Found problems: 85335
We consider a game on an infinite chessboard similar to that of solitaire: If two adjacent fields are occupied by pawns and the next field is empty (the three fields lie on a vertical or horizontal line), then we may remove these two pawns and put one of them on the third field. Prove that if in the initial position pawns fill a $3k \times n$ rectangle, then it is impossible to reach a position with only one pawn on the board.
Find the smallest number of the form $1...1$ in its decimal expression which is divisible by $\underbrace{\hbox{3...3}}_{\hbox{100}}$,.
Find all couples $(x,y)$ over the positive integers such that:
$7^x+x^4+47=y^2$
A number of flowers are distributed between $n$ persons so that the first of them, Andreas, gets one flower, the other gets two flowers, the third gets three flowers, etc., to $n$-th person who gets $n$ flowers. Andreas then walks around shaking hands with each other of the others, in any order. In order to do so, he receives a flower from everyone which he hangs on to and which has more flowers than himself at the moment they shake hands. Which is the smallest number of flowers Andreas can have after shaking hands with everyone?
Let $x_1$, $x_2$, ..., $x_5$ be nonnegative real numbers such that $x_1+x_2+x_3+x_4+x_5=5$. Determine the maximum value of $x_1x_2+x_2x_3+x_3x_4+x_4x_5$.
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Take five good haikus
Scramble their lines randomly
What are the chances
That you end up with
Five completely good haikus
(With five, seven, five)?
Your answer will be
m over n where m,n
Are numbers such that
m,n positive
Integers where gcd
Of m,n is 1.
Take this answer and
Add the numerator and
Denominator.
[i]Proposed by Jeff Lin[/i]
Let $n\ge2$ be an integer. A regular $(2n+1)-gon$ is divided in to $2n-1$ triangles by diagonals which do not meet except at the vertices. Prove that at least three of these triangles are isosceles.
Points $A_1, A_2,\ldots, A_n$ are found on a circle in this order. Each point $A_i$ has exactly $i$ coins. A move consists in taking two coins from two points (may be the same point) and moving them to adjacent points (one move clockwise and another counter-clockwise). Find all possible values of $n$ for which it is possible after a finite number of moves to obtain a configuration with each point $A_i$ having $n+1-i$ coins.
Juli has a deck of $54$ cards and proposes the following game to Bruno. Juli places the cards in a row, some face-up and others face-down. Bruno can repeatedly perform the following move: select a card and flip it along with its two neighbors (turning face-up cards face-down, and vice versa for face-down cards). Bruno wins if, through this process, he manages to turn all the cards face up. Otherwise, Juli wins. Determine which player has a winning strategy and explain it.
[b]Note:[/b] When Bruno selects the first or the last card in the row, he flips only two cards. In all other cases, he flips three cards.
Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.
Compute \[\sum_{k=0}^{100}\left\lfloor\dfrac{2^{100}}{2^{50}+2^k}\right\rfloor.\] (Here, if $x$ is a real number, then $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.)
Three spheres $s_1$, $s_2$, $s_3$ intersect along one circle $\omega$. Let $A $be an arbitrary point lying on the circle $\omega$. Ray $AB$ intersects spheres $s_1$, $s_2$, $s_3$ at points $B_1$, $B_2$, $B_3$, respectively, ray $AC$ intersects spheres $s_1$, $s_2$, $s_3$ at points $C_1$, $C_2$, $C_3$, respectively ($B_i \ne A_i$, $C_i \ne A_i$, $i=1,2,3$). It is known that $B_2$ is the midpoint of the segment $B_1B_3$. Prove that $C_2$ is the midpoint of the segment $C_1C_3$.
