Found problems: 85335
Find all pairs $\left(m,n\right)$ of positive integers, with $m,n\geq2$, such that $a^n-1$ is divisible by $m$ for each $a\in \left\{1,2,3,\ldots,n\right\}$.
Describe all quadrilaterals with perpendicular diagonals which are both inscribed and circumscribed.
Let $P$ be a point in the plane of a triangle $ABC$, different from its circumcenter. Prove that the triangle whose vertices are the projections of $P$ on the perpendicular bisectors of the sides of $ABC$, is similar to $ABC$.
Let $a$ and $b$ be integer solutions to $17a+6b=13$. What is the smallest possible positive value for $a-b$?
Two players play the following game. The first player starts by writing either $0$ or $1$ and then, on his every move, chooses either $0$ or $1$ and writes it to the right of the existing digits until there are $1999$ digits. Each time the first player puts down a digit (except the first one) , the second player chooses two digits among those already written and swaps them. Can the second player guarantee that after his last move the line of digits will be symmetrical about the middle digit?
(I Izmestiev)
Find $\sum_{k=1}^\infty\frac{k^2-2}{(k+2)!}$.
For a given a positive integer $n$, find all integers $x_1, x_2,... , x_n$ subject to $0 < x_1 < x_2 < ...< x_n < x_{n+1}$ and $$x_nx_{n+1} \le 2(x_1 + x_2 + ... + x_n).$$
Two circles intersect at the points $A$ and $B$. Tangent lines drawn to both of the circles at the point $A$ intersect the circles at the points $M$ and $N$. The lines $BM$ and $BN$ intersect the circles once more at the points $P$ and $Q$ respectively. Prove that the segments $MP$ and $NQ$ are equal.
(I Nagel)
Given a positive integer $n$, let $p(n)$ be the product of the non-zero digits of $n$. (If $n$ has only one digits, then $p(n)$ is equal to that digit.) Let \[ S=p(1)+p(2)+p(3)+\cdots+p(999). \] What is the largest prime factor of $S$?
$(MON 2)$ Given reals $x_0, x_1, \alpha, \beta$, find an expression for the solution of the system \[x_{n+2} -\alpha x_{n+1} -\beta x_n = 0, \qquad n= 0, 1, 2, \ldots\]
A regular heptagon $A_1A_2A_3A_4A_5A_6A_7$ is given. Straight $A_2A_3$ and $A_5A_6$ intersect at point $X$, and straight lines $A_3A_5$ and $A_1A_6$ intersect at point $Y$. Prove that lines $A_1A_2$ and $XY$ are parallel.
Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.
Six people sit at a circular table (in the shape of a regular hexagon) such that no two friends sit next to or across from each other. Find, with proof, the maximum number of unordered pairs of people that can be friends.
Let $u_k$ be a sequence of integers, and let $V_n$ be the number of those which are less than or equal to $n$. Show that if
$$\sum_{k=1}^{\infty} \frac{1}{u_k } < \infty,$$
then
$$\lim_{n \to \infty} \frac{ V_{n}}{n}=0.$$
Circles $\omega_1$ and $\omega_2$ touch externally at the point $O$. Points $A$ and $B$ on the circle $\omega_1$ and points $C$ and $D$ on the circle $\omega_2$ are such that $AC$ and $BD$ are common external tangents to circles. Line $AO$ intersects segment $CD$ at point $M$ and straight line $CO$ intersexts $\omega_1$ again at point $N$. Prove that the points $B$, $M$ and $N$ lie on the same straight line.
Given a triangle$ABC$ with $AB<BC<AC$ inscribed in circle $(c)$. The circle $c(A,AB)$ (with center $A$ and radius $AB$) interects the line $BC$ at point $D$ and the circle $(c)$ at point $H$. The circle $c(A,AC)$ (with center $A$ and radius $AC$) interects the line $BC$ at point $Z$ and the circle $(c)$ at point $E$. Lines $ZH$ and $ED$ intersect at point $T$. Prove that the circumscribed circles of triangles $TDZ$ and $TEH$ are equal.
Let $b$ and $c$ be fixed real numbers and let the ten points $(j,y_j )$ for $j=1,2,\ldots,10$ lie on the parabola $y =x^2 +bx+c.$ For $j=1,2,\ldots, 9$ let $I_j$ be the intersection of the tangents to the given parabola at $(j, y_j )$ and $(j+1, y_{j+1}).$ Determine the poynomial function $y=g(x)$ of least degree whose graph passes through all nine points $I_j .$
What is the sum of real roots of $(2+(2+(2+x)^2)^2)^2=2000$ ?
$ \textbf{(A)}\ -4
\qquad\textbf{(B)}\ -2
\qquad\textbf{(C)}\ 0
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ 4
$
Given a triangle $ABC$, let $P$ be a point inside the triangle such that $| BP | > | AP |, | BP | > | CP |$. Show that $\angle ABC <90^o$
Let $f(x) = 3x + 2.$ Prove that there exists $m \in \mathbb{N}$ such that $f^{100}(m)$ is divisible by $1988$.
Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$.
[i]Proposed by A. Golovanov, M. Ivanov, K. Kokhas[/i]
There are $100$ glasses, with $101,102,...,200$ cents.Two players play next game. In every move they can take some cents from one glass, but after move should be different number of cents in every glass. Who will win with right strategy?
What is the maximum possible value of $5-|6x-80|$ over all integers $x$?
$\textbf{(A) }{-}1\qquad\textbf{(B) }0\qquad\textbf{(C) }1\qquad\textbf{(D) }3\qquad\textbf{(E) }5$
The natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_n $ satisfy the condition $ 1 / a_1 + 1 / a_2 + \ldots + 1 / a_n = 1 $. Prove that all these numbers do not exceed $$ n ^ {2 ^ n} $$
$a_1,a_2,a_3,\dots$ and $b_1,b_2,b_3,\dots$ are sequences of positive integers. Show that we can find $m < n$ such that $a_m \leq a_n$ and $b_m \leq b_n$.