Found problems: 85335
Solve the following equation for $x, y \in \mathbb{N}$
\[x^3+y^3=101xy+101.\]
Prove that any real solution of $x^3+px+q=0$, where $p,q$ are real numbers, satisfies the inequality $4qx\le p^2$.
This test and the matching AMC 12P were developed for the use of a group of Taiwan schools, in early January of 2002. When Taiwan had taken the contests, the AMC released the questions here as a set of practice questions for the 2002 AMC 10 and AMC 12 contests.
Ash and Gary independently come up with their own lineups of $15$ fire, grass, and water monsters. Then, the first monster of both lineups will fight, with fire beating grass, grass beating water, and water beating fire. The defeated monster is then substituted with the next one from their team’s lineup; if there is a draw, both monsters get defeated.
Gary completes his lineup randomly, with each monster being equally likely to be any of the three types. Without seeing Gary’s lineup, Ash chooses a lineup that maximizes the probability p that his monsters are the last ones standing. Compute $p.$
Find all integer sequences of the form $ x_i, 1 \le i \le 1997$, that satisfy $ \sum_{k\equal{}1}^{1997} 2^{k\minus{}1} x_{k}^{1997}\equal{}1996\prod_{k\equal{}1}^{1997}x_k$.
Given are a prime $p$ and a positive integer $a$. Let $q$ be a prime divisor of $\frac{a^{p^3}-1}{a^{p^2}-1}$ and $q\neq p$. Prove that $q\equiv 1 ( \mod p^3)$.
[b]p1.[/b] A $10 \times 10$ array of squares is given. In each square, a student writes the product of the row number and the column number of the square (the upper left hand corner of this array is shown below). Determine the sum of the $100$ integers written in the array.
[img]https://cdn.artofproblemsolving.com/attachments/5/9/527fdf90529221f6d06af169de1728da296538.png[/img]
[b]p2.[/b] The equilateral triangle $DEF$ is inscribed in the equilateral triangle $ABC$ so that $ED$ is perpendicular to $BC$. If the area of $ABC$ equals one square unit, determine the area of $DEF$.
[img]https://cdn.artofproblemsolving.com/attachments/c/0/6e1a303a45fa89576e26bc8fd30ce6564aaad1.png[/img]
[b]p3.[/b] Consider a symmetric triangular set of points as shown (every point lies a distance of one unit from each of its neighbors). A collection of $m$ lines has the property that for every point in the arrangement, there is at least one line in the collection that passes through that point. Prove or disprove that $m \ge 10$.
[img]https://cdn.artofproblemsolving.com/attachments/0/9/540f2781312f86672df1578bfe4f68b51d3b2c.png[/img]
[b]p4.[/b] Let $P$ be a convex polygon drawn on graph paper (defined as the grid of all lines with equations $x = a$ and $y = b$, with $a$ and $b$ integers). We know that all the vertices of $P$ are at the intersections of grid lines and none of its sides is parallel to a grid line. Let $H$ be the sum of the lengths of the horizontal segments of the grid which are contained in the interior of $P$, and let $V$ be the sum of the lengths of the vertical segments of the grid in the interior of $P$. Prove that $H = V$ .
[b]p5.[/b] Peter, Paul, and Mary play the following game. Given a fixed positive integer $k$ which is at most $2013$, they randomly choose a subset $A$ of $\{1, 2,..., 2013\}$ with $k$ elements. The winner is Peter, Paul, or Mary, respectively, if the sum of the numbers in $A$ leaves a remainder of $0$, $1$, or $2$ when divided by $3$. Determine the values of $k$ for which this game is fair (i.e., such that the three possible outcomes are equally likely).
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
(J.-L.Ayme, France) Points $ P$, $ Q$ lie on the circumcircle $ \omega$ of triangle $ ABC$. The perpendicular bisector $ l$ to $ PQ$ intersects $ BC$, $ CA$, $ AB$ in points $ A'$, $ B'$, $ C'$. Let $ A"$, $ B"$, $ C"$ be the second common points of $ l$ with the circles $ A'PQ$, $ B'PQ$, $ C'PQ$. Prove that $ AA"$, $ BB"$, $ CC"$ concur.
Find the value of $\frac{\log_59\log_75\log_37}{\log_2\sqrt{6}}+\frac{1}{\log_9\sqrt{6}}$
$ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad\textbf{(E) }7 $
Bicycle license plates in Flatville each contain three letters. The first is chosen from the set $\{C,H,L,P,R\}$, the second from $\{A,I,O\}$, and the third from $\{D,M,N,T\}$.
When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added to one set and one to another set. What is the largest possible number of $\text{ADDITIONAL}$ license plates that can be made by adding two letters?
$ \text{(A)}\ 24\qquad\text{(B)}\ 30\qquad\text{(C)}\ 36\qquad\text{(D)}\ 40\qquad\text{(E)}\ 60 $
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f^{(19)}(n)+97f(n)=98n+232.\]
Let $M$ be a finite set of lattice points and $n$ be a positive integer. A $\textit{mine-avoiding path}$ is a path of lattice points with length $n$, beginning at $(0,0)$ and ending at a point on the line $x+y=n,$ that does not contain any point in $M$. Prove that if there exists a mine-avoiding path, then there exist at least $2^{n-|M|}$ mine-avoiding paths. [hide=*]A lattice point is a point $(x,y)$ where $x$ and $y$ are integers. A path of lattice points with length $n$ is a sequence of lattice points $P_0,P_1,\ldots, P_n$ in which any two adjacent points in the sequence have distance 1 from each other.[/hide]
[i]Ankit Bisain and Holden Mui[/i]
For which $n \in N$ is it possible to arrange a tennis tournament for doubles with $n$ players such that each player has every other player as an opponent exactly once?
