Found problems: 5802
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
The eyes of a magician are blindfolded while a person $A$ from the audience arranges $n$ identical coins in a row, some are heads and the others are tails. The assistant of the magician asks $A$ to write an integer between $1$ and $n$ inclusive and to show it to the audience. Having seen the number, the assistant chooses a coin and turns it to the other side (so if it was heads it becomes tails and vice versa) and does not touch anything else. Afterwards, the bandages are removed from the magician, he sees the sequence and guesses the written number by $A$. For which $n$ is this possible?
[hide=Spoiler hint] The original formulation asks: a) Show that if $n$ is possible, so is $2n$; b) Show that only powers of $2$ are possible; I have omitted this from the above formulation, for the reader's interest. [/hide]
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.
Define the sequence $x_1,x_2,\dots$ by the initial terms $x_1=2, x_2=4$, and the recurrence relation
\[x_{n+2}=3x_{n+1}-2x_n+\frac{2^n}{x_n} \quad \text{for} \quad n \ge 1.\]
Prove that $\lim_{n \to \infty} \frac{x_n}{2^n}$ exists and satisfies
\[\frac{1+\sqrt{3}}{2} \le \lim_{n \to \infty} \frac{x_n}{2^n} \le \frac{3}{2}.\]
Let $S$ be a set of $n$ points in the coordinate plane. Say that a pair of points is [i]aligned[/i] if the two points have the same $x$-coordinate or $y$-coordinate. Prove that $S$ can be partitioned into disjoint subsets such that (a) each of these subsets is a collinear set of points, and (b) at most $n^{3/2}$ unordered pairs of distinct points in $S$ are aligned but not in the same subset.
On the board are written $n$ natural numbers, $n\in \mathbb{N}$. In one move it is possible to choose two
equal written numbers and increase one by $1$ and decrease the other by $1$. Prove that in this
the game cannot be played more than $\frac{n^3}{6}$ moves.
Problem 4. Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel performs exactly one of the following moves:
(a) He clears every piece of rubbish from a single pile.
(b) He clears one piece of rubbish from each pile.
However, every evening, a demon sneaks into the warehouse and performs exactly one of the
following moves:
(a) He adds one piece of rubbish to each non-empty pile.
(b) He creates a new pile with one piece of rubbish.
What is the first morning when Angel can guarantee to have cleared all the rubbish from the
warehouse?
Let the function $f:N^*\to N^*$ such that
[b](1)[/b] $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$;
[b](2)[/b] $n\le f(n)\le n+2014 , \forall n\in N^*$
Show that: there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$.
(High School Affiliated to Nanjing Normal University )
In the dog dictionary the words are any sequence of letters $A$ and $U$ for example $AA$, $UAU$ and $AUAU$. For each word, your "profundity" will be the quantity of subwords we can obtain by the removal of some letters.
For each positive integer $n$, determine the largest "profundity" of word, in dog dictionary, can have with $n$ letters.
Note: The word $AAUUA$ has "profundity" $14$ because your subwords are $A, U, AU, AA, UU, UA, AUU, UUA, AAU, AUA, AAA, AAUU, AAUA, AUUA$.
Let $x_1, x_2, ..., x_n$ be positive numbers, with $n \ge 2$. Prove that
$$\left(x_1+\frac{1}{x_1}\right)\left(x_2+\frac{1}{x_2}\right)...\left(x_n+\frac{1}{x_n}\right)\ge \left(x_1+\frac{1}{x_2}\right)\left(x_2+\frac{1}{x_3}\right)...\left(x_{n-1}+\frac{1}{x_n}\right)\left(x_n+\frac{1}{x_1}\right)$$
Each cell of an $m\times n$ board is filled with some nonnegative integer. Two numbers in the filling are said to be [i]adjacent[/i] if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a [i]garden[/i] if it satisfies the following two conditions:
(i) The difference between any two adjacent numbers is either $0$ or $1$.
(ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to $0$.
Determine the number of distinct gardens in terms of $m$ and $n$.
Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.
$a_0 = a_1 = 1$ and ${a_{n+1} . a_{n-1}} = a_n . (a_n + 1)$ for all positive integers n.
prove that $a_n$ is one integer for all positive integers n.
Let $n\ge 3$ be a positive integer. In the plane $n$ points which are not all collinear are marked. Find the least possible number of triangles whose vertices are all marked.
(Recall that the vertices of a triangle are not collinear.)
Find all injective functions $ f : \mathbb{N} \to \mathbb{N}$ which satisfy
\[ f(f(n)) \le\frac{n \plus{} f(n)}{2}\]
for each $ n \in \mathbb{N}$.
Let $P(x) = x^4-x^3+x$.
a) Prove that for all positive real numbers $a$, the polynomial $P(x) - a$ has a unique positive zero.
b) A sequence $(a_n)$ is defined by $a_1 = \dfrac{1}{3}$ and for all $n \geq 1$, $a_{n+1}$ is the positive zero of the polynomial $P(x) - a_n$. Prove that the sequence $(a_n)$ converges, and find the limit of the sequence.
Let $n\geq1$ be an integer and let $t_1<t_2<\dots<t_n$ be positive integers. In a group of $t_n+1$ people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following two conditions to hold at the same time:
(i) The number of games played by each person is one of $t_1,t_2,\dots,t_n$.
(ii) For every $i$ with $1\leq i\leq n$, there is someone who has played exactly $t_i$ games of chess.
Suppose $\frac{1}{2} < s < 1$ . An insect flying on $[0,1]$ . If it is on point $a$ , it jump into point $ a\times s$ or $(a-1) \times s +1$ . For every real number $0 \le c \le 1$, Prove that insect can jump that after some jumps , it has a distance less than $\frac {1}{1402}$ from point $c$.
[i]Proposed by Navid Safaei [/i]
Let $n \geq 3$ be an integer. Let $f$ be a function from the set of all integers to itself with the following property: If the integers $a_1,a_2,\ldots,a_n$ form an arithmetic progression, then the numbers
$$f(a_1),f(a_2),\ldots,f(a_n)$$
form an arithmetic progression (possibly constant) in some order. Find all values for $n$ such that the only functions $f$ with this property are the functions of the form $f(x)=cx+d$, where $c$ and $d$ are integers.
For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$.
[*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list]
[i]Proposed by Bruno Le Floch, France[/i]
In boxes labeled $0$, $1$, $2$, $\dots$, we place integers according to the following rules:
$\bullet$ If $p$ is a prime number, we place it in box $1$.
$\bullet$ If $a$ is placed in box $m_a$ and $b$ is placed in box $m_b$, then $ab$ is placed in the box labeled $am_b+bm_a$.
Find all positive integers $n$ that are placed in the box labeled $n$.
Let $ f_1,f_2,\dots,f_n$ be regular functions on a domain of the complex plane, linearly independent over the complex field. Prove that the functions $ f_i\overline{f}_k, \;1 \leq i,k \leq n$, are also linearly independent.
[i]L. Lempert[/i]
Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal.
[i]David Yang.[/i]
Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties:
(a). $x_1 < x_2 < \cdots < x_{n-1}$ ;
(b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$;
(c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.
Let a function $f$ satisfy $f(1) = 1$ and $f(1)+ f(2)+...+ f(n) = n^2f(n)$ for all $n \in N$. Determine $f(1995)$.