}{97},\frac{50}{98}$ are written on the board.In each step,two random numbers $a$ and $b$ are chosen and deleted.Then,the number $2ab-a-b+1$ is written instead.What will be the number remained on the board after the last step.
Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$.
[i]Proposed by Hojoo Lee, Korea[/i]
Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.
The cells of an $8\times 8$ chessboard are all coloured in white. A move consists in inverting the colours of a rectangle $1 \times 3$ horizontal or vertical (the white cells become black and conversely). Is it possible to colour all the cells of the chessboard in black in a finite number of moves ?
You are cheating at a trivia contest. For each question, you can peek at each of the $n > 1$ other contestants' guesses before writing down your own. For each question, after all guesses are submitted, the emcee announces the correct answer. A correct guess is worth $0$ points. An incorrect guess is worth $-2$ points for other contestants, but only $-1$ point for you, since you hacked the scoring system. After announcing the correct answer, the emcee proceeds to read the next question. Show that if you are leading by $2^{n - 1}$ points at any time, then you can surely win first place.
[i]Linus Hamilton[/i]
Let \(n\ge3\) be a fixed integer, and let \(\alpha\) be a fixed positive real number. There are \(n\) numbers written around a circle such that there is exactly one \(1\) and the rest are \(0\)'s. An [i]operation[/i] consists of picking a number \(a\) in the circle, subtracting some positive real \(x\le a\) from it, and adding \(\alpha x\) to each of its neighbors.
Find all pairs \((n,\alpha)\) such that all the numbers in the circle can be made equal after a finite number of operations.
[i]Proposed by Anthony Wang[/i]
We say that a permutation $(a_1, \dots, a_n)$ of $(1, 2, \dots, n)$ is good if the sums $a_1 + a_2 + \dots + a_i$ are all distinct modulo $n$. Prove that there exists a positive integer $n$ such that there are at least $2020$ good permutations of $(1, 2, \dots, n)$.
[i]Proposed by Ariel García[/i]
A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities
$$P(x, y, z)=P(x, y, xy-z)=P(x, zx-y, z)=P(yz-x, y, z).$$
Prove that there exists a polynomial $F(t)$ in one variable such that
$$P(x,y,z)=F(x^2+y^2+z^2-xyz).$$
Let $ A\in\mathcal{M}_4\left(\mathbb{R}\right) $ be an invertible matrix whose trace is equal to the trace of its adjugate, which is nonzero. Show that $ A^2+I $ is singular if and only if there exists a nonzero matrix in $ \mathcal{M}_4\left( \mathbb{R} \right) $ that anti-commutes with it.
Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$.
Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$
Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.
A number of robots are placed on the squares of a finite, rectangular grid of squares. A square can hold any number of robots. Every edge of each square of the grid is classified as either passable or impassable. All edges on the boundary of the grid are impassable. You can give any of the commands up, down, left, or right.
All of the robots then simultaneously try to move in the specified direction. If the edge adjacent to a robot in that direction is passable, the robot moves across the edge and into the next square. Otherwise, the robot remains on its current square. You can then give another command of up, down, left, or right, then another, for as long as you want. Suppose that for any individual robot, and any square on the grid, there is a finite sequence of commands that will move that robot to that square. Prove that you can also give a finite sequence of commands such that all of the robots end up on the same square at the same time.