Found problems: 5802
There are $n$ students each having $r$ positive integers. Their $nr$ positive integers are all different. Prove that we can divide the students into $k$ classes satisfying the following conditions.
(a) $ k \le 4r $
(b) If a student $A$ has the number $m$, then the student $B$ in the same class can't have a number $l$ such that
\[ (m-1)! < l < (m+1)!+1 \]
Each vertex of a finite graph can be coloured either black or white. Initially all vertices are black. We are allowed to pick a vertex $P$ and change the colour of $P$ and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a
sequence of operations of this type?
Note: A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex $A$ and vertex $B,$ then $A$ and $B$ are neighbours of each other.
On the computer screen there are initially two $1$'s written. The [i] insert [/i] program causes the sum of those numbers to be inserted between each pair of numbers by pressing the $Enter$ key.
In the first step a number is inserted and we obtain $1-2-1$; In the second step two numbers are inserted and we have $1-3-2-3-1$; In the third, four numbers are inserted and you have $1-4-3-5-2-5-3-4-1$; etc Find the sum of all the numbers that appear on the screen at the end of step number $25$.
Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that:
$$f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q$$
holds for all $p,q\in\mathbb{P}$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
The Devil and the Man play a game. Initially, the Man pays some cash $s$ to the Devil. Then he lists some $97$ triples $\{i,j,k\}$ consisting of positive integers not exceeding $100$. After that, the Devil draws some convex polygon $A_1A_2...A_{100}$ with area $100$ and pays to the Man, the sum of areas of all triangles $A_iA_jA_k$. Determine the maximal value of $s$ which guarantees that the Man receives at least as much cash as he paid.
[i]Proposed by Nikolai Beluhov, Bulgaria[/i]
Santa Claus has at least $n$ gifts for $n$ children. For $i\in\{1,2, ... , n\}$, the $i$-th child considers $x_i > 0$ of these items to be desirable. Assume that
\[\dfrac{1}{x_1}+\cdots+\dfrac{1}{x_n}\le1.\]
Prove that Santa Claus can give each child a gift that this child likes.
Let $\mathbb{R}_+$ be the set of positive real numbers. Find all functions $f\colon \mathbb{R}_+ \to \mathbb{R}_+$ such that
\[f(xy + x + y) + f \left( \frac1x \right) f\left( \frac1y \right) = 1\]
for every $x$, $y\in \mathbb{R}_+$.
[i]Proposed by Li4 and Untro368.[/i]
Let $m$ and $n$ be positive integers with $m\le 2000$ and $k=3-\frac{m}{n}$. Find the smallest positive value of $k$.
Find all functions $f : \mathbb{Z} \to\mathbb{ Z}$ such that
\[ n^2+4f(n)=f(f(n))^2 \]
for all $n\in \mathbb{Z}$.
[i]Proposed by Sahl Khan, UK[/i]
Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system:
\[a_1^2 + a_1 - 1 = a_2\]
\[ a_2^2 + a_2 - 1 = a_3\]
\[\hspace*{3.3em} \vdots \]
\[a_{n}^2 + a_n - 1 = a_1\]
Given a natural number $n\ge 3$, determine all strictly increasing sequences $a_1<a_2<\cdots<a_n$ such that $\text{gcd}(a_1,a_2)=1$ and for any pair of natural numbers $(k,m)$ satisfy $n\ge m\ge 3$, $m\ge k$, $$\frac{a_1+a_2+\cdots +a_m}{a_k}$$ is a positive integer.
Find all functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ such that $$\dfrac{f(x)f(y)}{f(xy)} = \dfrac{\left( \sqrt{f(x)} + \sqrt{f(y)} \right)^2}{f(x+y)}$$ holds for all positive rational numbers $x, y$.
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ that satisfy the conditions:
$i) f(f(x)) = xf(x) - x^2 + 2,\forall x\in\mathbb{Z}$
$ii) f$ takes all integer values
Let $S = \{1,2,\dots,2014\}$. For each non-empty subset $T \subseteq S$, one of its members is chosen as its representative. Find the number of ways to assign representatives to all non-empty subsets of $S$ so that if a subset $D \subseteq S$ is a disjoint union of non-empty subsets $A, B, C \subseteq S$, then the representative of $D$ is also the representative of one of $A$, $B$, $C$.
