Found problems: 85335
Prove that $n^{11}-n$ is divisible by $11$.
In a triangle $ABC$ of the area $S$, point $H$ is the orthocenter, $D,E,F$ are the feet of the altitudes from $A,B,C$, and $P,Q,R$ are the reflections of $A,B,C$ in $BC,CA,AB$, respectively. The triangles $DEF$ and $PQR$ have the same area $T$. Given that $T > \frac{3}{5}S$, prove that $T = S$.
Let $f:[0,1] \to [0,1]$ a increasing continuous function, diferentiable in $(0,1)$ and with derivative smaller than 1 in every point. The sequence of sets $A_1,A_2,A_3,\dots$ is define as: $A_1 = f([0,1])$, and for $n \geq 2, A_n = f(A_{n-1}).$ Prove that $\displaystyle \lim_{n\to+\infty} d(A_n) = 0$, where $d(A)$ is the diameter of the set $A$.
Note: The diameter of a set $X$ is define as $d(X) = \sup_{x,y\in X} |x-y|.$
You are given $20$ weights such that any object of integer weight $m$, $1 \le m \le1997$, can be balanced by placing it on one pan of a balance and a subset of the weights on the other pan. What is the minimal value of largest of the $20$ weights if the weights are
(a) all integers;
(b) not necessarily integers?
(M Rasin)
Let $P = \{P_1, P_2, ..., P_{1997}\}$ be a set of $1997$ points in the interior of a circle of radius 1, where $P_1$ is the center of the circle. For each $k=1.\ldots,1997$, let $x_k$ be the distance of $P_k$ to the point of $P$ closer to $P_k$, but different from it. Show that $(x_1)^2 + (x_2)^2 + ... + (x_{1997})^2 \le 9.$
Let $m$ and $n$ be natural numbers with $mn$ even. Jetze is going to cover an $m \times n$ board (consisting of $m$ rows and $n$ columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer $V$ (in terms of $m$ and $n$) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than $V$, no matter how Jetze covers the board.
There exists infinitely many positive integers such that
$a^3 + 1990b^3 = c^4$.
For every positive integer $k>1$ prove that there exist a real number $x$ so that for every positive integer $n<1398$:
$$\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.$$
[i]Proposed by Mohammad Amin Sharifi[/i]
Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.
A four-element set $\{a, b, c, d\}$ of positive integers is called [i]good[/i] if there are two of them such that their product is a mutiple of the greatest common divisor of the remaining two. For example, the set $\{2, 4, 6, 8\}$ is good since the greatest common divisor of $2$ and $6$ is $2$, and it divides $4\times 8=32$.
Find the greatest possible value of $n$, such that any four-element set with elements less than or equal to $n$ is good.
[i]Proposed by Victor and Isaías de la Fuente[/i]
At a length of $104$ miles, the Danyang-Kushan Bridge holds the title for being the longest bridge in the world. A car travels at a constant speed of $39$ miles per hour across the Danyang-Kushan Bridge. How long does it take the car to travel across the entire bridge?
$\textbf{(A) }\text{2 hours, 12 minutes} \qquad \textbf{(B) }\text{2 hours, 20 minutes} \qquad \textbf{(C) }\text{2 hours, 25 minutes}$\\
$\textbf{(D) }\text{2 hours, 30 minutes} \qquad \textbf{(E) }\text{2 hours, 40 minutes}$
Calculate $$\sqrt{6 + \sqrt{6 + \sqrt{6 +... }}}+\frac{6}{1+ \frac{6}{1+...}}$$
Given any five nonnegative real numbers with the sum $1$, show that they can be arranged around a circle in such a way that the five products of two consecutive numbers sum up to at most $1/5$.
Let $I$ be the incenter of a triangle $ABC$. The lines passing through $A$ and parallel to $BI, CI$ meet the perpendicular bisector to $AI$ at points $S, T$ respectively. Let $Y$ be the common point of $BT$ and $CS$, and $A^*$ be a point such that $BICA^*$ is a parallelogram. Prove that the midpoint of segment $YA^*$ lies on the excircle of the triangle touching the side $BC$.
