Found problems: 85335
Find the positive integer $n$ such that \[ \underbrace{f(f(\cdots f}_{2013 \ f\text{'s}}(n)\cdots ))=2014^2+1 \] where $f(n)$ denotes the $n$th positive integer which is not a perfect square.
[i]Proposed by David Stoner[/i]
The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is
$ \textbf{(A)}\ \Big\{\frac{5}{2}\Big\}\qquad\textbf{(B)}\ \big\{x\ |\ 2 \le x \le 3\big\}\qquad\textbf{(C)}\ \big\{x\ |\ 2\le x < 3\big\}\qquad \\ \textbf{(D)}\ \Big\{x\ |\ 2 < x \le 3\Big\}\qquad\textbf{(E)}\ \Big\{x\ |\ 2 < x < 3\Big\} $
Evaluate $ \int_e^{e^2} \frac {(\ln x)^7\minus{}7!}{(\ln x)^8}\ dx.$
Sorry, I have deleted my first post because that was wrong.
kunny
Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for evert integer $n \ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$. How many times $111$ appears on this sequence?
Number $a$ is such that $\forall a_1, a_2, a_3, a_4 \in \mathbb{R}$, there are integers $k_1, k_2, k_3, k_4$ such that $\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a$. Find the minimum of $a$.
Four teams $A$, $B$, $C$ and $D$ play a football tournament in which each team plays exactly two times against each of the remaining three teams (there are $12$ matches). In each matchif it's a tie each team gets $1$ point and if it isn't a tie then the winner gets $3$ points and the loser gets $0$ points.
At the end of the tournament the teams $A$, $B$ and $C$ have $8$ points each. Determine all possible points of team $D$.
Let $\mathcal{P}$ be the set of all polynomials $p(x)=x^4+2x^2+mx+n$, where $m$ and $n$ range over the positive reals. There exists a unique $p(x) \in \mathcal{P}$ such that $p(x)$ has a real root, $m$ is minimized, and $p(1)=99$. Find $n$.
[i]Proposed by [b]AOPS12142015[/b][/i]
Find all possible real values for $a, b$ and $c$ such that
(a) $a + b + c = 51$,
(b) $abc = 4000$,
(c) $0 < a \le 10$ and $c \ge 25$.
Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)
A triangle with vertices $(6,5)$, $(8,-3)$, and $(9,1)$ is reflected about the line $x=8$ to create a second triangle. What is the area of the union of the two triangles?
$\textbf{(A) }9\qquad
\textbf{(B) }\dfrac{28}{3}\qquad
\textbf{(C) }10\qquad
\textbf{(D) }\dfrac{31}{3}\qquad
\textbf{(E) }\dfrac{32}{3}\qquad$
Prove that there are infinitely many positive integers $n$ such that the number of positive divisors in $2^n-1$ is greater than $n$.
Given a rectangular grid with some cells containing one letter, we say a row or column is [i]edible [/i] if it has more than one cell with a letter and all such cells contain the same letter. Given such a grid, the hungry, hungry letter monster repeats the following procedure: he nds all edible rows and all edible columns and simultaneously eats all the letters in those rows and columns, removing those letters from the grid and leaving those cells empty. He continues this until no more edible rows and columns remain. Call a grid a [i]meal [/i] if the letter monster can eat all of its letters using this procedure.
In the $7$ by $7$ grid to the right, ll each empty space with one letter so that the grid is a meal and there are a total of eight Us, nine Ss, ten As, eleven Ms, and eleven Ts. Some letters have been given to you.
You do not need to prove that your answer is the only one possible; you merely need to find an answer that satis fies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justi cation acceptable.)
[img]https://cdn.artofproblemsolving.com/attachments/9/a/d1886720796e4befd9d3ce0cbd2868d1b649d1.png[/img]
[b]p1.[/b] Joe owns stock. On Monday morning on October $20$th, $2008$, his stocks were worth $\$250,000$. The value of his stocks, for each day from Monday to Friday of that week, increased by $10\%$, increased by $5\%$, decreased by $5\%$, decreased by $15\%$, and decreased by $20\%$, though not necessarily in that order. Given this information, let $A$ be the largest possible value of his stocks on that Friday evening, and let $B$ be the smallest possible value of his stocks on that Friday evening. What is $A - B$?
[b]p2.[/b] What is the smallest positive integer $k$ such that $2k$ is a perfect square and $3k$ is a perfect cube?
[b]p3.[/b] Two competitive ducks decide to have a race in the first quadrant of the $xy$ plane. They both start at the origin, and the race ends when one of the ducks reaches the line $y = \frac12$ . The first duck follows the graph of $y = \frac{x}{3}$ and the second duck follows the graph of $y = \frac{x}{5}$ . If the two ducks move in such a way that their $x$-coordinates are the same at any time during the race, find the ratio of the speed of the first duck to that of the second duck when the race ends.
[b]p4.[/b] There were grammatical errors in this problem as stated during the contest. The problem should have said:
You play a carnival game as follows: The carnival worker has a circular mat of radius 20 cm, and on top of that is a square mat of side length $10$ cm, placed so that the centers of the two mats coincide. The carnival worker also has three disks, one each of radius $1$ cm, $2$ cm, and $3$ cm. You start by paying the worker a modest fee of one dollar, then choosing two of the disks, then throwing the two disks onto the mats, one at a time, so that the center of each disk lies on the circular mat. You win a cash prize if the center of the large disk is on the square AND the large disk touches the small disk, otherwise you just lost the game and you get no money. How much is the cash prize if choosing the two disks randomly and then throwing the disks randomly (i.e. with uniform distribution) will, on average, result in you breaking even?