Three circles $\omega_1$, $\omega_2$ and $\omega_3$ pass through one common point, say $P$. The tangent line to $\omega_1$ at $P$ intersects $\omega_2$ and $\omega_3$ for the second time at points $P_{1,2}$ and $P_{1,3}$, respectively. Points $P_{2,1}$, $P_{2,3}$, $P_{3,1}$ and $P_{3,2}$ are similarly defined. Prove that the perpendicular bisector of segments $P_{1,2}P_{1,3}$, $P_{2,1}P_{2,3}$ and $P_{3,1}P_{3,2}$ are concurrent.
[i]Proposed by Mahdi Etesamifard[/i]
Consider all triples $(x,y,p)$ of positive integers, where $p$ is a prime number, such that $4x^2 + 8y^2 + (2x-3y)p-12xy = 0$. Which below number is a perfect square number for every such triple $(x,y, p)$?
A. $4y + 1$ B. $2y + 1$ C. $8y + 1$ D. $5y - 3$ E. $8y - 1$
There exists a polynomial $P$ of degree $5$ with the following property: if $z$ is a complex number such that $z^5+2004z=1$, then $P(z^2)=0$. Calculate the quotient $\tfrac{P(1)}{P(-1)}$.
Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$
[i]Proposed by Christopher Bradley, United Kingdom[/i]
Compute
$$\int_0^6\frac{x-3}{x^2-6x-7}dx.$$
In the acute-angled triangle $ABC$, $P$ is the midpoint of segment $BC$ and the point $K$ is the foot of the altitude from $A$. $D$ is a point on segment $AP$ such that $\angle BDC=90$. Let $(ADK) \cap BC=E$ and $(ABC) \cap AE=F$. Prove that $\angle AFD=90$.
Let $ABCD$ a convex quadrilateral with $AB=BC=CD$, with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$.
Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?
Let $A{}$ be a point in the Cartesian plane. At each step, Ann tells Bob a number $0< a\leqslant 1$ and he then moves $A{}$ in one of the four cardinal directions, at his choice, by a distance of $a{}$. This process cotinues as long as Ann wishes. Amongst every $100$ consecutive moves, each of the four possible moves should have been made at least once. Ann's goal is to force Bob to eventually choose a point at a distance greater than $100$ from the initial position of $A{}$. Can Ann achieve her goal?
Let $ABCD$ be a parallelogram such that $AB = 35$ and $BC = 28$. Suppose that $BD \perp BC$. Let $\ell_1$ be the reflection of $AC$ across the angle bisector of $\angle BAD$, and let $\ell_2$ be the line through $B$ perpendicular to $CD$. $\ell_1$ and $\ell_2$ intersect at a point $P$. If $PD$ can be expressed in simplest form as $\frac{m}{n}$, find $m + n$.
Two circles $\omega_1$ and $\omega_2$ meeting at point $A$ and a line $a$ are given. Let $BC$ be an arbitrary chord of $\omega_2$ parallel to $a$, and $E$, $F$ be the second common points of $AB$ and $AC$ respectively with $\omega_1$. Find the locus of common points of lines $BC$ and $EF$.
Answer the following questions:
(1) Let $f(x)$ be a function such that $f''(x)$ is continuous and $f'(a)=f'(b)=0$ for some $a<b$.
Prove that $f(b)-f(a)=\int_a^b \left(\frac{a+b}{2}-x\right)f''(x)dx$.
(2) Consider the running a car on straight road. After a car which is at standstill at a traffic light started at time 0, it stopped again at the next traffic light apart a distance $L$ at time $T$. During the period, prove that there is an instant for which the absolute value of the acceleration of the car is more than or equal to $\frac{4L}{T^2}.$
Let $n$ be a positive integer, and denote by $\mathbb{Z}_n$ the ring of integers modulo $n$. Suppose that there exists a function $f:\mathbb{Z}_n\to\mathbb{Z}_n$ satisfying the following three properties:
(i) $f(x)\neq x$,
(ii) $f(f(x))=x$,
(iii) $f(f(f(x+1)+1)+1)=x$ for all $x\in\mathbb{Z}_n$.
Prove that $n\equiv 2 \pmod4$.
(Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Germany)