Let $n$ and $k$ be positive integers. Two infinite sequences $\{s_i\}_{i\geq 1}$ and $\{t_i\}_{i\geq 1}$ are [i]equivalent[/i] if, for all positive integers $i$ and $j$, $s_i = s_j$ if and only if $t_i = t_j$. A sequence $\{r_i\}_{i\geq 1}$ has [i]equi-period[/i] $k$ if $r_1, r_2, \ldots $ and $r_{k+1}, r_{k+2}, \ldots$ are equivalent.
Suppose $M$ infinite sequences with equi-period $k$ whose terms are in the set $\{1, \ldots, n\}$ can be chosen such that no two chosen sequences are equivalent to each other. Determine the largest possible value of $M$ in terms of $n$ and $k$.
A cricket randomly hops between $4$ leaves, on each turn hopping to one of the other $3$ leaves with equal probability. After $4$ hops what is the probability that the cricket has returned to the leaf where it started?
$\textbf{(A)}~\displaystyle\frac{2}{9}\qquad\textbf{(B)}~\displaystyle\frac{19}{80}\qquad\textbf{(C)}~\displaystyle\frac{20}{81}\qquad\textbf{(D)}~\displaystyle\frac{1}{4}\qquad\textbf{(E)}~\displaystyle\frac{7}{27}$
Let $a < b$ be positive integers. Prove that there is a positive integer $n{}$ and a polynomial of the form \[\pm1\pm x\pm x^2\pm\cdots\pm x^n,\]divisible by the polynomial $1+x^a+x^b$.
Find all functions $f : Z \to Z$ such that $f(-1) = f(1)$ and $f(x)+ f(y) = f(x+2xy)+ f(y-2xy)$ for all $x,y \in Z$
The numbers $1$ through $16$ are to be written in the cells of a $4\times 4$ board.
(a) Prove that this can be done in such a way that any two numbers in cells that share a side differ by at most $4$.
(b) Prove that this cannot be done in such a way that any two numbers in cells that share a side differ by at most $3$.
Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations:
\[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.
Find all $a$ such that for any positive integer $n$, the number $an(n+2)(n+3)(n+4)$ is an integer. (Author: O. Podlipski)
[url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=427802](similar to Problem 5 of grade 9)[/url]
Same problem for grades 10 and 11
Consider the ellipsoid$$\frac{x^2}{a^2}+\frac{y^2}{a^2}+\frac{z^2}{b^2}=1$$($a$ and $b > 0$) and the ellipse $E$ which is the intersection of the ellipsoid with the plane of equation$$mx + ny + pz = 0$$where the point $P = [m, n, p]$ is a random point from the unit sphere $(m^2 + n^2 + p^2 = 1)$. Consider the random variable $A_E$ the area of the ellipse $E$. If the point $P$ is chosen with uniform distribution with respect to the area on the unit sphere, what is the expectation of $A_E$ ?
The point $A_1$ on the perimeter of a convex quadrilateral $ABCD$ is such that the line $AA_1$ divides the quadrilateral into two parts of equal area. The points $B_1$, $C_1$, $D_1$ are defined similarly.
Prove that the area of the quadrilateral $A_1B_1C_1D_1$ is greater than a quarter of the area of $ABCD$.
[i]L. Emelyanov [/i]
Two circles $ \omega_{1}$ and $ \omega_{2}$ intersect in points $ A$ and $ B$. Let $ PQ$ and $ RS$ be segments of common tangents to these circles (points $ P$ and $ R$ lie on $ \omega_{1}$, points $ Q$ and $ S$ lie on $ \omega_{2}$). It appears that $ RB\parallel PQ$. Ray $ RB$ intersects $ \omega_{2}$ in a point $ W\ne B$. Find $ RB/BW$.
[i]S. Berlov [/i]
Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.