Found problems: 85335
In an $m\times n$ rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all $(m,n)$ such that the first person has a winning strategy.
The $2013$ numbers
$$\frac{1}{1\times 2}, \frac{1}{2\times 3},\frac{1}{3\times 4},...,\frac{1}{2013 \times 2014}$$
are arranged randomly on a circle.
(a) Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{4000}$ .
(b) Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{10000}$ .
Find the minimal value of the function
\[\begin{array}{c}\ f(x) =\sqrt{15 - 12 \cos x} + \sqrt{4 -2 \sqrt 3 \sin x}+\sqrt{7-4\sqrt 3 \sin x} +\sqrt{10-4 \sqrt 3 \sin x - 6 \cos x}\end{array}\]
For all natural $n$, an $n$-staircase is a figure consisting of unit squares, with one square in the first row, two squares in the second row, and so on, up to $n$ squares in the $n^{th}$ row, such that all the left-most squares in each row are aligned vertically.
Let $f(n)$ denote the minimum number of square tiles requires to tile the $n$-staircase, where the side lengths of the square tiles can be any natural number. e.g. $f(2)=3$ and $f(4)=7$.
(a) Find all $n$ such that $f(n)=n$.
(b) Find all $n$ such that $f(n) = n+1$.
Point $P$ is in the interior of $\triangle ABC$. The side lengths of $ABC$ are $AB = 7$, $BC = 8$, $CA = 9$. The three foots of perpendicular lines from $P$ to sides $BC$, $CA$, $AB$ are $D$, $E$, $F$ respectively. Suppose the minimal value of $\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$ can be written as $\frac{a}{b}\sqrt{c}$, where $\gcd(a,b) = 1$ and $c$ is square free, calculate $abc$.
[asy]
size(120); pathpen = linewidth(0.7); pointfontpen = fontsize(10); // pointpen = black;
pair B=(0,0), C=(8,0), A=IP(CR(B,7),CR(C,9)), P = (2,1.6), D=foot(P,B,C), E=foot(P,A,C), F=foot(P,A,B);
D(A--B--C--cycle); D(P--D); D(P--E); D(P--F);
D(MP("A",A,N)); D(MP("B",B)); D(MP("C",C)); D(MP("D",D)); D(MP("E",E,NE)); D(MP("F",F,NW)); D(MP("P",P,SE));
[/asy]
$(4^{-1} - 3^{-1})^{-1} =$
$ \textbf{(A)}\ -12\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ \frac{1}{12}\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 12 $
Let $n \ge 4$ be a positive integer and there exist $n$ positive integers that are arranged on a circle such that:
$\bullet$ The product of each pair of two non-adjacent numbers is divisible by $2015 \cdot 2016$.
$\bullet$ The product of each pair of two adjacent numbers is not divisible by $2015 \cdot 2016$.
Find the maximum value of $n$
Evaluate the following definite integral.
\[\int_{-1}^{1}\frac{e^{2x}+1-(x+1)(e^{x}+e^{-x})}{x(e^{x}-1)}dx\]
The incircle $ \omega $ of a quadrilateral $ ABCD $ touches $ AB, BC, CD, DA $ at $ E, F, G, H $, respectively. Choose an arbitrary point $ X$ on the segment $ AC $ inside $ \omega $. The segments $ XB, XD $ meet $ \omega $ at $ I, J $ respectively. Prove that $ FJ, IG, AC $ are concurrent.
Let $ABC$ a triangle. Let $D$ be a point on the circumcircle of this triangle and let $E , F$ be the feet of the perpendiculars from $A$ on $DB, DC$, respectively. Finally, let $N$ be the midpoint of $EF$. Let $M \ne N$ be the midpoint of the side $BC$ . Prove that the lines $NA$ and $NM$ are perpendicular.
Find
$$\prod^{2017}_{k=1} e^{\pi ik/2017}2 cos \left( \frac{\pi k}{2017} \right)$$
Let $n > 2$ be an integer. The numbers $1, 2, . . . , n$ are written at the vertices of an $n$-gon in that order. A move consists of choosing two adjacent vertices and adding $1$ to the numbers written there. Determine all n for which it is possible to achieve that all numbers are identical after a finite number of moves.
Let $ABC$ be a triangle with $AB < BC$. The bisector of angle $C$ meets the line parallel to $AC$ and passing through $B$, at point $P$. The tangent at $B$ to the circumcircle of $ABC$ meets this bisector at point $R$. Let $R'$ be the reflection of $R$ with respect to $AB$. Prove that $\angle R'P B = \angle RPA$.
