Found problems: 5802
In the nation of Onewaynia, certain pairs of cities are connected by roads. Every road connects exactly two cities (roads are allowed to cross each other, e.g., via bridges). Some roads have a traffic capacity of 1 unit and other roads have a traffic capacity of 2 units. However, on every road, traffic is only allowed to travel in one direction. It is known that for every city, the sum of the capacities of the roads connected to it is always odd. The transportation minister needs to assign a direction to every road. Prove that he can do it in such a way that for every city, the difference between the sum of the capacities of roads entering the city and the sum of the capacities of roads leaving the city is always exactly one.
[i]Proposed by Zuming Feng and Yufei Zhao[/i]
Find all functions $ f: \mathbb{Z}\setminus\{0\}\to \mathbb{Q}$ such that for all $ x,y \in \mathbb{Z}\setminus\{0\}$:
\[ f \left( \frac{x+y}{3}\right) =\frac{f(x)+f(y)}{2}, \; \; x, y \in \mathbb{Z}\setminus\{0\}\]
All the grids of a $m\times n$ chess board ($m,n\geq 3$), are colored either with red or with blue. Two adjacent grids (having a common side) are called a "good couple" if they have different colors. Suppose there are $S$ "good couples". Explain how to determine whether $S$ is odd or even. Is it prescribed by
some specific color grids? Justify your answers.
Let $N$ be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing $1$, and thereafter if a player has written $n$ on a certain move, his adversary is allowed to write $n+1$ or $2n$ as long as he/she does not obtain a number greater than $N$. The player who writes $N$ wins.
$(a)$ Determine which player has a winning strategy for $N=2005$.
$(b)$ Determine which player has a winning strategy for $N=2004$.
$(c)$ Find for how many integers $N\le 2005$ Barbara has a winning strategy.
Let $f(x) = (x^2+3x+2)^{\cos(\pi x)}$. Find the sum of all positive integers $n$ for which \[\left| \sum_{k=1}^n \log_{10} f(k) \right| = 1.\]
Let $f:\mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\}$ be defined by $f(0)=0$,
$$f(2n+1)=2f(n)$$
for $n \ge 0$ and
$$f(2n)=2f(n)+1$$
for $n \ge 1$
If $g(n)=f(f(n))$, prove that $g(n-g(n))=0$ for all $n \ge 0$.
$n\ge 3$ guests met at a party. Some of them know each other but there is no quartet of different guests $a, b, c, d$ such that in pairs $\lbrace a, b \rbrace, \lbrace b, c \rbrace, \lbrace c, d \rbrace, \lbrace d, a \rbrace$ guests know each other but in pairs $\lbrace a, c \rbrace, \lbrace b, d \rbrace$ guests don't know each other. We say a nonempty set of guests $X$ is an [i]ingroup[/i], when guests from $X$ know each other pairwise and there are no guests not from $X$ knowing all guests from $X$. Prove that there are at most $\frac{n(n-1)}{2}$ different ingroups at that party.
In a meeting, there are $2011$ scientists attending. We know that, every scientist know at least $1509$ other ones. Prove that a group of five scientists can be formed so that each one in this group knows $4$ people in his group.
Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as
\[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \]
where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.
Richard rolls a fair six-sided die repeatedly until he rolls his twentieth prime number or his second even number. Compute the probability that his last roll is prime.
Call a non-constant polynomial [i]real[/i] if all its coecients are real. Let $P$ and $Q$ be polynomials with complex coefficients such that the composition $P \circ Q$ is real. Show that if the leading coefficient of $Q$ and its constant term are both real, then $P$ and $Q$ are real.
Consider the power series expansion \[\dfrac{1}{1-2x-x^2}=\sum_{n=0}^\infty a_nx^n.\] Prove that, for each integer $n\geq 0$, there is an integer $m$ such that \[a_n^2+a_{n+1}^2=a_m.\]
Find all nondecreasing functions $f:\mathbb R\to \mathbb R$ such that, for all $x,y\in \mathbb R$, $$f(f(x))+f(y)=f(x+f(y))+1.$$
[i]Proposed by Carl Schildkraut[/i]
[b]2.[/b] Let $n \geq 3$ be an integer. Prove that for all integers $k$, with $1 \leq k \leq \binom{n}{2}$, there exists a set $A$ with $n$ distinct positive integer elements such that the set $B = \{\gcd(x, y): x, y \in A, x \neq y \}$ (gotten from the greatest common divisor of all pairs of distinct elements from $A$) contains exactly $k$ distinct elements.
