Found problems: 5923
What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?
A sequence $a_0, a_1, \dots , a_n, \dots$ of positive integers is constructed as follows:
[list]
[*] If the last digit of $a_n$ is less than or equal to $5$, then this digit is deleted and $a_{n+1}$ is the number consisting of the remaining digits. (If $a_{n+1}$ contains no digits, the process stops.)
[*] Otherwise, $a_{n+1}= 9a_n$.
[/list]
Can one choose $a_0$ so that this sequence is infinite?
In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.
An arithmetic sequence of exactly $10$ positive integers has the property that any two elements are relatively prime. Compute the smallest possible sum of the $10$ numbers.
[i]Proposed by Kyle Lee[/i]
Find all sequences ${{a}_{1}},{{a}_{2}},...,{{a}_{2000}}$ of real numbers such that $\sum\limits_{n=1}^{2000}{{{a}_{n}}=1999}$ and such that $\frac{1}{2}<{{a}_{n}}<1$ and ${{a}_{n+1}}={{a}_{n}}(2-{{a}_{n}})$ for all $n\ge 1$.
Find all sequences $a_{0}, a_{1},\ldots, a_{n}$ of real numbers such that $a_{n}\neq 0$, for which the following statement is true:
If $f: \mathbb{R}\to\mathbb{R}$ is an $n$ times differentiable function
and $x_{0}<x_{1}<\ldots <x_{n}$ are real numbers such that
$f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0$ then there is $h\in (x_{0}, x_{n})$ for which \[a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.\]
The sequence $ (x_n)$, $ n\in\mathbb{N}^*$ is defined by $ |x_1|<1$, and for all $ n \ge 1$, \[ x_{n\plus{}1} \equal{}\frac{\minus{}x_n \plus{}\sqrt{3\minus{}3x_n^2}}{2}\] (a) Find the necessary and sufficient condition for $ x_1$ so that each $ x_n > 0$.
(b) Is this sequence periodic? And why?
[b]p1.[/b] Frankie the frog likes to hop. On his first hop, he hops $1$ meter. On each successive hop, he hops twice as far as he did on the previous hop. For example, on his second hop, he hops $2$ meters, and on his third hop, he hops $4$ meters. How many meters, in total, has he travelled after $6$ hops?
[b]p2.[/b] Anton flips $5$ fair coins. The probability that he gets an odd number of heads can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$.
[b]p3.[/b] April discovers that the quadratic polynomial $x^2 + 5x + 3$ has distinct roots $a$ and $b$. She also discovers that the quadratic polynomial $x^2 + 7x + 4$ has distinct roots $c$ and $d$. Compute $$ac + bc + bd + ad + a + b.$$
[b]p4.[/b] A rectangular picture frame that has a $2$ inch border can exactly fit a $10$ by $7$ inch photo. What is the total area of the frame's border around the photo, in square inches?
[b]p5.[/b] Compute the median of the positive divisors of $9999$.
[b]p6.[/b] Kaity only eats bread, pizza, and salad for her meals. However, she will refuse to have salad if she had pizza for the meal right before. Given that she eats $3$ meals a day (not necessarily distinct), in how many ways can we arrange her meals for the day?
[b]p7.[/b] A triangle has side lengths $3$, $4$, and $x$, and another triangle has side lengths $3$, $4$, and $2x$. Assuming both triangles have positive area, compute the number of possible integer values for $x$.
[b]p8.[/b] In the diagram below, the largest circle has radius $30$ and the other two white circles each have a radius of $15$. Compute the radius of the shaded circle.
[img]https://cdn.artofproblemsolving.com/attachments/c/1/9eaf1064b2445edb15782278fc9c6efd1440b0.png[/img]
[b]p9.[/b] What is the remainder when $2022$ is divided by $9$?
[b]p10.[/b] For how many positive integers $x$ less than $2022$ is $x^3 - x^2 + x - 1$ prime?
[b]p11.[/b] A sphere and cylinder have the same volume, and both have radius $10$. The height of the cylinder can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$.
[b]p12.[/b] Amanda, Brianna, Chad, and Derrick are playing a game where they pass around a red flag. Two players "interact" whenever one passes the flag to the other. How many different ways can the flag be passed among the players such that
(1) each pair of players interacts exactly once, and
(2) Amanda both starts and ends the game with the flag?
