Found problems: 85335
In Middle-Earth, nine cities form a $3$ by $3$ grid. The top left city is the capital of Gondor and the bottom right city is the capital of Mordor. How many ways can the remaining cities be divided among the two nations such that all cities in a country can be reached from its capital via the grid-lines without passing through a city of the other country?
a) Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that\[f(x-f(y))=f(f(x))-f(y)-1\]holds for all $x,y\in\mathbb{Z}$. (It is [url=https://artofproblemsolving.com/community/c6h1268817p6621849]2015 IMO Shortlist A2 [/url])
b) The same question for if \[f(x-f(y))=f(f(x))-f(y)-2\] for all integers $x,y$
A quadruple $(p, a, b, c)$ of positive integers is called a Leiden quadruple if
- $p$ is an odd prime number,
- $a, b$, and $c$ are distinct and
- $ab + 1, bc + 1$ and $ca + 1$ are divisible by $p$.
a) Prove that for every Leiden quadruple $(p, a, b, c)$ we have $p + 2 \le \frac{a+b+c}{3}$ .
b) Determine all numbers $p$ for which a Leiden quadruple $(p, a, b, c)$ exists with $p + 2 = \frac{a+b+c}{3} $
Find all positive integers $n$ such that for all positive integers $m$, $1<m<n$, relatively prime to $n$, $m$ must be a prime number.
The longest diagonal of a convex hexagon is $2$. Is there necessarily a side or diagonal in this hexagon whose length does not exceed $1$?
Given natural numbers $k$ and $n, k \le n, n \ge 3,$ find the set of all values in the interval $(0, \pi)$ that the $k^{th}-$largest among the interior angles of a convex $n$-gon can take.
[b]p1.[/b] Arrange the whole numbers $1$ through $15$ in a row so that the sum of any two adjacent numbers is a perfect square. In how many ways this can be done?
[b]p2.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 - q^2$ is divisible by $24$.
[b]p3.[/b] If a polyleg has even number of legs he always tells truth. If he has an odd number of legs he always lies.
Once a green polyleg told a dark-blue polyleg ”- I have $8$ legs. And you have only $6$ legs!”
The offended dark-blue polyleg replied ”-It is me who has $8$ legs, and you have only $7$ legs!”
A violet polyleg added ”-The dark-blue polyleg indeed has $8$ legs. But I have $9$ legs!”
Then a stripped polyleg started ”None of you has $8$ legs. Only I have $8$ legs!”
Which polyleg has exactly $8$ legs?
[b][b]p4.[/b][/b] There is a small puncture (a point) in the wall (as shown in the figure below to the right). The housekeeper has a small flag of the following form (see the figure left). Show on the figure all the points of the wall where you can hammer in a nail such that if you hang the flag it will close up the puncture.
[img]https://cdn.artofproblemsolving.com/attachments/a/f/8bb55a3fdfb0aff8e62bc4cf20a2d3436f5d90.png[/img]
[b]p5.[/b] Assume $ a, b, c$ are odd integers. Show that the quadratic equation $ax^2 + bx + c = 0$ has no rational solutions.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
A college student drove his compact car $ 120$ miles home for the weekend and averaged $ 30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $ 20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?
$ \textbf{(A)}\ 22 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 26 \qquad \textbf{(E)}\ 28$
Points $ A,B,C,D,E$ and $ F$ lie, in that order, on $ \overline{AF}$, dividing it into five segments, each of length 1. Point $ G$ is not on line $ AF$. Point $ H$ lies on $ \overline{GD}$, and point $ J$ lies on $ \overline{GF}$. The line segments $ \overline{HC}, \overline{JE},$ and $ \overline{AG}$ are parallel. Find $ HC/JE$.
$ \text{(A)}\ 5/4 \qquad
\text{(B)}\ 4/3 \qquad
\text{(C)}\ 3/2 \qquad
\text{(D)}\ 5/3 \qquad
\text{(E)}\ 2$
Ed has five identical green marbles and a large supply of identical red marbles. He arranges the green marbles and some of the red marbles in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves equals the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is GGRRRGGRG. Let $m$ be the maximum number of red marbles for which Ed can make such an arrangement, and let $N$ be the number of ways in which Ed can arrange the $m+5$ marbles to satisfy the requirement. Find the remainder when $N$ is divided by 1000.
For a positive integer $n,$ let $P_n$ be the set of sequences of $2n$ elements, each $0$ or $1,$ where there are exactly $n$ $1$’s and $n$ $0$’s. I choose a sequence uniformly at random from $P_n.$ Then, I partition this sequence into maximal blocks of consecutive $0$’s and $1$’s. Define $f(n)$ to be the expected value of the sum of squares of the block lengths of this uniformly random sequence. What is the largest integer value that $f(n)$ can take on?
What percentage of twenty thousand is a quarter of a million?
An arbitrary circle can intersect the graph $ y \equal{} \sin x$ in
$ \textbf{(A)} \text{ at most 2 points} \qquad \textbf{(B)} \text{ at most 4 points} \qquad$
$ \textbf{(C)} \text{ at most 6 points} \qquad \textbf{(D)} \text{ at most 8 points} \qquad$
$ \textbf{(E)} \text{ more than 16 points}$
Let $ABC$ be an isosceles triangle with $\angle A=\angle B=80^\circ$. A straight line passes through $B$
and through the circumcenter of the triangle and intersects the side $AC$ at $D$. Prove that $AB=CD$.