Consider a triangle $ABC$ with altitudes $AD, BE$, and $CF$, and orthocenter $H$. Let the perpendicular line from $H$ to $EF$ intersects $EF, AB$ and $AC$ at $P, T$ and $L$, respectively. Point $K$ lies on the side $BC$ such that $BD=KC$. Let $\omega$ be a circle that passes through $H$ and $P$, that is tangent to $AH$. Prove that circumcircle of triangle $ATL$ and $\omega$ are tangent, and $KH$ passes through the tangency point.
Let $s=\frac{AB+BC+AC}{2}$ be half-perimeter of triangle $ABC$. Let $L$ and $N$be a point's on ray's $AB$ and $CB$, for which $AL=CN=s$. Let $K$ is point, symmetric of point $B$ by circumcenter of $ABC$. Prove, that perpendicular from $K$ to $NL$ passes through incenter of $ABC$.
Solution for problem [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]
How many integers with four different digits are there between $1,000$ and $9,999$ such that the absolute value of the difference between the first digit and the last digit is $2$?
$\textbf {(A) } 672 \qquad \textbf {(B) } 784 \qquad \textbf {(C) } 840 \qquad \textbf {(D) } 896 \qquad \textbf {(E) } 1008$
Let $(a_n), (b_n)$, $n = 1,2,...$ be two sequences of integers defined by $a_1 = 1, b_1 = 0$ and for $n \geq 1$
$a_{n+1} = 7a_n + 12b_n + 6$
$b_{n+1} = 4a_n + 7b_n + 3$
Prove that $a_n^2$ is the difference of two consecutive cubes.
For $n\in\mathbb{N}$ prove that
\[\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n}\leq\frac{1}{\sqrt{2n+1}}.\]
Find all continuously differentiable functions $ f: \mathbb{R}\to\mathbb{R}$ such that for every rational number $ q,$ the number $ f(q)$ is rational and has the same denominator as $ q.$ (The denominator of a rational number $ q$ is the unique positive integer $ b$ such that $ q\equal{}a/b$ for some integer $ a$ with $ \gcd(a,b)\equal{}1.$) (Note: $ \gcd$ means greatest common divisor.)
Determine all integers $n\ge 2$ with the following property: there exist nonzero real numbers $x_1, x_2, \ldots, x_n,y$ such that
\[(x_1+x_2+\ldots+x_k)(x_{k+1}+x_{k+2}+\ldots+x_n)=y\]
for all $k\in\{1,2,\ldots,n-1\}$.
Let $ S$ be the set of odd integers greater than $ 1$. For each $ x \in S$, denote by $ \delta (x)$ the unique integer satisfying the inequality $ 2^{\delta (x)}<x<2^{\delta (x) \plus{}1}$. For $ a,b \in S$, define:
$ a \ast b\equal{}2^{\delta (a)\minus{}1} (b\minus{}3)\plus{}a.$
Prove that if $ a,b,c \in S$, then:
$ (a)$ $ a \ast b \in S$ and
$ (b)$ $ (a \ast b)\ast c\equal{}a \ast (b \ast c)$.
A line in the plane of a triangle $T$ is called an [i]equalizer[/i] if it divides $T$ into two regions having equal area and equal perimeter. Find positive integers $a>b>c,$ with $a$ as small as possible, such that there exists a triangle with side lengths $a,b,c$ that has exactly two distinct equalizers.
Let $R$ and $T$ be points respectively on sides $[BC]$ and $[CD]$ of a square $ABCD$ with side length $6$ such that $|CR|+|RT|+|TC|=12$. What is $\tan (\widehat{RAT})$
$
\textbf{(A)}\ 2\sqrt 3
\qquad\textbf{(B)}\ \sqrt 3
\qquad\textbf{(C)}\ \dfrac 13
\qquad\textbf{(D)}\ \dfrac 12
\qquad\textbf{(E)}\ 1
$
Circle $B$, which has radius 2008, is tangent to horizontal line $A$ at point $P$. Circle $C_1$ has radius 1 and is tangent both to circle $B$ and to line $A$ at a point to the right of point $P$. Circle $C_2$ has radius larger than 1 and is tangent to line $A$ and both circles B and $C_1$. For $n>1$, circle $C_n$ is tangent to line $A$ and both circles $B$ and $C_{n-1}$. Find the largest value of n such that this sequence of circles can be constructed through circle $C_n$ where the n circles are all tangent to line $A$ at points to the right of $P$.
[asy]
size(300);
draw((-10,0)--(10,0));
draw(arc((0,10),10,210,330));
label("$P$",(0,0),S);
pair C=(0,10),X=(12,3);
for(int kk=0;kk<6;++kk)
{
pair Y=(X.x-X.y,X.y);
for(int k=0;k<20;++k)
Y+=(abs(Y-X)-X.y-Y.y,abs(Y-C)-10-Y.y)/3;
draw(circle(Y,Y.y));
X=Y;
}[/asy]
Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$.
[i]Proposed by Hojoo Lee, Korea[/i]