[b]p13.[/b] Compute the value of $$\dfrac{12}{1 + \dfrac{12}{1+ \dfrac{12}{1+...}}}$$
[b]p14.[/b] Compute the sum of all positive integers $a$ such that $a^2 - 505$ is a perfect square.
[b]p15.[/b] Alissa, Billy, Charles, Donovan, Eli, Faith, and Gerry each ask Sara a question. Sara must answer exactly $5$ of them, and must choose an order in which to answer the questions. Furthermore, Sara must answer Alissa and Billy's questions. In how many ways can Sara complete this task?
[b]p16.[/b] The integers $-x$, $x^2 - 1$, and $x3$ form a non-decreasing arithmetic sequence (in that order). Compute the sum of all possible values of $x^3$.
[b]p17.[/b] Moor and his $3$ other friends are trying to split burgers equally, but they will have $2$ left over. If they find another friend to split the burgers with, everyone can get an equal amount. What is the fewest number of burgers that Moor and his friends could have started with?
[b]p18.[/b] Consider regular dodecagon $ABCDEFGHIJKL$ below. The ratio of the area of rectangle $AFGL$ to the area of the dodecagon can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$.
[img]https://cdn.artofproblemsolving.com/attachments/8/3/c38c10a9b2f445faae397d8a7bc4c8d3ed0290.png[/img]
[b]p19.[/b] Compute the remainder when $3^{4^{5^6}}$ is divided by $4$.
[b]p20.[/b] Fred is located at the middle of a $9$ by $11$ lattice (diagram below). At every second, he randomly moves to a neighboring point (left, right, up, or down), each with probability $1/4$. The probability that he is back at the middle after exactly $4$ seconds can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$.
[img]https://cdn.artofproblemsolving.com/attachments/7/c/f8e092e60f568ab7b28964d23b2ee02cdba7ad.png[/img]
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
The sequence $\{x_n\}$ is given by
$$x_n=\frac14\left(\left(2+\sqrt3\right)^{2n-1}+\left(2-\sqrt3\right)^{2n-1}\right),\qquad n\in\mathbb N.$$Prove that each $x_n$ is equal to the sum of squares of two consecutive integers.
[b]p1.[/b] Fill in each box with an integer from $1$ to $9$. Each number in the right column is the product of the numbers in its row, and each number in the bottom row is the product of the numbers in its column. Some numbers may be used more than once, and not every number from $1$ to $9$ is required to be used.
[img]https://cdn.artofproblemsolving.com/attachments/c/0/0212181d87f89aac374f75f1f0bde6d0600037.png[/img]
[b]p2.[/b] A set $X$ is called [b]prime-difference free [/b] (henceforth pdf) if for all $x, y \in X$, $|x - y|$ is not prime. Find the number n such that the following both hold.
$\bullet$ There is a pdf set of size $n$ that is a subset of $\{1,..., 2016\}$, and
$\bullet$ There is no pdf set of size $n + 1$ that is a subset of $\{1,..., 2016\}$.
[b]p3.[/b] Let $X_1,...,X_{15}$ be a sequence of points in the $xy$-plane such that $X_1 = (10, 0)$ and $X_{15} = (0, 10)$. Prove that for some $i \in \{1, 2,..., 14\}$, the midpoint of $X_iX_{i+1}$ is of distance greater than $1/2$ from the origin.
[b]p4.[/b] Suppose that $s_1, s_2,..., s_{84}$ is a sequence of letters from the set $\{A,B,C\}$ such that every four-letter sequence from $\{A,B,C\}$ occurs exactly once as a consecutive subsequence $s_k$, $s_{k+1}$, $s_{k+2}$, $s_{k+3}$. Suppose that $$(s_1, s_2, s_3, s_4, s_5) = (A,B,B,C,A).$$ What is $s_{84}$? Prove your answer.
[b]p5.[/b] Determine (with proof) whether or not there exists a sequence of positive real numbers $a_1, a_2, a_3,...$ with both of the following properties:
$\bullet$ $\sum^n_{i=1} a_i \le n^2$, for all $n \ge 1$, and
$\bullet$ $\sum^n_{i=1} \frac{1}{a_i} \le 2016$, for all $n \ge 1$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
A sequence infinity $a_{1}, a_{2},...,$ is $generadora$ if: $a_{1}=1,2$ and $a_{n+1}$ is obtained by placing a digit 1 on the left or a digit 2 on the right for all natural n. Prove that there is an infinite $generadora$ sequence such that it does not contain any multiples of 7.