[b]p5.[/b] Four boys and four girls arrive at the Highball High School Senior Ball without a date. The principal, seeking to rectify the situation, asks each of the boys to rank the four girls in decreasing order of preference as a prom date and asks each girl to do the same for the four boys. None of the boys know any of the girls and vice-versa (otherwise they would have probably found each other before the prom), so all eight teenagers write their rankings randomly. Because the principal lacks the mathematical chops to pair the teenagers together according to their stated preference, he promptly ignores all eight of the lists and randomly pairs each of the boys with a girl. What is the probability that no boy ends up with his third or his fourth choice, and no girl ends up with her third or fourth choice?
[b]p6.[/b] In the diagram below, $ABCDEFGH$ is a rectangular prism, $\angle BAF = 30^o$ and $\angle DAH = 60^o$. What is the cosine of $\angle CEG$?
[img]https://cdn.artofproblemsolving.com/attachments/a/1/1af1a7d5d523884703b9ff95aaf301bcc18140.png[/img]
[b]p7.[/b] Two cows play a game where each has one playing piece, they begin by having the two pieces on opposite vertices of an octahedron, and the two cows take turns moving their piece to an adjacent vertex. The winner is the first player who moves its piece to the vertex occupied by its opponent’s piece. Because cows are not the most intelligent of creatures, they move their pieces randomly. What is the probability that the first cow to move eventually wins?
[b]p8.[/b] Find the last two digits of $$\sum^{2008}_{k=1}k {2008 \choose k}.$$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Initially, on the blackboard are written all natural numbers from $1$ to $20$. A move consists of selecting $2$ numbers $a<b$ written on the blackboard such that their difference is at least $2$, erasing these numbers and writting $a+1$ and $b-1$ instead. What is the maximum numbers of moves one can perform?
Let $\triangle ABC$ be an acute-angled triangle with $AB \not= AC$. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ be the midpoint of the side $BC$. Let $D$ be a point on the side $AB$ and $E$ a point on the side $AC$ such that $AE=AD$ and the points $D$, $H$, $E$ are on the same line. Prove that the line $HM$ is perpendicular to the common chord of the circumscribed circles of triangle $\triangle ABC$ and triangle $\triangle ADE$.
The number $770$ is written on a blackboard. Melody repeatedly performs moves, where a move consists of subtracting either $40$ or $41$ from the number on the board. She performs moves until the number is not positive, and then she stops. Let $N$ be the number of sequences of moves that Melody could perform. Suppose $N = a\cdot 2^b$ where $a$ is an odd positive integer and $b$ is a nonnegative integer. Compute $100a+b.$
Let $n$ be a positive integer with $n>1$. Prove that \[\frac{1}{2}+\cdots+\frac{1}{n}\] is not an integer.
Point $P$ is chosen inside triangle $ABC$ so that $\angle APC+\angle ABC=180^o$ and $BC=AP.$ On the side $AB$, a point $K$ is chosen such that $AK = KB + PC$. Prove that $CK \perp AB$.
Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$
In an acute-angled $ABC$, $\angle A>60^{\circ}$, $H$ is its orthocenter. $M,N$ are two points on $AB,AC$ respectively, such that $\angle HMB=\angle HNC=60^{\circ}$. Let $O$ be the circumcenter of triangle $HMN$. $D$ is a point on the same side with $A$ of $BC$ such that $\triangle DBC$ is an equilateral triangle. Prove that $H,O,D$ are collinear.
Let $\{b_n\}_{n\geq 1}^{\infty}$ be a sequence of positive integers. The sequence $\{a_n\}_{n\geq 1}^{\infty}$ is defined as follows: $a_1$ is a fixed positive integer and
\[a_{n+1}=a_n^{b_n}+1 ,\qquad \forall n\geq 1.\]
Find all positive integers $m\geq 3$ with the following property: If the sequence $\{a_n\mod m\}_{n\geq 1 }^{\infty}$ is eventually periodic, then there exist positive integers $q,u,v$ with $2\leq q\leq m-1$, such that the sequence $\{b_{v+ut}\mod q\}_{t\geq 1}^{\infty}$ is purely periodic.
Perimeter of triangle $ABC$ is $1$. Circle $\omega$ touches side $BC$, continuation of side $AB$ at $P$ and continuation of side $AC$ in $Q$. Line through midpoints $AB$ and $AC$ intersects circumcircle of $APQ$ at $X$ and $Y$.
Find length of $XY$.
Let $G, H$ be the centroid and orthocentre of $\vartriangle ABC$ which has an obtuse angle at $\angle B$. Let $\omega$ be the circle with diameter $AG$. $\omega$ intersects $\odot(ABC)$ again at $L \ne A$. The tangent to $\omega$ at $L$ intersects $\odot(ABC)$ at $K \ne L$. Given that $AG = GH$, prove $\angle HKG = 90^o$
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[i]Sam Bealing, United Kingdom[/i]
Given a $ n \times n$ grid board . How many ways can an $X$ and an $O$ be placed in such a way that they are not in adjacent squares?
Given right $n$-gon wit the point $O$ -- its centre. All the vertices are marked either with $+1$ or $-1$. We may change all the signs in the vertices of regular $k$-gon ($2 \le k \le n$) with the same centre $O$. (By $2$-gon we understand a segment, being halved by $O$.) Prove that in a), b) and c) cases there exists such a set of $(+1)$s and $(-1)$s, that we can never obtain a set of $(+1)$s only.
a) $n = 15$,
b) $n = 30$,
c) $n > 2$,
d) Let us denote $K(n)$ the maximal number of $(+1)$ and $(-1)$ sets such, that it is impossible to obtain one set from another. Prove, for example, that $K(200) = 2^{80}$