Found problems: 85335
For a natural number $x$ we define $f(x)$ to be the sum of all natural numbers less than $x$ and coprime with it. Let $m$ and $n$ be some natural numbers where $n$ is odd. Prove that there exist $x$, which is a multiple of $m$ and for which $f(x)$ is a perfect n-th power.
We say that a rook is "attacking" another rook on a chessboard if the two rooks are in the same row or column of the chessboard and there is no piece directly between them. Let $n$ be the maximum number of rooks that can be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other. How many ways can $n$ rooks be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other?
There are $N{}$ mess-loving clerks in the office. Each of them has some rubbish on the desk. The mess-loving clerks leave the office for lunch one at a time (after return of the preceding one). At that moment all those remaining put half of rubbish from their desks on the desk of the one who left. Can it so happen that after all of them have had lunch the amount of rubbish at the desk of each one will be the same as before lunch if a) $N = 2{}$ and b) $N = 10$?
[i]Alexey Zaslavsky[/i]
Two buckets each have four balls; two red balls and two white balls in the first, and two red balls and two blue balls in the second. At first, a bucket is selected, then a ball in the bucket is selected, with both buckets and balls inside the selected bucket having equal probability of being chosen. Then, without replacement of the first ball, the process is repeated once more. Determine the probability that the first ball drawn being red if the second ball drawn was blue.
What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?
Let $a,b $ be two relatively prime positive integers.Also let $m,n $ be positive integers with $n> m $.\\
Prove that\\
$lcm [am+b,a (m+1)+b,...,an+b]\ge (n+1)\cdot \binom {n}{m}$
[i]Proposed by Navid Safaei[/i]
The game Prongle is played with a special deck of cards: on each card is a nonempty set of distinct colors. No two cards in the deck contain the exact same set of colors. In this game, a “Prongle” is a set of at least $2$ cards such that each color is on an even number of cards in the set. Let k be the maximum possible number of prongles in a set of $2019$ cards. Find $\lfloor \log 2 (k) \rfloor$.
Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the equality
$$ \int_a^b f(x)dx=f(b)-f(a), $$
for any real numbers $ a,b. $
[i]Cosmin Nitu[/i]
Let $a_{1}=1$, $a_{n}=\frac{1}{n} \sum_{k=1}^{n-1}a_{k}a_{n-k}$ for $n\geq 2$. Show that
i) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}<2^{-\frac{1}{2}}$;
ii) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}\geq \frac{2}{3}$
Determine the least positive integer $n$ with the following property – for every 3-coloring of numbers $1,2,\ldots,n$ there are two (different) numbers $a,b$ of the same color such that $|a-b|$ is a perfect square.
Let $A$, $B$, $C$, $D$, $E$, $F$ be $6$ points on a circle in that order. Let $X$ be the intersection of $AD$ and $BE$, $Y$ is the intersection of $AD$ and $CF$, and $Z$ is the intersection of $CF$ and $BE$. $X$ lies on segments $BZ$ and $AY$ and $Y$ lies on segment $CZ$. Given that $AX = 3$, $BX = 2$, $CY = 4$, $DY = 10$, $EZ = 16$, and $FZ = 12$, find the perimeter of triangle $XYZ$.
Let $T$ be the set of all 2-digit numbers whose digits are in $\{1,2,3,4,5,6\}$ and the tens digit is strictly smaller than the units digit. Suppose $S$ is a subset of $T$ such that it contains all six digits and no three numbers in $S$ use all six digits. If the cardinality of $S$ is $n$, find all possible values of $n$.
Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuous and bijective function, such that $$\lim_{x\rightarrow\infty}\frac{f^{-1}(f(x)/x)}{x}=1.$$
[list=a]
[*] Show that $\lim_{x\rightarrow\infty}\frac{f(x)}{x}=\infty$ and $\lim_{x\rightarrow\infty}\frac{f^{-1}(ax)}{f^{-1}(x)}=1$ for any $a>0$.