A continuous function $f : [a,b] \to [a,b]$ is called piecewise monotone if $[a, b]$ can be subdivided into finitely many subintervals
$$I_1 = [c_0,c_1], I_2 = [c_1,c_2], \dots , I_\ell = [ c_{\ell - 1},c_\ell ]$$
such that $f$ restricted to each interval $I_j$ is strictly monotone, either increasing or decreasing. Here we are assuming that $a = c_0 < c_1 < \cdots < c_{\ell - 1} < c_\ell = b$. We are also assuming that each $I_j$ is a maximal interval on which $f$ is strictly monotone. Such a maximal interval is called a lap of the function $f$, and the number $\ell = \ell (f)$ of distinct laps is called the lap number of $f$. If $f : [a,b] \to [a,b]$ is a continuous piecewise-monotone function, show that the sequence $( \sqrt[n]{\ell (f^n )})$ converges; here $f^n$ means $f$ composed with itself $n$-times, so $f^2 (x) = f(f(x))$ etc.
An integer sequence is defined by \[{ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.\] Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.
Consider a grid of black and white squares with $3$ rows and $n$ columns. If there is a non-empty sequence of white squares $s_1$, $...$, $s_m$ such that $s_1$ is in the top row and $s_m$ is in the bottom row and consecutive squares in the sequence share an edge, then we say that the grid percolates. Let $T_n$ be the number of grids which do not percolate. There exists constants a, b such that $\frac{T_n}{ab^n}$ approaches $ 1$ as $n$ approaches $\infty$. Then $b$ is expressible as $(x+ \sqrt{y})/z$ for squarefree $y$ and coprime $x, z$. Find $x + y + z$.
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers in which $\lim_{n\to\infty} a_n = 0$ such that there is a constant $c >0$ so that for all $n \geq 1$, $|a_{n+1}-a_n| \leq c\cdot a_n^2$. Show that exists $d>0$ with $na_n \geq d, \forall n \geq 1$.
Let $ \{f_n \}_{n=0}^{\infty}$ be a uniformly bounded sequence of real-valued measurable functions defined on $ [0,1]$ satisfying \[ \int_0^1 f_n^2=1.\] Further, let $ \{ c_n \}$ be a sequence of real numbers with \[ \sum_{n=0}^{\infty} c_n^2= +\infty.\] Prove that some re-arrangement of the series $ \sum_{n=0}^{\infty} c_nf_n$ is divergent on a set of positive measure.
[i]J. Komlos[/i]
A [i]tile[/i] $T$ is a union of finitely many pairwise disjoint arcs of a unit circle $K$. The [i]size[/i] of $T$,
denoted by $|T|$, is the sum of the lengths of the arcs $T$ consists of, divided by $2\pi$. A [i]copy[/i] of $T$ is
a tile $T'$ obtained by rotating $T$ about the centre of $K$ through some angle. Given a positive
real number $\varepsilon < 1$, does there exist an infinite sequence of tiles $T_1,T_2,\ldots,T_n,\ldots$ satisfying the following two conditions simultaneously:
1) $|T_n| > 1 - \varepsilon$ for all $n$;
2) The union of all $T_n'$ (as $n$ runs through the positive integers) is a proper subset of $K$ for any choice of the copies $T_1'$, $T_2'$, $\ldots$, $T_n', \ldots$?
[hide=Note] In the extralist the problem statement had the clause "three conditions" rather than two, but only two are presented, the ones you see. I am quite confident this is a typo or that the problem might have been reformulated after submission.[/hide]
A sequence $\{a_n\}_{n\geq0}$ obeys the recurrence $a_n=1+a_{n-1}+\alpha a_{n-2}$ for all $n\geq2$ and for some $\alpha>0$. Given that $a_0=1$ and $a_1=2$, compute the value of $\alpha$ for which
$$\sum_{n=0}^{\infty}\frac{a_n}{2^n}=10$$
Real number $C > 1$ is given. Sequence of positive real numbers $a_1, a_2, a_3, \ldots$, in which $a_1=1$ and $a_2=2$, satisfy the conditions
\[a_{mn}=a_ma_n, \] \[a_{m+n} \leq C(a_m + a_n),\]
for $m, n = 1, 2, 3, \ldots$. Prove that $a_n = n$ for $n=1, 2, 3, \ldots$.