How many functions $\{f : 1,2, \cdots, 2013\} \rightarrow \{1,2, \cdots, 2013\}$ satisfy $f(j) < f(i) + j - i$ for all integers $i,j$ such that $1 \leq i < j \leq 2013$ ?
Let $n$ be positive integer. Define a sequence $\{a_k\}$ by
\[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\]
(1) Find $a_2$ and $a_3$.
(2) Find the general term $a_k$.
(3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$.
50 points
An arbitrary positive number $a$ is given. A sequence ${a_n}$ is defined by equalities $a_1=\frac{a}{a+1}$ and $a_{n+1}=\frac{aa_n}{a^2+a_n-aa_n}$ for all $n \geq 1$
Find the minimal constant $C$ such that inequality $$a_1+a_1a_2+\ldots+a_1\ldots a_m<C$$ holds for all positive integers $m$ regardless of $a$
On a circle there are $2n+1$ points, dividing it into equal arcs ($n\ge 2$). Two players take turns to erase one point. If after one player's turn, it turned out that all the triangles formed by the remaining points on the circle were obtuse, then the player wins and the game ends.
Who has a winning strategy: the starting player or his opponent?
There are $n$ boxes which is numbere from $1$ to $n$. The box with number $1$ is open, and the others are closed. There are $m$ identical balls ($m\geq n$). One of the balls is put into the open box, then we open the box with number $2$. Now, we put another ball to one of two open boxes, then we open the box with number $3$. Go on until the last box will be open. After that the remaining balls will be randomly put into the boxes. In how many ways this arrangement can be done?
Let $\mathbb{Z}^+$ denote the set of all positive integers. Find all surjective functions $f:\mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ that satisfy all of the following conditions: for all $a,b,c \in \mathbb{Z}^+$,
(i)$f(a,b) \leq a+b$;
(ii)$f(a,f(b,c))=f(f(a,b),c)$
(iii)Both $\binom{f(a,b)}{a}$ and $\binom{f(a,b)}{b}$ are odd numbers.(where $\binom{n}{k}$ denotes the binomial coefficients)
Let $A$ be a set of positive integers satisfying the following properties:
(i) if $m$ and $n$ belong to $A$, then $m+n$ belong to $A$;
(ii) there is no prime number that divides all elements of $A$.
(a) Suppose $n_1$ and $n_2$ are two integers belonging to $A$ such that $n_2-n_1 >1$. Show that you can find two integers $m_1$ and $m_2$ in $A$ such that $0< m_2-m_1 < n_2-n_1$
(b) Hence show that there are two consecutive integers belonging to $A$.
(c) Let $n_0$ and $n_0+1$ be two consecutive integers belonging to $A$. Show that if $n\geq n_0^2$ then $n$ belongs to $A$.
Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ with $c_1 = a, c_k = b$ such that $c_i, c_{i+1}$ are adjacent for $i=1,2,\ldots,k-1$.
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Find the maximal number $M$ such that there exists a coloring admitting $M$ pairwise disconnected squares.
Let $f: \mathbb{N} \to \mathbb{N}$ be an arbitrary function. Prove that there exist two positive integers $x$ and $y$ which satisfy $f(x+y) \le f(2x+f(y))$.
[i](Proposed by David Anghel, Romania)[/i]
Let $ U$ be a real normed space such that, for an finite-dimensional, real normed space $ X,U$ contains a subspace isometrically isomorphic to $ X$. Prove that every (not necessarily closed) subspace $ V$ of $ U$ of finite codimension has the same property. (We call $ V$ of finite codimension if there exists a finite-dimensional subspace $ N$ of $ U$ such that $ V\plus{}N\equal{}U$.)
[i]A. Bosznay[/i]
We are given one red and $k>1$ blue cells, and a pack of $2n$ cards, enumerated by the numbers from $1$ to $2n$. Initially, the pack is situated on the red cell and arranged in an arbitrary order. In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by $1$, or onto an empty cell. Given $k$, what is the maximal $n$ for which it is always possible to move all the cards onto a blue cell?