Let $x, y$ and $z$ be three positive real numbers for which
\[x^2+y^2+z^2=xy+yz+zx.\]
Find the value of $\frac{\sqrt x}{\sqrt x + \sqrt y+ \sqrt z}.$
Let $X_1X_2X_3X_4$ be a quadrilateral inscribed in circle $\Omega$ such that $\triangle{X_1X_2X_3}$ has side lengths $13,14,15$ in some order. For $1 \le i \le 4$, let $l_i$ denote the tangent to $\Omega$ at $X_i$, and let $Y_i$ denote the intersection of $l_i$ and $l_{i+1}$ (indices taken modulo $4$). Find the least possible area of $Y_1Y_2Y_3Y_4$.
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$
Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
Line $\frac{x}{4}+\frac{y}{3}=1$ and ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ intersect at $A$ and $B$. A point on the ellipse $P$ satisties that the area of $\triangle PAB$ is $3$. The number of such points is
$\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4$
Show that there exists a triangle $ABC$ such that $a \ne b$ and $a+t_a = b+t_b$, where $t_a,t_b$ are the medians corresponding to $a,b$, respectively. Also prove that there exists a number $k$ such that every such triangle satisfies $a+t_a = b+t_b = k(a+b)$. Finally, find all possible ratios $a : b$ in such triangles.
Let $p,q$ be prime numbers such that their sum isn't divisible by $3$. Find the all $(p,q,r,n)$ positive integer quadruples satisfy:
$$p+q=r(p-q)^n$$
[i]Proposed by Şahin Emrah[/i]
$p>2$ persons participate at a chess tournament, two players play at most one game against each other. After $n$ games were made, no more game is running and in every subset of three players, we can find at least two that havem't played against each other.
Show that $n \leq \frac{p^{2}}4$.
Suppose that points $A$ and $B$ lie on circle $\Omega$, and suppose that points $C$ and $D$ are the trisection points of major arc $AB$, with $C$ closer to $B$ than $A$. Let $E$ be the intersection of line $AB$ with the line tangent to $\Omega$ at $C$. Suppose that $DC = 8$ and $DB = 11$. If $DE = a\sqrt{b}$ for integers $a$ and $b$ with $b$ squarefree, find $a + b$.
$ABC$ is a right-angled triangle, with $\angle ABC = 90^{\circ}$. $B'$ is the reflection of $B$ over $AC$. $M$ is the midpoint of $AC$. We choose $D$ on $\overrightarrow{BM}$, such that $BD = AC$. Prove that $B'C$ is the angle bisector of $\angle MB'D$.
NOTE: An important condition not mentioned in the original problem is $AB<BC$. Otherwise, $\angle MB'D$ is not defined or $B'C$ is the external bisector.
Let $\mathcal M_n(\mathbb R)$ denote the set of all real $n\times n$ matrices. Find all surjective functions $f:\mathcal M_n(\mathbb R)\to\{0,1,\ldots,n\}$ which satisfy
$$f(XY)\le\min\{f(X),f(Y)\}$$for all $X,Y\in\mathcal M_n(\mathbb R)$.
An [i]annulus[/i] is the region between two concentric circles. The concentric circles in the figure have radii $ b$ and $ c$, with $ b > c$. Let $ \overline{OX}$ be a radius of the larger circle, let $ \overline{XZ}$ be tangent to the smaller circle at $ Z$, and let $ \overline{OY}$ be the radius of the larger circle that contains $ Z$. Let $ a \equal{} XZ$, $ d \equal{} YZ$, and $ e \equal{} XY$. What is the area of the annulus?
$ \textbf{(A)}\ \pi a^2 \qquad \textbf{(B)}\ \pi b^2 \qquad \textbf{(C)}\ \pi c^2 \qquad \textbf{(D)}\ \pi d^2 \qquad \textbf{(E)}\ \pi e^2$
[asy]unitsize(1.4cm);
defaultpen(linewidth(.8pt));
dotfactor=3;
real r1=1.0, r2=1.8;
pair O=(0,0), Z=r1*dir(90), Y=r2*dir(90);
pair X=intersectionpoints(Z--(Z.x+100,Z.y), Circle(O,r2))[0];
pair[] points={X,O,Y,Z};
filldraw(Circle(O,r2),mediumgray,black);
filldraw(Circle(O,r1),white,black);
dot(points);
draw(X--Y--O--cycle--Z);
label("$O$",O,SSW,fontsize(10pt));
label("$Z$",Z,SW,fontsize(10pt));
label("$Y$",Y,N,fontsize(10pt));
label("$X$",X,NE,fontsize(10pt));
defaultpen(fontsize(8pt));
label("$c$",midpoint(O--Z),W);
label("$d$",midpoint(Z--Y),W);
label("$e$",midpoint(X--Y),NE);
label("$a$",midpoint(X--Z),N);
label("$b$",midpoint(O--X),SE);[/asy]