Actually, this is an MO I participated in :) but it's really hard to get problems from this year if you don't know some people.
P1. Let $f$ be a function satisfying $f(x + 1) + f(x - 1) = \sqrt{2} f(x)$, for all reals $x$. If $f(x - 1) = a$ and $f(x) = b$, determine the value of $f(x + 4)$.
[hide=Remarks]We found out that this is the modified version of a problem from LMNAS UGM 2008, Senior High School Level, on its First Round. This is also the same with Arthur Engel's "Problem Solving Strategies" Book, Example Problem E2.[/hide]
P2. The sequence of "Sanga" numbers is formed by the following procedure.
i. Pick a positive integer $n$.
ii. The first term of the sequence $(U_1)$ is $9n$.
iii. For $k \geq 2$, $U_k = U_{k-1} - 17$.
Sanga$[r]$ is the "Sanga" sequence whose smallest positive term is $r$.
As an example, for $n = 3$, the "Sanga" sequence which is formed is $27, 10, -7, -24, -41, \ldots.$ Since the smallest positive term of such sequence is $10$, for $n = 3$, the sequence formed is called Sanga$[10]$. For $n \leq 100$, determine the sum of all $n$ which makes the sequence Sanga$[4]$.
P3. The cube $ABCD.EFGH$ has an edge length of 6 cm. Point $R$ is on the extension of line (segment) $EH$ with $EH : ER = 1 : 2$, such that triangle $AFR$ cuts edge $GH$ at point $P$ and cuts edge $DH$ at $Q$. Determine the area of the region bounded by the quadrilateral $AFPQ$.
[url=https://artofproblemsolving.com/community/q1h2395046p19649729]P4[/url]. Ten skydivers are planning to form a circle formation when they are in the air by holding hands with both adjacent skydivers. If each person has 2 choices for the colour of his/her uniform to be worn, that is, red or white, determine the number of different colour formations that can be constructed.
P5. After pressing the start button, a game machine works according to the following procedure.
i. It picks 7 numbers randomly from 1 to 9 (these numbers are integers, not stated but corrected) without showing it on screen.
ii. It shows the product of the seven chosen numbes on screen.
iii. It shows a calculator menu (it does not function as a calculator) on screen and asks the player whether the sum of the seven chosen numbers is odd or even.
iv. Shows the seven chosen numbers and their sum and products.
v. Releases a prize if the guess of the player was correct or shows the message "Try again" on screen if the guess by the player was incorrect. (Although the player is not allowed to guess with those numbers, and the machine's procedures are started all over again.)
Kiki says that this game is really easy since the probability of winning is greater than $90$%. Explain, whether you agree with Kiki.
A game consists in pushing a flat stone along a sequence of squares $S_0, S_1, S_2, . . .$ that are arranged in linear order. The stone is initially placed on square $S_0$. When the stone stops on a square $S_k$ it is pushed again in the same direction and so on until it reaches $S_{1987}$ or goes beyond it; then the game stops. Each time the stone is pushed, the probability that it will advance exactly $n$ squares is $\frac{1}{2^n}$. Determine the probability that the stone will stop exactly on square $S_{1987}.$
The sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, \ldots$ consists of 1’s separated by blocks of 2’s with n 2’s in the nth block. The sum of the first $1234$ terms of this sequence is
$\text{(A)}\ 1996 \qquad \text{(B)}\ 2419 \qquad \text{(C)}\ 2429 \qquad \text{(D)}\ 2439 \qquad \text{(E)}\ 2449$
Let $k$ be an integer $\geq 3$. Sequence $\{a_n\}$ satisfies that $a_k = 2k$ and for all $n > k$, we have
\[a_n =
\begin{cases}
a_{n-1}+1 & \text{if } (a_{n-1},n) = 1 \\
2n & \text{if } (a_{n-1},n) > 1
\end{cases}
\]
Prove that there are infinitely many primes in the sequence $\{a_n - a_{n-1}\}$.
Let be the sequence $ \left( a_n \right)_{n\ge 0} $ of positive real numbers defined by
$$ a_n=1+\frac{a_{n-1}}{n} ,\quad\forall n\ge 1. $$
Calculate $ \lim_{n\to\infty } a_n ^n . $
[i]Florian Dumitrel[/i]