[i]Warut Suksompong, Thailand[/i]
Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that
$ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$
(Proposed by Gerhard Woeginger, Austria)
Let $2=p_1<p_2<\ldots<p_n<\ldots$ be all prime numbers.
Prove that for any positive integer $n \geq 3$ there exist at least $p_n+n-1$ prime numbers, that do not exceed $p_1p_2\ldots p_n$
[i]I. Voronovich[/i]
We say that a natural number $n$ is [i]charrua[/i] if it satisfy simultaneously the following conditions:
- Every digit of $n$ is greater than 1.
- Every time that four digits of $n$ are multiplied, it is obtained a divisor of $n$
Show that every natural number $k$ there exists a [i]charrua[/i] number with more than $k$ digits.
A positive integer $n$ is said to be a [i]perfect power[/i] if $n=a^b$ for some integers $a,b$ with $b>1$.
$(\text{a})$ Find $2004$ perfect powers in arithmetic progression.
$(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.
Determine all non-constant monic polynomials $f(x)$ with integer coefficients for which there exists a natural number $M$ such that for all $n \geq M$, $f(n)$ divides $f(2^n) - 2^{f(n)}$
[i] Proposed by Anant Mudgal [/i]
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
For each natural number $n\geq 2$, solve the following system of equations in the integers $x_1, x_2, ..., x_n$:
$$(n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n$$
where
$$S=x_1^2+x_2^2+\dots+x_n^2.$$
Ana and Banana are playing a game. First Ana picks a word, which is defined to be a nonempty sequence of capital English letters. (The word does not need to be a valid English word.) Then Banana picks a nonnegative integer $k$ and challenges Ana to supply a word with exactly $k$ subsequences which are equal to Ana's word. Ana wins if she is able to supply such a word, otherwise she loses.
For example, if Ana picks the word "TST", and Banana chooses $k=4$, then Ana can supply the word "TSTST" which has 4 subsequences which are equal to Ana's word.
Which words can Ana pick so that she wins no matter what value of $k$ Banana chooses?
(The subsequences of a string of length $n$ are the $2^n$ strings which are formed by deleting some of its characters, possibly all or none, while preserving the order of the remaining characters.)
[i]Proposed by Kevin Sun
Let $a_{1}=1$, $a_{2}=2$, $a_{3}$, $a_{4}$, $\cdots$ be the sequence of positive integers of the form $2^{\alpha}3^{\beta}$, where $\alpha$ and $\beta$ are nonnegative integers. Prove that every positive integer is expressible in the form \[a_{i_{1}}+a_{i_{2}}+\cdots+a_{i_{n}},\] where no summand is a multiple of any other.
Alice has an isosceles triangle $M_0N_0P$, where $M_0P=N_0P$ and $\angle M_0PN_0=\alpha^{\circ}$. (The angle is measured in degrees.) Given a triangle $M_iN_jP$ for nonnegative integers $i$ and $j$, Alice may perform one of two [i]elongations[/i]:
a) an $M$-[i]elongation[/i], where she extends ray $\overrightarrow{PM_i}$ to a point $M_{i+1}$ where $M_iM_{i+1}=M_iN_j$ and removes the point $M_i$.
b) an $N$-[i]elongation[/i], where she extends ray $\overrightarrow{PN_j}$ to a point $N_{j+1}$ where $N_jN_{j+1}=M_iN_j$ and removes the point $N_j$.
After a series of $5$ elongations, $k$ of which were $M$-elongations, Alice finds that triangle $M_kN_{5-k}P$ is an isosceles triangle. Given that $10\alpha$ is an integer, compute $10\alpha$.
[i]Proposed by Yannick Yao[/i]
Let be a nonzero real number $ a, $ and a natural number $ n. $ Prove the implication:
$$ \{ a \} +\left\{\frac{1}{a}\right\} =1 \implies \{ a^n \} +\left\{\frac{1}{a^n}\right\} =1 , $$
where $ \{\} $ is the fractional part.