Found problems: 85335
Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers
A positive number, if its fractional part, integeral part, and itself are geometric series, then the number is________.
Let $n \ge 6$ be an integer and $F$ be the system of the $3$-element subsets of the set $\{1, 2,...,n \}$ satisfying the following condition:
for every $1 \le i < j \le n$ there is at least $ \lfloor \frac{1}{3} n \rfloor -1$ subsets $A\in F$ such that $i, j \in A$.
Prove that for some integer $m \ge 1$ exist the mutually disjoint subsets $A_1, A_2 , ... , A_m \in F $ also, that $|A_1\cup A_2 \cup ... \cup A_m |\ge n-5 $
(Poland)
PS. just in case my translation does not make sense,
I leave the original in Slovak, in case someone understands something else
Points $X$ and $Y$ are chosen inside an acute triangle $ABC$ so that:
$$\angle AXB = \angle CYB = 180^\circ - \angle ABC, \text{ } \angle ABX = \angle CBY$$
Show that the points $X$ and $Y$ are equidistant from the center of the circumscribed circle of $\triangle ABC$.
[i]Proposed by Anton Trygub[/i]
If $m$ and $n$ are integers such that $3m + 4n = 100$, what is the smallest possible value of $\left| m - n \right|$ ?
Prove that there exist $16$ subsets of set $M = \{1,2,...,10000\}$ with the following property:
For every $z \in M$ there are eight of these subsets whose intersection is $\{z\}$.
Let $n\ge3$ be a positive integer. We say that a set $S$ of positive integers is good if $|S|=n$, no element of S is a multiple of n, and the sum of all elements of $S$ is not a multiple of $n$ either. Find, in terms of $n$, the least positive integer $d$ for which there exists a good set $S$ such that there are exactly d nonempty subsets of $S$ the sum of whose elements is a multiple of $n$.
Proposed by Aleksandar Makelov, Burgas, Bulgaria and Nikolai Beluhov, Stara Zagora, Bulgaria
Two polynomials $ f(x)=a_{100}x^{100}+a_{99}x^{99}+\dots+a_{1}x+a_{0}$ and $ g(x)=b_{100}x^{100}+b_{99}x^{99}+\dots+b_{1}x+b_{0}$ of degree $ 100$ differ from each other by a permutation of coefficients. It is known that $ a_{i}\ne b_{i}$ for $ i=0, 1, 2, \dots, 100$. Is it possible that $ f(x)\geq g(x)$ for all real $ x$?
Let $p$ be a prime number such that $p\mid (2^{2019}-1) .$ The sequence $a_1,a_2,...,a_n$ satisfies the following conditions: $a_0=2, a_1=1 ,a_{n+1}=a_n+\frac{p^2-1}{4}a_{n-1}$ $(n\geq 1).$ Prove that $p\nmid (a_n+1),$ for any $n\geq 0.$
Find the minimum and the maximum value of the expression $\sqrt{4 -a^2} +\sqrt{4 -b^2} +\sqrt{4 -c^2}$
where $a,b, c$ are positive real numbers satisfying the condition $a^2 + b^2 + c^2=6$
Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent.
[i]Proposed by Alireza Dadgarnia[/i]
Triangle $ABC$ has side lengths $13$, $14$ and $15$. Let $k, k_A,k_B,k_C$ be four circles of radius $ r$ inside the triangle such that $k_A$ is tangent to sides $AB$ and $AC$, $k_B$ is tangent to sides $BA$ and $BC$, $k_C$ is tangent to sides $CA$ and $CB$, and $k$ is externally tangent to circles $k_A$, $k_B$ and $k_C$. Let $r = m/n$ where $m$ and $n$ are coprime. Find $m + n$.
Four dice are thrown. What is the probability that the product of the number equals $ 36?$
Prove that the polynomial $P (x)$ with integer coefficients, taking odd values for $x = 0$ and $x= 1$, has no integer roots.
Let \[\begin{array}{ccccccccccc}A&=&5\cdot 6&-&6\cdot 7&+&7\cdot 8&-&\cdots&+&2003\cdot 2004,\\B&=&1\cdot 10&-&2\cdot 11&+&3\cdot 12&-&\cdots&+&1999\cdot 2008.\end{array}\]
Find the value of $A-B$.
Let the polynomial $P(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots +a_1x+a_0$ where $1011\leq a_i\leq 2021$ for all $i=0,1,2,...,21.$ Given that $P(x)$ has an integer root and there exists an positive real number$c$ such that $|a_{k+2}-a_k|\leq c$ for all $k=0,1,...,19.$
a) Prove that $P(x)$ has an only integer root.
b) Prove that $$\sum_{k=0}^{10}(a_{2k+1}-a_{2k})^2\leq 440c^2.$$
Find the minimum positive integer $k$ such that there exists a function $f$ from the set $\Bbb{Z}$ of all integers to $\{1, 2, \ldots k\}$ with the property that $f(x) \neq f(y)$ whenever $|x-y| \in \{5, 7, 12\}$.
How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?
$ \textbf{(A) } 21 \qquad \textbf{(B) } 34 \qquad \textbf{(C) } 51 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 150$
Given a positive integer $ n$, for all positive integers $ a_1, a_2, \cdots, a_n$ that satisfy $ a_1 \equal{} 1$, $ a_{i \plus{} 1} \leq a_i \plus{} 1$, find $ \displaystyle \sum_{i \equal{} 1}^{n} a_1a_2 \cdots a_i$.
there are some identical squares with sides parallel, in a plane. Among any $k+1$ of them, there are two with a point in common. Prove they can be divided into $2k-1$ sets, such that all the squares in one set aint pairwise disjoint.
For an integer $n$, define $f(n)$ to be the greatest integer $k$ such that $2^k$ divides $\binom{n}{m}$ for some $0 \le m \le n$. Compute $f(1) + f(2) + \cdots + f(2048)$.
In triangle ABC $\angle ABC=60^{o}$ and $O$ is the center of the circumscribed circle. The bisector $BL$ intersects the circumscribed circle at the point $W$. Prove that $OW$ is tangent to $(BOL)$
A right parallelepiped (i.e. a parallelepiped one of whose edges is perpendicular to a face) is given. Its vertices have integral coordinates, and no other points with integral coordinates lie on its faces or edges. Prove that the volume of this parallelepiped is a sum of three perfect squares.
[i]Proposed by A. Golovanov[/i]
Suppose $f$ and $g$ are differentiable functions such that \[xg(f(x))f^\prime(g(x))g^\prime(x)=f(g(x))g^\prime(f(x))f^\prime(x)\] for all real $x$. Moreover, $f$ is nonnegative and $g$ is positive. Furthermore, \[\int_0^a f(g(x))dx=1-\dfrac{e^{-2a}}{2}\] for all reals $a$. Given that $g(f(0))=1$, compute the value of $g(f(4))$.
In the system of base $n^2 + 1$ find a number $N$ with $n$ different digits such that:
[b](i)[/b] $N$ is a multiple of $n$. Let $N = nN'.$
[b](ii)[/b] The number $N$ and $N'$ have the same number $n$ of different digits in base $n^2 + 1$, none of them being zero.
[b]
(iii)[/b] If $s(C)$ denotes the number in base $n^2 + 1$ obtained by applying the permutation $s$ to the $n$ digits of the number $C$, then for each permutation $s, s(N) = ns(N').$