Christian Reiher and Reid Barton want to open a security box, they already managed to discover the algorithm to generate the key codes and they obtained the following information:
$i)$ In the screen of the box will appear a sequence of $n+1$ numbers, $C_0 = (a_{0,1},a_{0,2},...,a_{0,n+1})$
$ii)$ If the code $K = (k_1,k_2,...,k_n)$ opens the security box then the following must happen:
a) A sequence $C_i = (a_{i,1},a_{i,2},...,a_{i,n+1})$ will be asigned to each $k_i$ defined as follows:
$a_{i,1} = 1$ and $a_{i,j} = a_{i-1,j}-k_ia_{i,j-1}$, for $i,j \ge 1$
b) The sequence $(C_n)$ asigned to $k_n$ satisfies that $S_n = \sum_{i=1}^{n+1}|a_i|$ has its least possible value, considering all possible sequences $K$.
The sequence $C_0$ that appears in the screen is the following:
$a_{0,1} = 1$ and $a_0,i$ is the sum of the products of the elements of each of the subsets with $i-1$ elements of the set $A =$ {$1,2,3,...,n$}, $i\ge 2$, such that $a_{0, n+1} = n!$
Find a sequence $K = (k_1,k_2,...,k_n)$ that satisfies the conditions of the problem and show that there exists at least $n!$ of them.
Two equal-radii circles with centres $ O_1$ and $ O_2$ intersect each other at $ P$ and $ Q$, $ O$ is the midpoint of the common chord $ PQ$. Two lines $ AB$ and $ CD$ are drawn through $ P$ ( $ AB$ and $ CD$ are not coincide with $ PQ$ ) such that $ A$ and $ C$ lie on circle $ O_1$ and $ B$ and $ D$ lie on circle $ O_2$. $ M$ and $ N$ are the mipoints of segments $ AD$ and $ BC$ respectively. Knowing that $ O_1$ and $ O_2$ are not in the common part of the two circles, and $ M$, $ N$ are not coincide with $ O$.
Prove that $ M$, $ N$, $ O$ are collinear.
Find all pairs $(x, y)$ of rational numbers such that $y^2 =x^3 -3x+2$.
a,b,c>0 and $abc\ge 1$.Prove that:
$\dfrac{1}{a^3+2b^3+6}+\dfrac{1}{b^3+2c^3+6}+\dfrac{1}{c^3+2a^3+6} \le \dfrac{1}{3}$
There are $n!$ empty baskets in a row, labelled $1, 2, . . . , n!$. Caesar
first puts a stone in every basket. Caesar then puts 2 stones in every second basket.
Caesar continues similarly until he has put $n$ stones into every nth basket. In
other words, for each $i = 1, 2, . . . , n,$ Caesar puts $i$ stones into the baskets labelled
$i, 2i, 3i, . . . , n!.$
Let $x_i$ be the number of stones in basket $i$ after all these steps. Show that
$n! \cdot n^2 \leq \sum_{i=1}^{n!} x_i^2 \leq n! \cdot n^2 \cdot \sum_{i=1}^{n} \frac{1}{i} $
For $k=1,2,\ldots ,2011$ we denote $S_k=\frac{1}{k}+\frac{1}{k+1}+\cdots +\frac{1}{2011}$.
Compute the sum $S_1+S_1^2+S_2^2+\cdots +S_{2011}^2$.
Let $\alpha, \beta, \gamma$ be the angles of some triangle. Prove that there is a triangle whose sides are equal to $\sin \alpha$, $\sin \beta$, $\sin \gamma$.
Let $a, b, c$ and $d$ be four integers such that $7a + 8b = 14c + 28d$.
Prove that the product $a\cdot b$ is always divisible by $14$.
Is it possible to mark four points on the plane so that the distances between any point and three other points form an arithmetic progression? (V. Brayman)
Consider the set $A = \{1, 2, ..., n\}$. For each integer $k$, let $r_k$ be the largest quantity of different elements of $A$ that we can choose so that the difference between two numbers chosen is always different from $k$. Determine the highest value possible of $r_k$, where $1 \le k \le \frac{n}{2}$
Find all functions $f: R\to R$, such that $f (xf (y) + f (x)) = xy$ for all $x, y \in R $.
Let $P(n)$ be the number of functions $f: \mathbb{R} \to \mathbb{R}$, $f(x)=a x^2 + b x + c$, with $a,b,c \in \{1,2,\ldots,n\}$ and that have the property that $f(x)=0$ has only integer solutions. Prove that $n<P(n)<n^2$, for all $n \geq 4$.
[i]Laurentiu Panaitopol[/i]