[*] Give an example of function which satisfies the hypothesis.
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
The vertices of a convex $2550$-gon are colored black and white as follows: black, white, two black, two white, three black, three white, ..., 50 black, 50 white. Dania divides the polygon into quadrilaterals with diagonals that have no common points. Prove that there exists a quadrilateral among these, in which two adjacent vertices are black and the other two are white.
[i]D. Rudenko[/i]
Convex quadrilateral $ABCD$ is cyclic in circle $(O)$, $P$ is the intersection of the diagonals $AC$ and $BD$. Circle $(O_{1})$ passes through $P$ and $B$, circle $(O_{2})$ passes through $P$ and $A$, Circles $(O_{1})$ and $(O_{2})$ intersect at $P$ and $Q$. $(O_{1})$, $(O_{2})$ intersect $(O)$ at another points $E$, $F$ (besides $B$, $A$), respectively. Prove that $PQ$, $CE$, $DF$ are concurrent or parallel.
Let $ABCD$ be a convex quadrilateral. Assume line $AB$ and $CD$ intersect at $E$, and $B$ lies between $A$ and $E$. Assume line $AD$ and $BC$ intersect at $F$, and $D$ lies between $A$ and $F$. Assume the circumcircles of $\triangle BEC$ and $\triangle CFD$ intersect at $C$ and $P$. Prove that $\angle BAP=\angle CAD$ if and only if $BD\parallel EF$.
Let $N$ be a positive integer. Kingdom Wierdo has $N$ castles, with at most one road between each pair of cities. There are at most four guards on each road. To cost down, the King of Wierdos makes the following policy:
(1) For any three castles, if there are roads between any two of them, then any of these roads cannot have four guards.
(2) For any four castles, if there are roads between any two of them, then for any one castle among them, the roads from it toward the other three castles cannot all have three guards.
Prove that, under this policy, the total number of guards on roads in Kingdom Wierdo is smaller than or equal to $N^2$.
[i]Remark[/i]: Proving that the number of guards does not exceed $cN^2$ for some $c > 1$ independent of $N$ will be scored based on the value of $c$.
[i]Proposed by usjl[/i]
In triangle $ABC$, find the smallest possible value of $$|(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)|$$
A non-square rectangle is cut into $N$ rectangles of various shapes and sizes. Prove that one can always cut each of these rectangles into two rectangles so that one can construct a square and rectangle, each figure consisting of $N$ pieces.
[i](6 points)[/i]
Construct a triangle, given the feet of its altitudes. Express the sides of a triangle $Y$ in terms of the sides of the triangle $X$ formed by the feet of the altitudes of $Y$.
The radius of the circumcircle of triangle $\Delta$ is $R$ and the radius of the inscribed circle is $r$.
Prove that a circle of radius $R + r$ has an area more than $5$ times the area of triangle $\Delta$.
Three circles of radius $1$ are inscribed in a square of side length $s$, such that the circles do not overlap or coincide with each other. What is the minimum $s$ where such a configuration is possible?
Prove that there exists $m \in \mathbb{N}$ such that there exists an integral sequence $\lbrace a_n \rbrace$ which satisfies:
[b]I.[/b] $a_0 = 1, a_1 = 337$;
[b]II.[/b] $(a_{n + 1} a_{n - 1} - a_n^2) + \frac{3}{4}(a_{n + 1} + a_{n - 1} - 2a_n) = m, \forall$ $n \geq 1$;
[b]III. [/b]$\frac{1}{6}(a_n + 1)(2a_n + 1)$ is a perfect square $\forall$ $n \geq 1$.
On the board are written the natural numbers from $1$ to $2001$ inclusive. You have to delete some numbers so that among those that remain undeleted it is impossible to choose two different numbers such that the result of their multiplication is equal to one of the numbers that remain undeleted. What is the minimum number of numbers that must be deleted? For that amount, present an example showing which numbers are erased. Justify why, if fewer numbers are deleted, the desired property is not obtained.